3.1.23 \(\int \frac {x^5}{(a+b \sec (c+d x^2))^2} \, dx\) [23]

3.1.23.1 Optimal result

Integrand size = 18, antiderivative size = 1092 \[ \int \frac {x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=-\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {i b^3 x^4 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^4 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^4 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^4 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {b^3 x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {2 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^3 x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {2 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {i b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {2 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {i b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {2 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {b^2 x^4 \sin \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d x^2\right )\right )} \]

output

-2*I*b*polylog(3,-a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/(-a^2+b 
^2)^(1/2)+1/6*x^6/a^2+b^2*x^2*ln(1+a*exp(I*(d*x^2+c))/(b-I*(a^2-b^2)^(1/2) 
))/a^2/(a^2-b^2)/d^2+b^2*x^2*ln(1+a*exp(I*(d*x^2+c))/(b+I*(a^2-b^2)^(1/2)) 
)/a^2/(a^2-b^2)/d^2-I*b*x^4*ln(1+a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/ 
a^2/d/(-a^2+b^2)^(1/2)-I*b^3*polylog(3,-a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^( 
1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3-1/2*I*b^3*x^4*ln(1+a*exp(I*(d*x^2+c))/(b-( 
-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d+1/2*I*b^3*x^4*ln(1+a*exp(I*(d*x^2 
+c))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d-b^3*x^2*polylog(2,-a*exp 
(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2+b^3*x^2*polyl 
og(2,-a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2-I* 
b^2*polylog(2,-a*exp(I*(d*x^2+c))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3 
+I*b^3*polylog(3,-a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^ 
(3/2)/d^3+1/2*b^2*x^4*sin(d*x^2+c)/a/(a^2-b^2)/d/(b+a*cos(d*x^2+c))-I*b^2* 
polylog(2,-a*exp(I*(d*x^2+c))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3+2*I 
*b*polylog(3,-a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/d^3/(-a^2+b^2)^ 
(1/2)+2*b*x^2*polylog(2,-a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/ 
(-a^2+b^2)^(1/2)-2*b*x^2*polylog(2,-a*exp(I*(d*x^2+c))/(b+(-a^2+b^2)^(1/2) 
))/a^2/d^2/(-a^2+b^2)^(1/2)+I*b*x^4*ln(1+a*exp(I*(d*x^2+c))/(b-(-a^2+b^2)^ 
(1/2)))/a^2/d/(-a^2+b^2)^(1/2)-1/2*I*b^2*x^4/a^2/(a^2-b^2)/d
 

3.1.23.2 Mathematica [A] (verified)

Time = 6.73 (sec) , antiderivative size = 895, normalized size of antiderivative = 0.82 \[ \int \frac {x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\frac {\left (b+a \cos \left (c+d x^2\right )\right ) \sec ^2\left (c+d x^2\right ) \left (x^6 \left (b+a \cos \left (c+d x^2\right )\right )-\frac {3 b \left (b+a \cos \left (c+d x^2\right )\right ) \left (2 \left (1+e^{2 i c}\right ) \left (i b \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}-2 a^2 d e^{i c} x^2+b^2 d e^{i c} x^2\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (2 c+d x^2\right )}}{b e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+2 \left (1+e^{2 i c}\right ) \left (i b \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}+2 a^2 d e^{i c} x^2-b^2 d e^{i c} x^2\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (2 c+d x^2\right )}}{b e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+i \left (d x^2 \left (2 b d e^{2 i c} \sqrt {\left (-a^2+b^2\right ) e^{2 i c}} x^2+\left (1+e^{2 i c}\right ) \left (2 i b \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}-2 a^2 d e^{i c} x^2+b^2 d e^{i c} x^2\right ) \log \left (1+\frac {a e^{i \left (2 c+d x^2\right )}}{b e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+\left (1+e^{2 i c}\right ) \left (2 i b \sqrt {\left (-a^2+b^2\right ) e^{2 i c}}+2 a^2 d e^{i c} x^2-b^2 d e^{i c} x^2\right ) \log \left (1+\frac {a e^{i \left (2 c+d x^2\right )}}{b e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )\right )-2 \left (2 a^2-b^2\right ) e^{i c} \left (1+e^{2 i c}\right ) \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (2 c+d x^2\right )}}{b e^{i c}-\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )+2 \left (2 a^2-b^2\right ) e^{i c} \left (1+e^{2 i c}\right ) \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (2 c+d x^2\right )}}{b e^{i c}+\sqrt {\left (-a^2+b^2\right ) e^{2 i c}}}\right )\right )\right )}{\left (a^2-b^2\right ) d^3 \sqrt {\left (-a^2+b^2\right ) e^{2 i c}} \left (1+e^{2 i c}\right )}+\frac {3 b^2 x^4 \left (-b \sin (c)+a \sin \left (d x^2\right )\right )}{(a-b) (a+b) d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right )}\right )}{6 a^2 \left (a+b \sec \left (c+d x^2\right )\right )^2} \]

