\(\int \frac {x^2}{(a+b \tan (c+d \sqrt [3]{x}))^2} \, dx\) [62]

Optimal result

Integrand size = 20, antiderivative size = 1691 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=-\frac {6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}+\frac {6 b^2 x^{8/3}}{(a+i b) (i a+b)^2 d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {24 b^2 x^{7/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {84 i b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {24 b x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {24 b^2 x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {252 b^2 x^{5/3} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {84 b x^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {84 i b^2 x^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {630 i b^2 x^{4/3} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac {252 b x^{5/3} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {252 b^2 x^{5/3} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {1260 b^2 x \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}-\frac {630 b x^{4/3} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac {630 i b^2 x^{4/3} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^5}-\frac {1890 i b^2 x^{2/3} \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}+\frac {1260 b x \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^6}-\frac {1260 b^2 x \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^6}+\frac {1890 b^2 \sqrt [3]{x} \operatorname {PolyLog}\left (7,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^8}+\frac {1890 b x^{2/3} \operatorname {PolyLog}\left (7,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^7}-\frac {1890 i b^2 x^{2/3} \operatorname {PolyLog}\left (7,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^7}+\frac {945 i b^2 \operatorname {PolyLog}\left (8,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^9}-\frac {1890 b \sqrt [3]{x} \operatorname {PolyLog}\left (8,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^8}+\frac {1890 b^2 \sqrt [3]{x} \operatorname {PolyLog}\left (8,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^8}-\frac {945 b \operatorname {PolyLog}\left (9,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^9}+\frac {945 i b^2 \operatorname {PolyLog}\left (9,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^9} \]

[Out]

630*I*b^2*x^(4/3)*polylog(5,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^5+630*I*b^2*x^(4/3)*polylog
(4,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^5+4/3*b*x^3/(I*a-b)/(a-I*b)^2+252*b^2*x^(5/3)*polylo
g(4,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^4-1260*b^2*x*polylog(5,-(a-I*b)*exp(2*I*(c+d*x^(1/3
)))/(a+I*b))/(a^2+b^2)^2/d^6-1260*b^2*x*polylog(6,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^6+189
0*b^2*x^(1/3)*polylog(7,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^8+1890*b^2*x^(1/3)*polylog(8,-(
a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^8-945*b*polylog(9,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)
)/(a-I*b)^2/(a+I*b)/d^9-6*I*b^2*x^(8/3)/(a^2+b^2)^2/d+24*b^2*x^(7/3)*ln(1+(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*
b))/(a^2+b^2)^2/d^2-24*b^2*x^(7/3)*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^2+252*b^2*
x^(5/3)*polylog(3,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^4+945*I*b^2*polylog(8,-(a-I*b)*exp(2*
I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^9+945*I*b^2*polylog(9,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^
2)^2/d^9-84*I*b^2*x^2*polylog(3,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^3-1890*I*b^2*x^(2/3)*po
lylog(6,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^7-1890*I*b^2*x^(2/3)*polylog(7,-(a-I*b)*exp(2*I
*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^7+6*b^2*x^(8/3)/(a+I*b)/(I*a+b)^2/d/(I*a-b+(I*a+b)*exp(2*I*(c+d*x^(1/3)
)))+24*b*x^(7/3)*polylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^2+84*b*x^2*polylog(3,-
(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d^3-252*b*x^(5/3)*polylog(4,-(a-I*b)*exp(2*I*(c+d*x^
(1/3)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^4-630*b*x^(4/3)*polylog(5,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a-I*b
)^2/(a+I*b)/d^5+1260*b*x*polylog(6,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(I*a-b)/(a-I*b)^2/d^6+1890*b*x^(2/
3)*polylog(7,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d^7+6*b*x^(8/3)*ln(1+(a-I*b)*exp(2*I*(
c+d*x^(1/3)))/(a+I*b))/(a-I*b)^2/(a+I*b)/d-1890*b*x^(1/3)*polylog(8,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(
I*a-b)/(a-I*b)^2/d^8-6*I*b^2*x^(8/3)*ln(1+(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d-84*I*b^2*x^2*p
olylog(2,-(a-I*b)*exp(2*I*(c+d*x^(1/3)))/(a+I*b))/(a^2+b^2)^2/d^3-4/3*b^2*x^3/(a^2+b^2)^2+1/3*x^3/(a-I*b)^2

Rubi [A] (verified)

