3.1.57 \(\int \frac {x^2}{a+b \tan (c+d \sqrt [3]{x})} \, dx\) [57]

3.1.57.1 Optimal result

Integrand size = 20, antiderivative size = 511 \[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {x^3}{3 (a+i b)}+\frac {3 b x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d}-\frac {12 i b x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^2}+\frac {42 b x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^3}+\frac {126 i b x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^4}-\frac {315 b x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^5}-\frac {630 i b x \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^6}+\frac {945 b x^{2/3} \operatorname {PolyLog}\left (7,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^7}+\frac {945 i b \sqrt [3]{x} \operatorname {PolyLog}\left (8,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) d^8}-\frac {945 b \operatorname {PolyLog}\left (9,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) d^9} \]

output

1/3*x^3/(a+I*b)+3*b*x^(8/3)*ln(1+(a^2+b^2)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^ 
2)/(a^2+b^2)/d-12*I*b*x^(7/3)*polylog(2,-(a^2+b^2)*exp(2*I*(c+d*x^(1/3)))/ 
(a+I*b)^2)/(a^2+b^2)/d^2+42*b*x^2*polylog(3,-(a^2+b^2)*exp(2*I*(c+d*x^(1/3 
)))/(a+I*b)^2)/(a^2+b^2)/d^3+126*I*b*x^(5/3)*polylog(4,-(a^2+b^2)*exp(2*I* 
(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2)/d^4-315*b*x^(4/3)*polylog(5,-(a^2+b^2) 
*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2)/d^5-630*I*b*x*polylog(6,-(a^2 
+b^2)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2)/d^6+945*b*x^(2/3)*polylo 
g(7,-(a^2+b^2)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2)/d^7+945*I*b*x^( 
1/3)*polylog(8,-(a^2+b^2)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2)/d^8- 
945/2*b*polylog(9,-(a^2+b^2)*exp(2*I*(c+d*x^(1/3)))/(a+I*b)^2)/(a^2+b^2)/d 
^9
 

3.1.57.2 Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\frac {2 a d^9 x^3+2 i b d^9 x^3+18 b d^8 x^{8/3} \log \left (1+\frac {(a+i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+72 i b d^7 x^{7/3} \operatorname {PolyLog}\left (2,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+252 b d^6 x^2 \operatorname {PolyLog}\left (3,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-756 i b d^5 x^{5/3} \operatorname {PolyLog}\left (4,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-1890 b d^4 x^{4/3} \operatorname {PolyLog}\left (5,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+3780 i b d^3 x \operatorname {PolyLog}\left (6,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )+5670 b d^2 x^{2/3} \operatorname {PolyLog}\left (7,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-5670 i b d \sqrt [3]{x} \operatorname {PolyLog}\left (8,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )-2835 b \operatorname {PolyLog}\left (9,\frac {(-a-i b) e^{-2 i \left (c+d \sqrt [3]{x}\right )}}{a-i b}\right )}{6 \left (a^2+b^2\right ) d^9} \]

input

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

output

(2*a*d^9*x^3 + (2*I)*b*d^9*x^3 + 18*b*d^8*x^(8/3)*Log[1 + (a + I*b)/((a - 
I*b)*E^((2*I)*(c + d*x^(1/3))))] + (72*I)*b*d^7*x^(7/3)*PolyLog[2, (-a - I 
*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + 252*b*d^6*x^2*PolyLog[3, (-a 
- I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - (756*I)*b*d^5*x^(5/3)*Poly 
Log[4, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] - 1890*b*d^4*x^(4 
/3)*PolyLog[5, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + (3780*I 
)*b*d^3*x*PolyLog[6, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3))))] + 5 
670*b*d^2*x^(2/3)*PolyLog[7, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x^(1/3) 
)))] - (5670*I)*b*d*x^(1/3)*PolyLog[8, (-a - I*b)/((a - I*b)*E^((2*I)*(c + 
 d*x^(1/3))))] - 2835*b*PolyLog[9, (-a - I*b)/((a - I*b)*E^((2*I)*(c + d*x 
^(1/3))))])/(6*(a^2 + b^2)*d^9)
 

3.1.57.3 Rubi [A] (verified)

Time = 2.24 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {4234, 3042, 4215, 2620, 3011, 7163, 7163, 7163, 7163, 7163, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 4234