input

Integrate[x^5/(a + b*Sec[c + d*x^2])^2,x]
 

output

((b + a*Cos[c + d*x^2])*Sec[c + d*x^2]^2*(x^6*(b + a*Cos[c + d*x^2]) - (3* 
b*(b + a*Cos[c + d*x^2])*(2*(1 + E^((2*I)*c))*(I*b*Sqrt[(-a^2 + b^2)*E^((2 
*I)*c)] - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*PolyLog[2, -((a*E^(I*(2 
*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2*(1 + E^(( 
2*I)*c))*(I*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*x^2 - b^2*d 
*E^(I*c)*x^2)*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 
 + b^2)*E^((2*I)*c)]))] + I*(d*x^2*(2*b*d*E^((2*I)*c)*Sqrt[(-a^2 + b^2)*E^ 
((2*I)*c)]*x^2 + (1 + E^((2*I)*c))*((2*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] 
 - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*Log[1 + (a*E^(I*(2*c + d*x^2)) 
)/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (1 + E^((2*I)*c))*((2*I) 
*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*x^2 - b^2*d*E^(I*c)*x^ 
2)*Log[1 + (a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I) 
*c)])]) - 2*(2*a^2 - b^2)*E^(I*c)*(1 + E^((2*I)*c))*PolyLog[3, -((a*E^(I*( 
2*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2*(2*a^2 - 
 b^2)*E^(I*c)*(1 + E^((2*I)*c))*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(b*E^ 
(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])))/((a^2 - b^2)*d^3*Sqrt[(-a^2 + 
 b^2)*E^((2*I)*c)]*(1 + E^((2*I)*c))) + (3*b^2*x^4*(-(b*Sin[c]) + a*Sin[d* 
x^2]))/((a - b)*(a + b)*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/( 
6*a^2*(a + b*Sec[c + d*x^2])^2)
 

3.1.23.3 Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 1090, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4692, 3042, 4679, 2009}

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.

\(\displaystyle \int \frac {x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx\)

\(\Big \downarrow \) 4692

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+b \sec \left (d x^2+c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+b \csc \left (d x^2+c+\frac {\pi }{2}\right )\right )^2}dx^2\)

\(\Big \downarrow \) 4679

\(\displaystyle \frac {1}{2} \int \left (-\frac {2 b x^4}{a^2 \left (b+a \cos \left (d x^2+c\right )\right )}+\frac {x^4}{a^2}+\frac {b^2 x^4}{a^2 \left (b+a \cos \left (d x^2+c\right )\right )^2}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {x^6}{3 a^2}+\frac {2 i b \log \left (\frac {e^{i \left (d x^2+c\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^4}{a^2 \sqrt {b^2-a^2} d}-\frac {i b^3 \log \left (\frac {e^{i \left (d x^2+c\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^4}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {2 i b \log \left (\frac {e^{i \left (d x^2+c\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^4}{a^2 \sqrt {b^2-a^2} d}+\frac {i b^3 \log \left (\frac {e^{i \left (d x^2+c\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^4}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {b^2 \sin \left (d x^2+c\right ) x^4}{a \left (a^2-b^2\right ) d \left (b+a \cos \left (d x^2+c\right )\right )}-\frac {i b^2 x^4}{a^2 \left (a^2-b^2\right ) d}+\frac {2 b^2 \log \left (\frac {e^{i \left (d x^2+c\right )} a}{b-i \sqrt {a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^2 \log \left (\frac {e^{i \left (d x^2+c\right )} a}{b+i \sqrt {a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^2}-\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^2}+\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^2+c\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {4 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}-\frac {2 i b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {4 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}+\frac {2 i b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}\right )\)