Time = 3.29 (sec) , antiderivative size = 1691, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3832, 3815, 2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222} \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\frac {4 b x^3}{3 (i a-b) (a-i b)^2}+\frac {x^3}{3 (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {6 b \log \left (\frac {e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) x^{8/3}}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 \log \left (\frac {e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}+\frac {6 b^2 x^{8/3}}{(a+i b) (i a+b)^2 d \left (i a+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}-b\right )}+\frac {24 b^2 \log \left (\frac {e^{2 i \left (c+d \sqrt [3]{x}\right )} (a-i b)}{a+i b}+1\right ) x^{7/3}}{\left (a^2+b^2\right )^2 d^2}+\frac {24 b \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{7/3}}{(i a-b) (a-i b)^2 d^2}-\frac {24 b^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{7/3}}{\left (a^2+b^2\right )^2 d^2}-\frac {84 i b^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^2}{\left (a^2+b^2\right )^2 d^3}+\frac {84 b \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^2}{(a-i b)^2 (a+i b) d^3}-\frac {84 i b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^2}{\left (a^2+b^2\right )^2 d^3}+\frac {252 b^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{5/3}}{\left (a^2+b^2\right )^2 d^4}-\frac {252 b \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{5/3}}{(i a-b) (a-i b)^2 d^4}+\frac {252 b^2 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{5/3}}{\left (a^2+b^2\right )^2 d^4}+\frac {630 i b^2 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{4/3}}{\left (a^2+b^2\right )^2 d^5}-\frac {630 b \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{4/3}}{(a-i b)^2 (a+i b) d^5}+\frac {630 i b^2 \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{4/3}}{\left (a^2+b^2\right )^2 d^5}-\frac {1260 b^2 \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x}{\left (a^2+b^2\right )^2 d^6}+\frac {1260 b \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x}{(i a-b) (a-i b)^2 d^6}-\frac {1260 b^2 \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x}{\left (a^2+b^2\right )^2 d^6}-\frac {1890 i b^2 \operatorname {PolyLog}\left (6,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{2/3}}{\left (a^2+b^2\right )^2 d^7}+\frac {1890 b \operatorname {PolyLog}\left (7,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{2/3}}{(a-i b)^2 (a+i b) d^7}-\frac {1890 i b^2 \operatorname {PolyLog}\left (7,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) x^{2/3}}{\left (a^2+b^2\right )^2 d^7}+\frac {1890 b^2 \operatorname {PolyLog}\left (7,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) \sqrt [3]{x}}{\left (a^2+b^2\right )^2 d^8}-\frac {1890 b \operatorname {PolyLog}\left (8,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) \sqrt [3]{x}}{(i a-b) (a-i b)^2 d^8}+\frac {1890 b^2 \operatorname {PolyLog}\left (8,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right ) \sqrt [3]{x}}{\left (a^2+b^2\right )^2 d^8}+\frac {945 i b^2 \operatorname {PolyLog}\left (8,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^9}-\frac {945 b \operatorname {PolyLog}\left (9,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^9}+\frac {945 i b^2 \operatorname {PolyLog}\left (9,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^9} \]

[In]

Int[x^2/(a + b*Tan[c + d*x^(1/3)])^2,x]

[Out]

((-6*I)*b^2*x^(8/3))/((a^2 + b^2)^2*d) + (6*b^2*x^(8/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^((2*I
)*(c + d*x^(1/3))))) + x^3/(3*(a - I*b)^2) + (4*b*x^3)/(3*(I*a - b)*(a - I*b)^2) - (4*b^2*x^3)/(3*(a^2 + b^2)^
2) + (24*b^2*x^(7/3)*Log[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (6*b*x^(8
/3)*Log[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - ((6*I)*b^2*x^(8/3)*L
og[1 + ((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - ((84*I)*b^2*x^2*PolyLog[2, -(((a
- I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (24*b*x^(7/3)*PolyLog[2, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (24*b^2*x^(7/3)*PolyLog[2, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (252*b^2*x^(5/3)*PolyLog[3, -(((a - I*b)*E^((2*I)*(
c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (84*b*x^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))
))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^3) - ((84*I)*b^2*x^2*PolyLog[3, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))
))/(a + I*b))])/((a^2 + b^2)^2*d^3) + ((630*I)*b^2*x^(4/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/
(a + I*b))])/((a^2 + b^2)^2*d^5) - (252*b*x^(5/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)
)])/((I*a - b)*(a - I*b)^2*d^4) + (252*b^2*x^(5/3)*PolyLog[4, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b
))])/((a^2 + b^2)^2*d^4) - (1260*b^2*x*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 +
b^2)^2*d^6) - (630*b*x^(4/3)*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a
+ I*b)*d^5) + ((630*I)*b^2*x^(4/3)*PolyLog[5, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2
)^2*d^5) - ((1890*I)*b^2*x^(2/3)*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^
2*d^7) + (1260*b*x*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^6)
- (1260*b^2*x*PolyLog[6, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^6) + (1890*b^2
*x^(1/3)*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^8) + (1890*b*x^(2/3)
*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^7) - ((1890*I)*b^2*x
^(2/3)*PolyLog[7, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^7) + ((945*I)*b^2*Poly
Log[8, -(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^9) - (1890*b*x^(1/3)*PolyLog[8, -
(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^8) + (1890*b^2*x^(1/3)*PolyLog[8,
-(((a - I*b)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^8) - (945*b*PolyLog[9, -(((a - I*b)*E^(
(2*I)*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^9) + ((945*I)*b^2*PolyLog[9, -(((a - I*b)*E^((2*
I)*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^9)

Rule 2215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2216

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2222

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*
((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*((a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1
)*Log[F])), x] - Dist[d*(m/(b*f*g*n*(p + 1)*Log[F])), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1
), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3815

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(
c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]