\(\displaystyle 3 \int \frac {x^{8/3}}{a+b \tan \left (c+d \sqrt [3]{x}\right )}d\sqrt [3]{x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 \int \frac {x^{8/3}}{a+b \tan \left (c+d \sqrt [3]{x}\right )}d\sqrt [3]{x}\)

\(\Big \downarrow \) 4215

\(\displaystyle 3 \left (2 i b \int \frac {e^{2 i \left (c+d \sqrt [3]{x}\right )} x^{8/3}}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}d\sqrt [3]{x}+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \int x^{7/3} \log \left (\frac {e^{2 i \left (c+d \sqrt [3]{x}\right )} \left (a^2+b^2\right )}{(a+i b)^2}+1\right )d\sqrt [3]{x}}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \int x^2 \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )d\sqrt [3]{x}}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \left (\frac {3 i \int x^{5/3} \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )d\sqrt [3]{x}}{d}-\frac {i x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \left (\frac {3 i \left (\frac {5 i \int x^{4/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )d\sqrt [3]{x}}{2 d}-\frac {i x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \left (\frac {3 i \left (\frac {5 i \left (\frac {2 i \int x \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )d\sqrt [3]{x}}{d}-\frac {i x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \left (\frac {3 i \left (\frac {5 i \left (\frac {2 i \left (\frac {3 i \int x^{2/3} \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )d\sqrt [3]{x}}{2 d}-\frac {i x \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \left (\frac {3 i \left (\frac {5 i \left (\frac {2 i \left (\frac {3 i \left (\frac {i \int \sqrt [3]{x} \operatorname {PolyLog}\left (7,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )d\sqrt [3]{x}}{d}-\frac {i x^{2/3} \operatorname {PolyLog}\left (7,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \left (\frac {3 i \left (\frac {5 i \left (\frac {2 i \left (\frac {3 i \left (\frac {i \left (\frac {i \int \operatorname {PolyLog}\left (8,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )d\sqrt [3]{x}}{2 d}-\frac {i \sqrt [3]{x} \operatorname {PolyLog}\left (8,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^{2/3} \operatorname {PolyLog}\left (7,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \left (\frac {3 i \left (\frac {5 i \left (\frac {2 i \left (\frac {3 i \left (\frac {i \left (\frac {\int \frac {\operatorname {PolyLog}\left (8,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{\sqrt [3]{x}}de^{2 i \left (c+d \sqrt [3]{x}\right )}}{4 d^2}-\frac {i \sqrt [3]{x} \operatorname {PolyLog}\left (8,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^{2/3} \operatorname {PolyLog}\left (7,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle 3 \left (2 i b \left (\frac {4 i \left (\frac {i x^{7/3} \operatorname {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}-\frac {7 i \left (\frac {3 i \left (\frac {5 i \left (\frac {2 i \left (\frac {3 i \left (\frac {i \left (\frac {\operatorname {PolyLog}\left (9,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{4 d^2}-\frac {i \sqrt [3]{x} \operatorname {PolyLog}\left (8,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^{2/3} \operatorname {PolyLog}\left (7,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x \operatorname {PolyLog}\left (6,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^{4/3} \operatorname {PolyLog}\left (5,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}-\frac {i x^{5/3} \operatorname {PolyLog}\left (4,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{d}-\frac {i x^2 \operatorname {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d}\right )}{2 d}\right )}{d \left (a^2+b^2\right )}-\frac {i x^{8/3} \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i \left (c+d \sqrt [3]{x}\right )}}{(a+i b)^2}\right )}{2 d \left (a^2+b^2\right )}\right )+\frac {x^3}{9 (a+i b)}\right )\)

input

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

output

3*(x^3/(9*(a + I*b)) + (2*I)*b*(((-1/2*I)*x^(8/3)*Log[1 + ((a^2 + b^2)*E^( 
(2*I)*(c + d*x^(1/3))))/(a + I*b)^2])/((a^2 + b^2)*d) + ((4*I)*(((I/2)*x^( 
7/3)*PolyLog[2, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/d 
 - (((7*I)/2)*(((-1/2*I)*x^2*PolyLog[3, -(((a^2 + b^2)*E^((2*I)*(c + d*x^( 
1/3))))/(a + I*b)^2)])/d + ((3*I)*(((-1/2*I)*x^(5/3)*PolyLog[4, -(((a^2 + 
b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/d + (((5*I)/2)*(((-1/2*I)*x 
^(4/3)*PolyLog[5, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)]) 
/d + ((2*I)*(((-1/2*I)*x*PolyLog[6, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3) 
)))/(a + I*b)^2)])/d + (((3*I)/2)*(((-1/2*I)*x^(2/3)*PolyLog[7, -(((a^2 + 
b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/d + (I*(((-1/2*I)*x^(1/3)*P 
olyLog[8, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)])/d + Pol 
yLog[9, -(((a^2 + b^2)*E^((2*I)*(c + d*x^(1/3))))/(a + I*b)^2)]/(4*d^2)))/ 
d))/d))/d))/d))/d))/d))/((a^2 + b^2)*d)))
 