input

Int[x^5/(a + b*Sec[c + d*x^2])^2,x]
 

output

(((-I)*b^2*x^4)/(a^2*(a^2 - b^2)*d) + x^6/(3*a^2) + (2*b^2*x^2*Log[1 + (a* 
E^(I*(c + d*x^2)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^ 
2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - 
b^2)*d^2) - (I*b^3*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2] 
)])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((2*I)*b*x^4*Log[1 + (a*E^(I*(c + d*x^2)) 
)/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + (I*b^3*x^4*Log[1 + ( 
a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - 
 ((2*I)*b*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2* 
Sqrt[-a^2 + b^2]*d) - ((2*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - I 
*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((2*I)*b^2*PolyLog[2, -((a*E^ 
(I*(c + d*x^2)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (2*b^3 
*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a 
^2 + b^2)^(3/2)*d^2) + (4*b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sq 
rt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (2*b^3*x^2*PolyLog[2, -((a 
*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) 
 - (4*b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/( 
a^2*Sqrt[-a^2 + b^2]*d^2) - ((2*I)*b^3*PolyLog[3, -((a*E^(I*(c + d*x^2)))/ 
(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((4*I)*b*PolyLog[ 
3, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2] 
*d^3) + ((2*I)*b^3*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + ...
 

3.1.23.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 4692

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

3.1.23.4 Maple [F]

\[\int \frac {x^{5}}{{\left (a +b \sec \left (d \,x^{2}+c \right )\right )}^{2}}d x\]

input

int(x^5/(a+b*sec(d*x^2+c))^2,x)
 

output

int(x^5/(a+b*sec(d*x^2+c))^2,x)
 

3.1.23.5 Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3050 vs. \(2 (958) = 1916\).

Time = 0.51 (sec) , antiderivative size = 3050, normalized size of antiderivative = 2.79 \[ \int \frac {x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \]

input

integrate(x^5/(a+b*sec(d*x^2+c))^2,x, algorithm="fricas")
 

output

1/12*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d^3*x^6*cos(d*x^2 + c) + 2*(a^4*b - 2*a^ 
2*b^3 + b^5)*d^3*x^6 + 6*(a^3*b^2 - a*b^4)*d^2*x^4*sin(d*x^2 + c) - 6*(2*I 
*a^3*b^2 - I*a*b^4 + (2*I*a^4*b - I*a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - 
b^2)/a^2)*polylog(3, -(b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c) + (a*cos(d*x^ 
2 + c) + I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))/a) - 6*(-2*I*a^3*b^2 
+ I*a*b^4 + (-2*I*a^4*b + I*a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2 
)*polylog(3, -(b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) + 
 I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))/a) - 6*(-2*I*a^3*b^2 + I*a*b^ 
4 + (-2*I*a^4*b + I*a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*polylo 
g(3, -(b*cos(d*x^2 + c) - I*b*sin(d*x^2 + c) + (a*cos(d*x^2 + c) - I*a*sin 
(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2))/a) - 6*(2*I*a^3*b^2 - I*a*b^4 + (2*I* 
a^4*b - I*a^2*b^3)*cos(d*x^2 + c))*sqrt(-(a^2 - b^2)/a^2)*polylog(3, -(b*c 
os(d*x^2 + c) - I*b*sin(d*x^2 + c) - (a*cos(d*x^2 + c) - I*a*sin(d*x^2 + c 
))*sqrt(-(a^2 - b^2)/a^2))/a) - 6*(I*a^2*b^3 - I*b^5 + (I*a^3*b^2 - I*a*b^ 
4)*cos(d*x^2 + c) + ((2*a^4*b - a^2*b^3)*d*x^2*cos(d*x^2 + c) + (2*a^3*b^2 
 - a*b^4)*d*x^2)*sqrt(-(a^2 - b^2)/a^2))*dilog(-(b*cos(d*x^2 + c) + I*b*si 
n(d*x^2 + c) + (a*cos(d*x^2 + c) + I*a*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/a 
^2) + a)/a + 1) - 6*(I*a^2*b^3 - I*b^5 + (I*a^3*b^2 - I*a*b^4)*cos(d*x^2 + 
 c) - ((2*a^4*b - a^2*b^3)*d*x^2*cos(d*x^2 + c) + (2*a^3*b^2 - a*b^4)*d*x^ 
2)*sqrt(-(a^2 - b^2)/a^2))*dilog(-(b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c...
 