Rule 3832

Int[(x_)^(m_.)*((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Tan[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^8}{(a+b \tan (c+d x))^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {x^8}{(a-i b)^2}-\frac {4 b^2 x^8}{(i a+b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2}+\frac {4 b x^8}{(a-i b)^2 \left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {x^3}{3 (a-i b)^2}+\frac {(12 b) \text {Subst}\left (\int \frac {x^8}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {x^8}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{(i a+b)^2} \\ & = \frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {x^8}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2}-\frac {(12 b) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^8}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{a^2+b^2}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^8}{\left (i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{a^2+b^2} \\ & = -\frac {6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {6 b x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^8}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a+i b)^2 (i a+b)}-\frac {(48 b) \text {Subst}\left (\int x^7 \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d}+\frac {\left (48 b^2\right ) \text {Subst}\left (\int \frac {x^7}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d} \\ & = -\frac {6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {6 b x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {24 b x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {(168 b) \text {Subst}\left (\int x^6 \operatorname {PolyLog}\left (2,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {\left (48 b^2\right ) \text {Subst}\left (\int \frac {e^{2 i c+2 i d x} x^7}{i a \left (1+\frac {i b}{a}\right )+i a \left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}} \, dx,x,\sqrt [3]{x}\right )}{(a-i b) (a+i b)^2 d}+\frac {\left (48 i b^2\right ) \text {Subst}\left (\int x^7 \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d} \\ & = -\frac {6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {24 b^2 x^{7/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}+\frac {24 b x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {24 b^2 x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {84 b x^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {(504 b) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (3,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {\left (168 b^2\right ) \text {Subst}\left (\int x^6 \log \left (1+\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {\left (168 b^2\right ) \text {Subst}\left (\int x^6 \operatorname {PolyLog}\left (2,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^2} \\ & = -\frac {6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {24 b^2 x^{7/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {84 i b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {24 b x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {24 b^2 x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {84 b x^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {84 i b^2 x^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {252 b x^{5/3} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {(1260 b) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (4,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {\left (504 i b^2\right ) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (2,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {\left (504 i b^2\right ) \text {Subst}\left (\int x^5 \operatorname {PolyLog}\left (3,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^3} \\ & = -\frac {6 i b^2 x^{8/3}}{\left (a^2+b^2\right )^2 d}-\frac {6 b^2 x^{8/3}}{(a-i b)^2 (a+i b) d \left (i a-b+(i a+b) e^{2 i \left (c+d \sqrt [3]{x}\right )}\right )}+\frac {x^3}{3 (a-i b)^2}+\frac {4 b x^3}{3 (i a-b) (a-i b)^2}-\frac {4 b^2 x^3}{3 \left (a^2+b^2\right )^2}+\frac {24 b^2 x^{7/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {6 b x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d}-\frac {6 i b^2 x^{8/3} \log \left (1+\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d}-\frac {84 i b^2 x^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {24 b x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^2}-\frac {24 b^2 x^{7/3} \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {252 b^2 x^{5/3} \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}+\frac {84 b x^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^3}-\frac {84 i b^2 x^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {252 b x^{5/3} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(i a-b) (a-i b)^2 d^4}+\frac {252 b^2 x^{5/3} \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {630 b x^{4/3} \operatorname {PolyLog}\left (5,-\frac {(a-i b) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{a+i b}\right )}{(a-i b)^2 (a+i b) d^5}+\frac {(2520 b) \text {Subst}\left (\int x^3 \operatorname {PolyLog}\left (5,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{(a-i b)^2 (a+i b) d^5}-\frac {\left (1260 b^2\right ) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (3,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^4}-\frac {\left (1260 b^2\right ) \text {Subst}\left (\int x^4 \operatorname {PolyLog}\left (4,-\frac {\left (1-\frac {i b}{a}\right ) e^{2 i c+2 i d x}}{1+\frac {i b}{a}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (a^2+b^2\right )^2 d^4} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.27 (sec) , antiderivative size = 1136, normalized size of antiderivative = 0.67 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\frac {-\frac {i b \left (18 (a+i b) b (i a+b) d^8 x^{8/3}+4 a (a+i b) (i a+b) d^9 x^3+72 (a-i b) b d^7 \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) x^{7/3} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+18 a (a-i b) d^8 \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) x^{8/3} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+63 b (i a+b) \left (b \left (-1+e^{2 i c}\right )+i a \left (1+e^{2 i c}\right )\right ) \left (-4 i d^6 x^2 \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-12 d^5 x^{5/3} \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+15 i \left (2 d^4 x^{4/3} \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-4 i d^3 x \operatorname {PolyLog}\left (5,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-6 d^2 x^{2/3} \operatorname {PolyLog}\left (6,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+6 i d \sqrt [3]{x} \operatorname {PolyLog}\left (7,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+3 \operatorname {PolyLog}\left (8,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )\right )\right )+9 a (a-i b) \left (-i b \left (-1+e^{2 i c}\right )+a \left (1+e^{2 i c}\right )\right ) \left (8 i d^7 x^{7/3} \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+28 d^6 x^2 \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-84 i d^5 x^{5/3} \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-105 \left (2 d^4 x^{4/3} \operatorname {PolyLog}\left (5,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-4 i d^3 x \operatorname {PolyLog}\left (6,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-6 d^2 x^{2/3} \operatorname {PolyLog}\left (7,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+6 i d \sqrt [3]{x} \operatorname {PolyLog}\left (8,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+3 \operatorname {PolyLog}\left (9,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )\right )\right )\right )}{d^9 \left (b-b e^{2 i c}-i a \left (1+e^{2 i c}\right )\right )}+\frac {(a-i b)^2 (a+i b) x^3 (a \cos (c)-b \sin (c))}{a \cos (c)+b \sin (c)}+\frac {9 (a-i b)^2 (a+i b) b^2 x^{8/3} \sin \left (d \sqrt [3]{x}\right )}{d (a \cos (c)+b \sin (c)) \left (a \cos \left (c+d \sqrt [3]{x}\right )+b \sin \left (c+d \sqrt [3]{x}\right )\right )}}{3 (a-i b)^3 (a+i b)^2} \]

[In]

Integrate[x^2/(a + b*Tan[c + d*x^(1/3)])^2,x]

[Out]