3.1.57.3.1 Defintions of rubi rules used

rule 2620

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] - Si 
mp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + I*b)), x] + Simp[2*I*b   In 
t[(c + d*x)^m*(E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^Simp[2 
*I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2 
, 0] && IGtQ[m, 0]
 

rule 4234

Int[(x_)^(m_.)*((a_.) + (b_.)*Tan[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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 7163

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] - Simp[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]
 

3.1.57.4 Maple [F]

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

input

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

output

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

3.1.57.5 Fricas [F]

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

input

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

output

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

3.1.57.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.1.57.7 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. 1315 vs. \(2 (430) = 860\).

Time = 0.68 (sec) , antiderivative size = 1315, normalized size of antiderivative = 2.57 \[ \int \frac {x^2}{a+b \tan \left (c+d \sqrt [3]{x}\right )} \, dx=\text {Too large to display} \]

input

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

output

1/210*(315*(2*(d*x^(1/3) + c)*a/(a^2 + b^2) + 2*b*log(b*tan(d*x^(1/3) + c) 
 + a)/(a^2 + b^2) - b*log(tan(d*x^(1/3) + c)^2 + 1)/(a^2 + b^2))*c^8 + 2*( 
35*(d*x^(1/3) + c)^9*(a - I*b) - 315*(d*x^(1/3) + c)^8*(a - I*b)*c + 1260* 
(d*x^(1/3) + c)^7*(a - I*b)*c^2 - 2940*(d*x^(1/3) + c)^6*(a - I*b)*c^3 + 4 
410*(d*x^(1/3) + c)^5*(a - I*b)*c^4 - 4410*(d*x^(1/3) + c)^4*(a - I*b)*c^5 
 + 2940*(d*x^(1/3) + c)^3*(a - I*b)*c^6 - 1260*(d*x^(1/3) + c)^2*(a - I*b) 
*c^7 - 12*(420*I*(d*x^(1/3) + c)^8*b - 1920*I*(d*x^(1/3) + c)^7*b*c + 3920 
*I*(d*x^(1/3) + c)^6*b*c^2 - 4704*I*(d*x^(1/3) + c)^5*b*c^3 + 3675*I*(d*x^ 
(1/3) + c)^4*b*c^4 - 1960*I*(d*x^(1/3) + c)^3*b*c^5 + 735*I*(d*x^(1/3) + c 
)^2*b*c^6 - 210*I*(d*x^(1/3) + c)*b*c^7)*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)) 
 - 1260*(16*I*(d*x^(1/3) + c)^7*b - 64*I*(d*x^(1/3) + c)^6*b*c + 112*I*(d* 
x^(1/3) + c)^5*b*c^2 - 112*I*(d*x^(1/3) + c)^4*b*c^3 + 70*I*(d*x^(1/3) + c 
)^3*b*c^4 - 28*I*(d*x^(1/3) + c)^2*b*c^5 + 7*I*(d*x^(1/3) + c)*b*c^6 - I*b 
*c^7)*dilog((I*a + b)*e^(2*I*d*x^(1/3) + 2*I*c)/(-I*a + b)) + 6*(420*(d*x^ 
(1/3) + c)^8*b - 1920*(d*x^(1/3) + c)^7*b*c + 3920*(d*x^(1/3) + c)^6*b*c^2 
 - 4704*(d*x^(1/3) + c)^5*b*c^3 + 3675*(d*x^(1/3) + c)^4*b*c^4 - 1960*(d*x 
^(1/3) + c)^3*b*c^5 + 735*(d*x^(1/3) + c)^2*b*c^6 - 210*(d*x^(1/3) + c)*b* 
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) ...
 

3.1.57.8 Giac [F]

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

input

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

output

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

3.1.57.9 Mupad [F(-1)]

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

input

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

output

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