3.1.23.6 Sympy [F]

\[ \int \frac {x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{5}}{\left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

input

integrate(x**5/(a+b*sec(d*x**2+c))**2,x)
 

output

Integral(x**5/(a + b*sec(c + d*x**2))**2, x)
 

3.1.23.7 Maxima [F]

\[ \int \frac {x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^5/(a+b*sec(d*x^2+c))^2,x, algorithm="maxima")
 

output

1/6*((a^4 - a^2*b^2)*d*x^6*cos(2*d*x^2 + 2*c)^2 + 4*(a^2*b^2 - b^4)*d*x^6* 
cos(d*x^2 + c)^2 + (a^4 - a^2*b^2)*d*x^6*sin(2*d*x^2 + 2*c)^2 + 4*(a^2*b^2 
 - b^4)*d*x^6*sin(d*x^2 + c)^2 + 4*(a^3*b - a*b^3)*d*x^6*cos(d*x^2 + c) + 
6*a*b^3*x^4*sin(d*x^2 + c) + (a^4 - a^2*b^2)*d*x^6 + 2*(2*(a^3*b - a*b^3)* 
d*x^6*cos(d*x^2 + c) - 3*a*b^3*x^4*sin(d*x^2 + c) + (a^4 - a^2*b^2)*d*x^6) 
*cos(2*d*x^2 + 2*c) - 6*((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^4*b 
^2 - a^2*b^4)*d*cos(d*x^2 + c)^2 + (a^6 - a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 
+ 4*(a^5*b - a^3*b^3)*d*sin(2*d*x^2 + 2*c)*sin(d*x^2 + c) + 4*(a^4*b^2 - a 
^2*b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c) + (a^6 - 
 a^4*b^2)*d + 2*(2*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c) + (a^6 - a^4*b^2)*d) 
*cos(2*d*x^2 + 2*c))*integrate(2*(2*(2*a^2*b^2 - b^4)*d*x^5*cos(d*x^2 + c) 
^2 + 2*(2*a^2*b^2 - b^4)*d*x^5*sin(d*x^2 + c)^2 + (2*a^3*b - a*b^3)*d*x^5* 
cos(d*x^2 + c) + 2*a*b^3*x^3*sin(d*x^2 + c) + ((2*a^3*b - a*b^3)*d*x^5*cos 
(d*x^2 + c) - 2*a*b^3*x^3*sin(d*x^2 + c))*cos(2*d*x^2 + 2*c) + (2*a*b^3*x^ 
3*cos(d*x^2 + c) + (2*a^3*b - a*b^3)*d*x^5*sin(d*x^2 + c) + 2*a^2*b^2*x^3) 
*sin(2*d*x^2 + 2*c))/((a^6 - a^4*b^2)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^4*b^2 
- a^2*b^4)*d*cos(d*x^2 + c)^2 + (a^6 - a^4*b^2)*d*sin(2*d*x^2 + 2*c)^2 + 4 
*(a^5*b - a^3*b^3)*d*sin(2*d*x^2 + 2*c)*sin(d*x^2 + c) + 4*(a^4*b^2 - a^2* 
b^4)*d*sin(d*x^2 + c)^2 + 4*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c) + (a^6 - a^ 
4*b^2)*d + 2*(2*(a^5*b - a^3*b^3)*d*cos(d*x^2 + c) + (a^6 - a^4*b^2)*d)...
 

3.1.23.8 Giac [F]

\[ \int \frac {x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^5/(a+b*sec(d*x^2+c))^2,x, algorithm="giac")
 

output

integrate(x^5/(b*sec(d*x^2 + c) + a)^2, x)
 

3.1.23.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^5}{{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2} \,d x \]

input

int(x^5/(a + b/cos(c + d*x^2))^2,x)
 

output

int(x^5/(a + b/cos(c + d*x^2))^2, x)