(((-I)*b*(18*(a + I*b)*b*(I*a + b)*d^8*x^(8/3) + 4*a*(a + I*b)*(I*a + b)*d^9*x^3 + 72*(a - I*b)*b*d^7*((-I)*b*
(-1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c)))*x^(7/3)*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + 1
8*a*(a - I*b)*d^8*((-I)*b*(-1 + E^((2*I)*c)) + a*(1 + E^((2*I)*c)))*x^(8/3)*Log[1 + (a + I*b)/((a - I*b)*E^((2
*I)*(c + d*x^(1/3))))] + 63*b*(I*a + b)*(b*(-1 + E^((2*I)*c)) + I*a*(1 + E^((2*I)*c)))*((-4*I)*d^6*x^2*PolyLog
[2, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 12*d^5*x^(5/3)*PolyLog[3, (-a - I*b)/((a - I*b)*E^((2*
I)*(c + d*x^(1/3))))] + (15*I)*(2*d^4*x^(4/3)*PolyLog[4, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - (
4*I)*d^3*x*PolyLog[5, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 6*d^2*x^(2/3)*PolyLog[6, (-a - I*b)/
((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + (6*I)*d*x^(1/3)*PolyLog[7, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1
/3))))] + 3*PolyLog[8, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))])) + 9*a*(a - I*b)*((-I)*b*(-1 + E^((2
*I)*c)) + a*(1 + E^((2*I)*c)))*((8*I)*d^7*x^(7/3)*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))]
+ 28*d^6*x^2*PolyLog[3, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - (84*I)*d^5*x^(5/3)*PolyLog[4, (-a
- I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 105*(2*d^4*x^(4/3)*PolyLog[5, (-a - I*b)/((a - I*b)*E^((2*I)*
(c + d*x^(1/3))))] - (4*I)*d^3*x*PolyLog[6, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 6*d^2*x^(2/3)*
PolyLog[7, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + (6*I)*d*x^(1/3)*PolyLog[8, (-a - I*b)/((a - I*b
)*E^((2*I)*(c + d*x^(1/3))))] + 3*PolyLog[9, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))]))))/(d^9*(b - b
*E^((2*I)*c) - I*a*(1 + E^((2*I)*c)))) + ((a - I*b)^2*(a + I*b)*x^3*(a*Cos[c] - b*Sin[c]))/(a*Cos[c] + b*Sin[c
]) + (9*(a - I*b)^2*(a + I*b)*b^2*x^(8/3)*Sin[d*x^(1/3)])/(d*(a*Cos[c] + b*Sin[c])*(a*Cos[c + d*x^(1/3)] + b*S
in[c + d*x^(1/3)])))/(3*(a - I*b)^3*(a + I*b)^2)

Maple [F]

\[\int \frac {x^{2}}{{\left (a +b \tan \left (c +d \,x^{\frac {1}{3}}\right )\right )}^{2}}d x\]

[In]

int(x^2/(a+b*tan(c+d*x^(1/3)))^2,x)

[Out]

int(x^2/(a+b*tan(c+d*x^(1/3)))^2,x)

Fricas [F]

\[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="fricas")

[Out]

integral(x^2/(b^2*tan(d*x^(1/3) + c)^2 + 2*a*b*tan(d*x^(1/3) + c) + a^2), x)

Sympy [F]

\[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int \frac {x^{2}}{\left (a + b \tan {\left (c + d \sqrt [3]{x} \right )}\right )^{2}}\, dx \]

[In]

integrate(x**2/(a+b*tan(c+d*x**(1/3)))**2,x)

[Out]

Integral(x**2/(a + b*tan(c + d*x**(1/3)))**2, x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8152 vs. \(2 (1362) = 2724\).

Time = 2.64 (sec) , antiderivative size = 8152, normalized size of antiderivative = 4.82 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(x^2/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="maxima")

[Out]

