Optimal. Leaf size=596 \[ \frac {b \text {Li}_2\left (-\frac {a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}-\frac {b \text {Li}_2\left (-\frac {a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}+\frac {b^2 \log \left (a \cos \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}+\frac {i b x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {i b x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d \sqrt {b^2-a^2}}+\frac {b^2 x^2 \sin \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \cos \left (c+d x^2\right )+b\right )}-\frac {b^3 \text {Li}_2\left (-\frac {a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^3 \text {Li}_2\left (-\frac {a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac {i b^3 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {x^4}{4 a^2} \]
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Rubi [A] time = 1.20, antiderivative size = 596, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4204, 4191, 3324, 3321, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac {b \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 \sqrt {b^2-a^2}}-\frac {b^3 \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}-\frac {b \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d^2 \sqrt {b^2-a^2}}+\frac {b^3 \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{2 a^2 d^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^2 \log \left (a \cos \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2-b^2\right )}+\frac {i b x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d \sqrt {b^2-a^2}}-\frac {i b^3 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{b-\sqrt {b^2-a^2}}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}-\frac {i b x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d \sqrt {b^2-a^2}}+\frac {i b^3 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{\sqrt {b^2-a^2}+b}\right )}{2 a^2 d \left (b^2-a^2\right )^{3/2}}+\frac {b^2 x^2 \sin \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a \cos \left (c+d x^2\right )+b\right )}+\frac {x^4}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3321
Rule 3324
Rule 4191
Rule 4204
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a+b \sec (c+d x))^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {x}{a^2}+\frac {b^2 x}{a^2 (b+a \cos (c+d x))^2}-\frac {2 b x}{a^2 (b+a \cos (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac {x^4}{4 a^2}-\frac {b \operatorname {Subst}\left (\int \frac {x}{b+a \cos (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {x}{(b+a \cos (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {x^4}{4 a^2}+\frac {b^2 x^2 \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 )}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {x}{b+a \cos (c+d x)} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{b+a \cos (c+d x)} \, dx,x,x^2\right )}{2 a \left (a^2-b^2\right ) d}\\ &=\frac {x^4}{4 a^2}+\frac {b^2 x^2 \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 )}-\frac {b^3 \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right )}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \cos \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}\\ &=\frac {x^4}{4 a^2}+\frac {i b x^2 \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 x^2 \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 {b^2 \log \left (b+a \cos \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \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 )}+\frac {b^3 \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac {b^3 \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac {(i b) \operatorname {Subst}\left (\int \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {(i b) \operatorname {Subst}\left (\int \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt {-a^2+b^2} d}\\ &=\frac {x^4}{4 a^2}-\frac {i b^3 x^2 \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^2 \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^2 \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^2 \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 {b^2 \log \left (b+a \cos \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \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 )}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac {x^4}{4 a^2}-\frac {i b^3 x^2 \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^2 \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^2 \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^2 \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 {b^2 \log \left (b+a \cos \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}+\frac {b \text {Li}_2\left (-\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 \text {Li}_2\left (-\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^2 x^2 \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 )}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=\frac {x^4}{4 a^2}-\frac {i b^3 x^2 \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^2 \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^2 \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^2 \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 {b^2 \log \left (b+a \cos \left (c+d x^2\right )\right )}{2 a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 \text {Li}_2\left (-\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^2}+\frac {b \text {Li}_2\left (-\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 \text {Li}_2\left (-\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^2}-\frac {b \text {Li}_2\left (-\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^2 x^2 \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 )}\\ \end {align*}
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Mathematica [A] time = 9.03, size = 1069, normalized size = 1.79 \[ \frac {\left (b+a \cos \left (d x^2+c\right )\right ) \left (b^2 c \sin \left (d x^2+c\right )-b^2 \left (d x^2+c\right ) \sin \left (d x^2+c\right )\right ) \sec ^2\left (d x^2+c\right )}{2 a (b-a) (a+b) d^2 \left (a+b \sec \left (d x^2+c\right )\right )^2}+\frac {b \cos ^2\left (\frac {1}{2} \left (d x^2+c\right )\right ) \left (b+a \cos \left (d x^2+c\right )\right ) \left (-\frac {2 \left (2 a^2-b^2\right ) c \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )}{\sqrt {-a-b}}\right )}{\sqrt {-a-b} \sqrt {a-b}}+b \left (\log \left (-\left (\left (b+a \cos \left (d x^2+c\right )\right ) \sec ^2\left (\frac {1}{2} \left (d x^2+c\right )\right )\right )\right )-\log \left (\sec ^2\left (\frac {1}{2} \left (d x^2+c\right )\right )\right )\right )-\frac {i \left (2 a^2-b^2\right ) \left (\log \left (i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )+1\right ) \log \left (\frac {i \left (\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )-\log \left (1-i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )\right ) \log \left (\frac {\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )}{i \sqrt {a-b}+\sqrt {a+b}}\right )+\log \left (1-i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )\right ) \log \left (\frac {i \left (\sqrt {a-b} \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )+\sqrt {a+b}\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )-\log \left (i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )+1\right ) \log \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )+\sqrt {a+b}}{i \sqrt {a-b}+\sqrt {a+b}}\right )-\text {Li}_2\left (\frac {\sqrt {a-b} \left (1-i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )\right )}{\sqrt {a-b}-i \sqrt {a+b}}\right )+\text {Li}_2\left (\frac {\sqrt {a-b} \left (1-i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )-\text {Li}_2\left (\frac {\sqrt {a-b} \left (i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )+1\right )}{\sqrt {a-b}-i \sqrt {a+b}}\right )+\text {Li}_2\left (\frac {\sqrt {a-b} \left (i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )+1\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \left (\left (2 a^2-b^2\right ) d x^2+a b \sin \left (d x^2+c\right )\right ) \left (\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )\right ) \left (\sqrt {a-b} \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )+\sqrt {a+b}\right ) \sec ^2\left (d x^2+c\right )}{2 a^2 \left (a^2-b^2\right ) d^2 \left (a+b \sec \left (d x^2+c\right )\right )^2 \left (a b \sin \left (d x^2+c\right )-\left (2 a^2-b^2\right ) \left (c-i \log \left (1-i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )\right )+i \log \left (i \tan \left (\frac {1}{2} \left (d x^2+c\right )\right )+1\right )\right )\right )}+\frac {\left (d x^2-c\right ) \left (d x^2+c\right ) \left (b+a \cos \left (d x^2+c\right )\right )^2 \sec ^2\left (d x^2+c\right )}{4 a^2 d^2 \left (a+b \sec \left (d x^2+c\right )\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.01, size = 1928, normalized size = 3.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.54, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a +b \sec \left (d \,x^{2}+c \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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