Optimal. Leaf size=663 \[ \frac {i \text {Li}_2\left (\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^3 \left (a^2-b^2\right )}+\frac {i \text {Li}_2\left (\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d^3 \left (a^2-b^2\right )}-\frac {i a \text {Li}_3\left (\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^3 \left (a^2-b^2\right )^{3/2}}+\frac {i a \text {Li}_3\left (\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d^3 \left (a^2-b^2\right )^{3/2}}-\frac {a x^2 \text {Li}_2\left (\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^2 \left (a^2-b^2\right )^{3/2}}+\frac {a x^2 \text {Li}_2\left (\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d^2 \left (a^2-b^2\right )^{3/2}}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^2 \left (a^2-b^2\right )}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{d^2 \left (a^2-b^2\right )}-\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 d \left (a^2-b^2\right )^{3/2}}+\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{2 d \left (a^2-b^2\right )^{3/2}}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}+\frac {i x^4}{2 d \left (a^2-b^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.30, antiderivative size = 663, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {3379, 3324, 3323, 2264, 2190, 2531, 2282, 6589, 4519, 2279, 2391} \[ -\frac {a x^2 \text {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^2 \left (a^2-b^2\right )^{3/2}}+\frac {a x^2 \text {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{d^2 \left (a^2-b^2\right )^{3/2}}+\frac {i \text {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^3 \left (a^2-b^2\right )}+\frac {i \text {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{d^3 \left (a^2-b^2\right )}-\frac {i a \text {PolyLog}\left (3,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^3 \left (a^2-b^2\right )^{3/2}}+\frac {i a \text {PolyLog}\left (3,\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{d^3 \left (a^2-b^2\right )^{3/2}}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^2 \left (a^2-b^2\right )}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{d^2 \left (a^2-b^2\right )}-\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 d \left (a^2-b^2\right )^{3/2}}+\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{2 d \left (a^2-b^2\right )^{3/2}}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}+\frac {i x^4}{2 d \left (a^2-b^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3323
Rule 3324
Rule 3379
Rule 4519
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{(a+b \sin (c+d x))^2} \, dx,x,x^2\right )\\ &=\frac {b x^4 \cos \left (c+d x^2\right )}{2 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^2\right )\right )}+\frac {a \operatorname {Subst}\left (\int \frac {x^2}{a+b \sin (c+d x)} \, dx,x,x^2\right )}{2 \left (a^2-b^2\right )}-\frac {b \operatorname {Subst}\left (\int \frac {x \cos (c+d x)}{a+b \sin (c+d x)} \, dx,x,x^2\right )}{\left (a^2-b^2\right ) d}\\ &=\frac {i x^4}{2 \left (a^2-b^2\right ) d}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^2\right )\right )}+\frac {a \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a^2-b^2}-\frac {b \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{a-\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx,x,x^2\right )}{\left (a^2-b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx,x,x^2\right )}{\left (a^2-b^2\right ) d}\\ &=\frac {i x^4}{2 \left (a^2-b^2\right ) d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^2\right )\right )}-\frac {(i a b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx,x,x^2\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {(i a b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx,x,x^2\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\left (a^2-b^2\right ) d^2}+\frac {\operatorname {Subst}\left (\int \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\left (a^2-b^2\right ) d^2}\\ &=\frac {i x^4}{2 \left (a^2-b^2\right ) d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^2\right )\right )}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{\left (a^2-b^2\right ) d^3}-\frac {i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{\left (a^2-b^2\right ) d^3}+\frac {(i a) \operatorname {Subst}\left (\int x \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac {(i a) \operatorname {Subst}\left (\int x \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\left (a^2-b^2\right )^{3/2} d}\\ &=\frac {i x^4}{2 \left (a^2-b^2\right ) d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}+\frac {i \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {a x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}+\frac {i \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {a x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^2\right )\right )}+\frac {a \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {a \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{\left (a^2-b^2\right )^{3/2} d^2}\\ &=\frac {i x^4}{2 \left (a^2-b^2\right ) d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}+\frac {i \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {a x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}+\frac {i \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {a x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^2\right )\right )}-\frac {(i a) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{\left (a^2-b^2\right )^{3/2} d^3}+\frac {(i a) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{\left (a^2-b^2\right )^{3/2} d^3}\\ &=\frac {i x^4}{2 \left (a^2-b^2\right ) d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}+\frac {i \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {a x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}+\frac {i \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {a x^2 \text {Li}_2\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {i a \text {Li}_3\left (\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}+\frac {i a \text {Li}_3\left (\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^2\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.33, size = 513, normalized size = 0.77 \[ \frac {-\frac {i a d^2 x^4 \log \left (1+\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}-a}\right )}{\sqrt {a^2-b^2}}+\frac {i a d^2 x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{\sqrt {a^2-b^2}}+\left (-\frac {2 a d x^2}{\sqrt {a^2-b^2}}+2 i\right ) \text {Li}_2\left (-\frac {i b e^{i \left (d x^2+c\right )}}{\sqrt {a^2-b^2}-a}\right )+\left (\frac {2 a d x^2}{\sqrt {a^2-b^2}}+2 i\right ) \text {Li}_2\left (\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )-\frac {2 i a \text {Li}_3\left (\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {2 i a \text {Li}_3\left (\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-2 d x^2 \log \left (1+\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}-a}\right )-2 d x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )+\frac {b d^2 x^4 \cos \left (c+d x^2\right )}{a+b \sin \left (c+d x^2\right )}+i d^2 x^4}{2 d^3 \left (a^2-b^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 1.26, size = 2487, normalized size = 3.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{{\left (b \sin \left (d x^{2} + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.15, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (a +b \sin \left (d \,x^{2}+c \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5}{{\left (a+b\,\sin \left (d\,x^2+c\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________