1/105*(315*(2*a*b*log(b*tan(d*x^(1/3) + c) + a)/(a^4 + 2*a^2*b^2 + b^4) - a*b*log(tan(d*x^(1/3) + c)^2 + 1)/(a
^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*(d*x^(1/3) + c)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*t
an(d*x^(1/3) + c)))*c^8 + (35*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*x^(1/3) + c)^9 - 315*(a^3 - I*a^2*b + a*b^2 -
I*b^3)*(d*x^(1/3) + c)^8*c + 1260*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*x^(1/3) + c)^7*c^2 - 2940*(a^3 - I*a^2*b
+ a*b^2 - I*b^3)*(d*x^(1/3) + c)^6*c^3 + 4410*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*x^(1/3) + c)^5*c^4 - 4410*(a
^3 - I*a^2*b + a*b^2 - I*b^3)*(d*x^(1/3) + c)^4*c^5 + 2940*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*x^(1/3) + c)^3*c
^6 - 1260*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(d*x^(1/3) + c)^2*c^7 - 2520*((I*a*b^2 + b^3)*c^7*cos(2*d*x^(1/3) +
2*c) - (a*b^2 - I*b^3)*c^7*sin(2*d*x^(1/3) + 2*c) + (I*a*b^2 - b^3)*c^7)*arctan2(-b*cos(2*d*x^(1/3) + 2*c) + a
*sin(2*d*x^(1/3) + 2*c) + b, a*cos(2*d*x^(1/3) + 2*c) + b*sin(2*d*x^(1/3) + 2*c) + a) - 24*(420*(I*a^2*b - a*b
^2)*(d*x^(1/3) + c)^8 + 960*(I*a*b^2 - b^3 + 2*(-I*a^2*b + a*b^2)*c)*(d*x^(1/3) + c)^7 + 3920*((I*a^2*b - a*b^
2)*c^2 + (-I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^6 + 2352*(2*(-I*a^2*b + a*b^2)*c^3 + 3*(I*a*b^2 - b^3)*c^2)*(d*x^
(1/3) + c)^5 + 3675*((I*a^2*b - a*b^2)*c^4 + 2*(-I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c)^4 + 980*(2*(-I*a^2*b + a*
b^2)*c^5 + 5*(I*a*b^2 - b^3)*c^4)*(d*x^(1/3) + c)^3 + 735*((I*a^2*b - a*b^2)*c^6 + 3*(-I*a*b^2 + b^3)*c^5)*(d*
x^(1/3) + c)^2 + 105*(2*(-I*a^2*b + a*b^2)*c^7 + 7*(I*a*b^2 - b^3)*c^6)*(d*x^(1/3) + c) + (420*(I*a^2*b + a*b^
2)*(d*x^(1/3) + c)^8 + 960*(I*a*b^2 + b^3 + 2*(-I*a^2*b - a*b^2)*c)*(d*x^(1/3) + c)^7 + 3920*((I*a^2*b + a*b^2
)*c^2 + (-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^6 + 2352*(2*(-I*a^2*b - a*b^2)*c^3 + 3*(I*a*b^2 + b^3)*c^2)*(d*x^(
1/3) + c)^5 + 3675*((I*a^2*b + a*b^2)*c^4 + 2*(-I*a*b^2 - b^3)*c^3)*(d*x^(1/3) + c)^4 + 980*(2*(-I*a^2*b - a*b
^2)*c^5 + 5*(I*a*b^2 + b^3)*c^4)*(d*x^(1/3) + c)^3 + 735*((I*a^2*b + a*b^2)*c^6 + 3*(-I*a*b^2 - b^3)*c^5)*(d*x
^(1/3) + c)^2 + 105*(2*(-I*a^2*b - a*b^2)*c^7 + 7*(I*a*b^2 + b^3)*c^6)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c)
- (420*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^8 + 960*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^7 +
3920*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)^6 - 2352*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2
- I*b^3)*c^2)*(d*x^(1/3) + c)^5 + 3675*((a^2*b - I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(d*x^(1/3) + c)^4 - 980
*(2*(a^2*b - I*a*b^2)*c^5 - 5*(a*b^2 - I*b^3)*c^4)*(d*x^(1/3) + c)^3 + 735*((a^2*b - I*a*b^2)*c^6 - 3*(a*b^2 -
I*b^3)*c^5)*(d*x^(1/3) + c)^2 - 105*(2*(a^2*b - I*a*b^2)*c^7 - 7*(a*b^2 - I*b^3)*c^6)*(d*x^(1/3) + c))*sin(2*
d*x^(1/3) + 2*c))*arctan2((2*a*b*cos(2*d*x^(1/3) + 2*c) - (a^2 - b^2)*sin(2*d*x^(1/3) + 2*c))/(a^2 + b^2), (2*
a*b*sin(2*d*x^(1/3) + 2*c) + a^2 + b^2 + (a^2 - b^2)*cos(2*d*x^(1/3) + 2*c))/(a^2 + b^2)) + 35*((a^3 - 3*I*a^2
*b - 3*a*b^2 + I*b^3)*(d*x^(1/3) + c)^9 - 9*(2*I*a*b^2 + 2*b^3 + (a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c)*(d*x^(
1/3) + c)^8 - 144*(-I*a*b^2 - b^3)*(d*x^(1/3) + c)*c^7 + 36*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^2 - 4*(-I*a
*b^2 - b^3)*c)*(d*x^(1/3) + c)^7 - 84*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^3 + 6*(I*a*b^2 + b^3)*c^2)*(d*x^(
1/3) + c)^6 + 126*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^4 - 8*(-I*a*b^2 - b^3)*c^3)*(d*x^(1/3) + c)^5 - 126*(
(a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^5 + 10*(I*a*b^2 + b^3)*c^4)*(d*x^(1/3) + c)^4 + 84*((a^3 - 3*I*a^2*b - 3
*a*b^2 + I*b^3)*c^6 - 12*(-I*a*b^2 - b^3)*c^5)*(d*x^(1/3) + c)^3 - 36*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c^7
+ 14*(I*a*b^2 + b^3)*c^6)*(d*x^(1/3) + c)^2)*cos(2*d*x^(1/3) + 2*c) - 1260*(32*(I*a^2*b - a*b^2)*(d*x^(1/3) +
c)^7 + 2*(-I*a^2*b + a*b^2)*c^7 + 64*(I*a*b^2 - b^3 + 2*(-I*a^2*b + a*b^2)*c)*(d*x^(1/3) + c)^6 + 7*(I*a*b^2
- b^3)*c^6 + 224*((I*a^2*b - a*b^2)*c^2 + (-I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^5 + 112*(2*(-I*a^2*b + a*b^2)*c^
3 + 3*(I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^4 + 140*((I*a^2*b - a*b^2)*c^4 + 2*(-I*a*b^2 + b^3)*c^3)*(d*x^(1/3)
+ c)^3 + 28*(2*(-I*a^2*b + a*b^2)*c^5 + 5*(I*a*b^2 - b^3)*c^4)*(d*x^(1/3) + c)^2 + 14*((I*a^2*b - a*b^2)*c^6
+ 3*(-I*a*b^2 + b^3)*c^5)*(d*x^(1/3) + c) + (32*(I*a^2*b + a*b^2)*(d*x^(1/3) + c)^7 + 2*(-I*a^2*b - a*b^2)*c^7
+ 64*(I*a*b^2 + b^3 + 2*(-I*a^2*b - a*b^2)*c)*(d*x^(1/3) + c)^6 + 7*(I*a*b^2 + b^3)*c^6 + 224*((I*a^2*b + a*b
^2)*c^2 + (-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^5 + 112*(2*(-I*a^2*b - a*b^2)*c^3 + 3*(I*a*b^2 + b^3)*c^2)*(d*x^
(1/3) + c)^4 + 140*((I*a^2*b + a*b^2)*c^4 + 2*(-I*a*b^2 - b^3)*c^3)*(d*x^(1/3) + c)^3 + 28*(2*(-I*a^2*b - a*b^
2)*c^5 + 5*(I*a*b^2 + b^3)*c^4)*(d*x^(1/3) + c)^2 + 14*((I*a^2*b + a*b^2)*c^6 + 3*(-I*a*b^2 - b^3)*c^5)*(d*x^(
1/3) + c))*cos(2*d*x^(1/3) + 2*c) - (32*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^7 - 2*(a^2*b - I*a*b^2)*c^7 + 64*(a*
b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^6 + 7*(a*b^2 - I*b^3)*c^6 + 224*((a^2*b - I*a*b^2)*c^2 -
(a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)^5 - 112*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^4
+ 140*((a^2*b - I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(d*x^(1/3) + c)^3 - 28*(2*(a^2*b - I*a*b^2)*c^5 - 5*(a*
b^2 - I*b^3)*c^4)*(d*x^(1/3) + c)^2 + 14*((a^2*b - I*a*b^2)*c^6 - 3*(a*b^2 - I*b^3)*c^5)*(d*x^(1/3) + c))*sin(
2*d*x^(1/3) + 2*c))*dilog((I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) - 1260*((a*b^2 - I*b^3)*c^7*cos(2*d*
x^(1/3) + 2*c) + (I*a*b^2 + b^3)*c^7*sin(2*d*x^(1/3) + 2*c) + (a*b^2 + I*b^3)*c^7)*log((a^2 + b^2)*cos(2*d*x^(
1/3) + 2*c)^2 + 4*a*b*sin(2*d*x^(1/3) + 2*c) + (a^2 + b^2)*sin(2*d*x^(1/3) + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2
)*cos(2*d*x^(1/3) + 2*c)) + 12*(420*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^8 + 960*(a*b^2 + I*b^3 - 2*(a^2*b + I*a*
b^2)*c)*(d*x^(1/3) + c)^7 + 3920*((a^2*b + I*a*b^2)*c^2 - (a*b^2 + I*b^3)*c)*(d*x^(1/3) + c)^6 - 2352*(2*(a^2*
b + I*a*b^2)*c^3 - 3*(a*b^2 + I*b^3)*c^2)*(d*x^(1/3) + c)^5 + 3675*((a^2*b + I*a*b^2)*c^4 - 2*(a*b^2 + I*b^3)*
c^3)*(d*x^(1/3) + c)^4 - 980*(2*(a^2*b + I*a*b^2)*c^5 - 5*(a*b^2 + I*b^3)*c^4)*(d*x^(1/3) + c)^3 + 735*((a^2*b
+ I*a*b^2)*c^6 - 3*(a*b^2 + I*b^3)*c^5)*(d*x^(1/3) + c)^2 - 105*(2*(a^2*b + I*a*b^2)*c^7 - 7*(a*b^2 + I*b^3)*
c^6)*(d*x^(1/3) + c) + (420*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^8 + 960*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*
(d*x^(1/3) + c)^7 + 3920*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)^6 - 2352*(2*(a^2*b - I*a*
b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^5 + 3675*((a^2*b - I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(d*
x^(1/3) + c)^4 - 980*(2*(a^2*b - I*a*b^2)*c^5 - 5*(a*b^2 - I*b^3)*c^4)*(d*x^(1/3) + c)^3 + 735*((a^2*b - I*a*b
^2)*c^6 - 3*(a*b^2 - I*b^3)*c^5)*(d*x^(1/3) + c)^2 - 105*(2*(a^2*b - I*a*b^2)*c^7 - 7*(a*b^2 - I*b^3)*c^6)*(d*
x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - (420*(-I*a^2*b - a*b^2)*(d*x^(1/3) + c)^8 + 960*(-I*a*b^2 - b^3 + 2*(I*
a^2*b + a*b^2)*c)*(d*x^(1/3) + c)^7 + 3920*((-I*a^2*b - a*b^2)*c^2 + (I*a*b^2 + b^3)*c)*(d*x^(1/3) + c)^6 + 23
52*(2*(I*a^2*b + a*b^2)*c^3 + 3*(-I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^5 + 3675*((-I*a^2*b - a*b^2)*c^4 + 2*(I*
a*b^2 + b^3)*c^3)*(d*x^(1/3) + c)^4 + 980*(2*(I*a^2*b + a*b^2)*c^5 + 5*(-I*a*b^2 - b^3)*c^4)*(d*x^(1/3) + c)^3
+ 735*((-I*a^2*b - a*b^2)*c^6 + 3*(I*a*b^2 + b^3)*c^5)*(d*x^(1/3) + c)^2 + 105*(2*(I*a^2*b + a*b^2)*c^7 + 7*(
-I*a*b^2 - b^3)*c^6)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*log(((a^2 + b^2)*cos(2*d*x^(1/3) + 2*c)^2 + 4*a*
b*sin(2*d*x^(1/3) + 2*c) + (a^2 + b^2)*sin(2*d*x^(1/3) + 2*c)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*d*x^(1/3) +
2*c))/(a^2 + b^2)) - 1587600*(a^2*b + I*a*b^2 + (a^2*b - I*a*b^2)*cos(2*d*x^(1/3) + 2*c) + (I*a^2*b + a*b^2)*s
in(2*d*x^(1/3) + 2*c))*polylog(9, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) - 453600*(-2*I*a*b^2 + 2*b^3
+ 7*(-I*a^2*b + a*b^2)*(d*x^(1/3) + c) + 4*(I*a^2*b - a*b^2)*c + (-2*I*a*b^2 - 2*b^3 + 7*(-I*a^2*b - a*b^2)*(
d*x^(1/3) + c) + 4*(I*a^2*b + a*b^2)*c)*cos(2*d*x^(1/3) + 2*c) + (2*a*b^2 - 2*I*b^3 + 7*(a^2*b - I*a*b^2)*(d*x
^(1/3) + c) - 4*(a^2*b - I*a*b^2)*c)*sin(2*d*x^(1/3) + 2*c))*polylog(8, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-
I*a + b)) + 151200*(21*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^2 + 7*(a^2*b + I*a*b^2)*c^2 + 12*(a*b^2 + I*b^3 - 2*(
a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c) - 7*(a*b^2 + I*b^3)*c + (21*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^2 + 7*(a^2*b
- I*a*b^2)*c^2 + 12*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c) - 7*(a*b^2 - I*b^3)*c)*cos(2*d*x^
(1/3) + 2*c) - (21*(-I*a^2*b - a*b^2)*(d*x^(1/3) + c)^2 + 7*(-I*a^2*b - a*b^2)*c^2 + 12*(-I*a*b^2 - b^3 + 2*(I
*a^2*b + a*b^2)*c)*(d*x^(1/3) + c) + 7*(I*a*b^2 + b^3)*c)*sin(2*d*x^(1/3) + 2*c))*polylog(7, (I*a + b)*e^(2*I*
d*x^(1/3) + 2*I*c)/(-I*a + b)) - 30240*(70*(I*a^2*b - a*b^2)*(d*x^(1/3) + c)^3 + 14*(-I*a^2*b + a*b^2)*c^3 + 6
0*(I*a*b^2 - b^3 + 2*(-I*a^2*b + a*b^2)*c)*(d*x^(1/3) + c)^2 + 21*(I*a*b^2 - b^3)*c^2 + 70*((I*a^2*b - a*b^2)*
c^2 + (-I*a*b^2 + b^3)*c)*(d*x^(1/3) + c) + (70*(I*a^2*b + a*b^2)*(d*x^(1/3) + c)^3 + 14*(-I*a^2*b - a*b^2)*c^
3 + 60*(I*a*b^2 + b^3 + 2*(-I*a^2*b - a*b^2)*c)*(d*x^(1/3) + c)^2 + 21*(I*a*b^2 + b^3)*c^2 + 70*((I*a^2*b + a*
b^2)*c^2 + (-I*a*b^2 - b^3)*c)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) - (70*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)
^3 - 14*(a^2*b - I*a*b^2)*c^3 + 60*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^2 + 21*(a*b^2 - I*b
^3)*c^2 + 70*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(6, (
I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) - 3780*(280*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^4 + 35*(a^2*b + I
*a*b^2)*c^4 + 320*(a*b^2 + I*b^3 - 2*(a^2*b + I*a*b^2)*c)*(d*x^(1/3) + c)^3 - 70*(a*b^2 + I*b^3)*c^3 + 560*((a
^2*b + I*a*b^2)*c^2 - (a*b^2 + I*b^3)*c)*(d*x^(1/3) + c)^2 - 112*(2*(a^2*b + I*a*b^2)*c^3 - 3*(a*b^2 + I*b^3)*
c^2)*(d*x^(1/3) + c) + (280*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^4 + 35*(a^2*b - I*a*b^2)*c^4 + 320*(a*b^2 - I*b^
3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^3 - 70*(a*b^2 - I*b^3)*c^3 + 560*((a^2*b - I*a*b^2)*c^2 - (a*b^2 -
I*b^3)*c)*(d*x^(1/3) + c)^2 - 112*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c))*cos(2*d*x
^(1/3) + 2*c) + (280*(I*a^2*b + a*b^2)*(d*x^(1/3) + c)^4 + 35*(I*a^2*b + a*b^2)*c^4 + 320*(I*a*b^2 + b^3 + 2*(
-I*a^2*b - a*b^2)*c)*(d*x^(1/3) + c)^3 + 70*(-I*a*b^2 - b^3)*c^3 + 560*((I*a^2*b + a*b^2)*c^2 + (-I*a*b^2 - b^
3)*c)*(d*x^(1/3) + c)^2 + 112*(2*(-I*a^2*b - a*b^2)*c^3 + 3*(I*a*b^2 + b^3)*c^2)*(d*x^(1/3) + c))*sin(2*d*x^(1
/3) + 2*c))*polylog(5, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) - 2520*(168*(-I*a^2*b + a*b^2)*(d*x^(1/
3) + c)^5 + 14*(I*a^2*b - a*b^2)*c^5 + 240*(-I*a*b^2 + b^3 + 2*(I*a^2*b - a*b^2)*c)*(d*x^(1/3) + c)^4 + 35*(-I
*a*b^2 + b^3)*c^4 + 560*((-I*a^2*b + a*b^2)*c^2 + (I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^3 + 168*(2*(I*a^2*b - a*b
^2)*c^3 + 3*(-I*a*b^2 + b^3)*c^2)*(d*x^(1/3) + c)^2 + 105*((-I*a^2*b + a*b^2)*c^4 + 2*(I*a*b^2 - b^3)*c^3)*(d*
x^(1/3) + c) + (168*(-I*a^2*b - a*b^2)*(d*x^(1/3) + c)^5 + 14*(I*a^2*b + a*b^2)*c^5 + 240*(-I*a*b^2 - b^3 + 2*
(I*a^2*b + a*b^2)*c)*(d*x^(1/3) + c)^4 + 35*(-I*a*b^2 - b^3)*c^4 + 560*((-I*a^2*b - a*b^2)*c^2 + (I*a*b^2 + b^
3)*c)*(d*x^(1/3) + c)^3 + 168*(2*(I*a^2*b + a*b^2)*c^3 + 3*(-I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^2 + 105*((-I*
a^2*b - a*b^2)*c^4 + 2*(I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c))*cos(2*d*x^(1/3) + 2*c) + (168*(a^2*b - I*a*b^2)*(
d*x^(1/3) + c)^5 - 14*(a^2*b - I*a*b^2)*c^5 + 240*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c)*(d*x^(1/3) + c)^4 +
35*(a*b^2 - I*b^3)*c^4 + 560*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)^3 - 168*(2*(a^2*b - I
*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^2 + 105*((a^2*b - I*a*b^2)*c^4 - 2*(a*b^2 - I*b^3)*c^3)*(
d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(4, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + 1260*(112
*(a^2*b + I*a*b^2)*(d*x^(1/3) + c)^6 + 7*(a^2*b + I*a*b^2)*c^6 + 192*(a*b^2 + I*b^3 - 2*(a^2*b + I*a*b^2)*c)*(
d*x^(1/3) + c)^5 - 21*(a*b^2 + I*b^3)*c^5 + 560*((a^2*b + I*a*b^2)*c^2 - (a*b^2 + I*b^3)*c)*(d*x^(1/3) + c)^4
- 224*(2*(a^2*b + I*a*b^2)*c^3 - 3*(a*b^2 + I*b^3)*c^2)*(d*x^(1/3) + c)^3 + 210*((a^2*b + I*a*b^2)*c^4 - 2*(a*
b^2 + I*b^3)*c^3)*(d*x^(1/3) + c)^2 - 28*(2*(a^2*b + I*a*b^2)*c^5 - 5*(a*b^2 + I*b^3)*c^4)*(d*x^(1/3) + c) + (
112*(a^2*b - I*a*b^2)*(d*x^(1/3) + c)^6 + 7*(a^2*b - I*a*b^2)*c^6 + 192*(a*b^2 - I*b^3 - 2*(a^2*b - I*a*b^2)*c
)*(d*x^(1/3) + c)^5 - 21*(a*b^2 - I*b^3)*c^5 + 560*((a^2*b - I*a*b^2)*c^2 - (a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)
^4 - 224*(2*(a^2*b - I*a*b^2)*c^3 - 3*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^3 + 210*((a^2*b - I*a*b^2)*c^4 - 2*
(a*b^2 - I*b^3)*c^3)*(d*x^(1/3) + c)^2 - 28*(2*(a^2*b - I*a*b^2)*c^5 - 5*(a*b^2 - I*b^3)*c^4)*(d*x^(1/3) + c))
*cos(2*d*x^(1/3) + 2*c) - (112*(-I*a^2*b - a*b^2)*(d*x^(1/3) + c)^6 + 7*(-I*a^2*b - a*b^2)*c^6 + 192*(-I*a*b^2
- b^3 + 2*(I*a^2*b + a*b^2)*c)*(d*x^(1/3) + c)^5 + 21*(I*a*b^2 + b^3)*c^5 + 560*((-I*a^2*b - a*b^2)*c^2 + (I*
a*b^2 + b^3)*c)*(d*x^(1/3) + c)^4 + 224*(2*(I*a^2*b + a*b^2)*c^3 + 3*(-I*a*b^2 - b^3)*c^2)*(d*x^(1/3) + c)^3 +
210*((-I*a^2*b - a*b^2)*c^4 + 2*(I*a*b^2 + b^3)*c^3)*(d*x^(1/3) + c)^2 + 28*(2*(I*a^2*b + a*b^2)*c^5 + 5*(-I*
a*b^2 - b^3)*c^4)*(d*x^(1/3) + c))*sin(2*d*x^(1/3) + 2*c))*polylog(3, (I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*
a + b)) - 35*((-I*a^3 - 3*a^2*b + 3*I*a*b^2 + b^3)*(d*x^(1/3) + c)^9 - 9*(2*a*b^2 - 2*I*b^3 - (I*a^3 + 3*a^2*b
- 3*I*a*b^2 - b^3)*c)*(d*x^(1/3) + c)^8 + 144*(a*b^2 - I*b^3)*(d*x^(1/3) + c)*c^7 + 36*((-I*a^3 - 3*a^2*b + 3
*I*a*b^2 + b^3)*c^2 + 4*(a*b^2 - I*b^3)*c)*(d*x^(1/3) + c)^7 + 84*((I*a^3 + 3*a^2*b - 3*I*a*b^2 - b^3)*c^3 - 6
*(a*b^2 - I*b^3)*c^2)*(d*x^(1/3) + c)^6 + 126*((-I*a^3 - 3*a^2*b + 3*I*a*b^2 + b^3)*c^4 + 8*(a*b^2 - I*b^3)*c^
3)*(d*x^(1/3) + c)^5 + 126*((I*a^3 + 3*a^2*b - 3*I*a*b^2 - b^3)*c^5 - 10*(a*b^2 - I*b^3)*c^4)*(d*x^(1/3) + c)^
4 + 84*((-I*a^3 - 3*a^2*b + 3*I*a*b^2 + b^3)*c^6 + 12*(a*b^2 - I*b^3)*c^5)*(d*x^(1/3) + c)^3 + 36*((I*a^3 + 3*
a^2*b - 3*I*a*b^2 - b^3)*c^7 - 14*(a*b^2 - I*b^3)*c^6)*(d*x^(1/3) + c)^2)*sin(2*d*x^(1/3) + 2*c))/(a^5 + I*a^4
*b + 2*a^3*b^2 + 2*I*a^2*b^3 + a*b^4 + I*b^5 + (a^5 - I*a^4*b + 2*a^3*b^2 - 2*I*a^2*b^3 + a*b^4 - I*b^5)*cos(2
*d*x^(1/3) + 2*c) - (-I*a^5 - a^4*b - 2*I*a^3*b^2 - 2*a^2*b^3 - I*a*b^4 - b^5)*sin(2*d*x^(1/3) + 2*c)))/d^9

Giac [F]

\[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \tan \left (d x^{\frac {1}{3}} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(a+b*tan(c+d*x^(1/3)))^2,x, algorithm="giac")

[Out]

integrate(x^2/(b*tan(d*x^(1/3) + c) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (a+b \tan \left (c+d \sqrt [3]{x}\right )\right )^2} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {tan}\left (c+d\,x^{1/3}\right )\right )}^2} \,d x \]

[In]

int(x^2/(a + b*tan(c + d*x^(1/3)))^2,x)

[Out]

int(x^2/(a + b*tan(c + d*x^(1/3)))^2, x)