Optimal. Leaf size=126 \[ \frac {a^2 x^4}{4}+\frac {i a b \text {Li}_2\left (-e^{2 i \left (d x^2+c\right )}\right )}{2 d^2}-\frac {a b x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d}+\frac {1}{2} i a b x^4+\frac {b^2 \log \left (\cos \left (c+d x^2\right )\right )}{2 d^2}+\frac {b^2 x^2 \tan \left (c+d x^2\right )}{2 d}-\frac {b^2 x^4}{4} \]
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Rubi [A] time = 0.24, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3747, 3722, 3719, 2190, 2279, 2391, 3720, 3475, 30} \[ \frac {a^2 x^4}{4}+\frac {i a b \text {Li}_2\left (-e^{2 i \left (d x^2+c\right )}\right )}{2 d^2}-\frac {a b x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d}+\frac {1}{2} i a b x^4+\frac {b^2 \log \left (\cos \left (c+d x^2\right )\right )}{2 d^2}+\frac {b^2 x^2 \tan \left (c+d x^2\right )}{2 d}-\frac {b^2 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2279
Rule 2391
Rule 3475
Rule 3719
Rule 3720
Rule 3722
Rule 3747
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b \tan (c+d x))^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^2 x+2 a b x \tan (c+d x)+b^2 x \tan ^2(c+d x)\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^4}{4}+(a b) \operatorname {Subst}\left (\int x \tan (c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \operatorname {Subst}\left (\int x \tan ^2(c+d x) \, dx,x,x^2\right )\\ &=\frac {a^2 x^4}{4}+\frac {1}{2} i a b x^4+\frac {b^2 x^2 \tan \left (c+d x^2\right )}{2 d}-(2 i a b) \operatorname {Subst}\left (\int \frac {e^{2 i (c+d x)} x}{1+e^{2 i (c+d x)}} \, dx,x,x^2\right )-\frac {1}{2} b^2 \operatorname {Subst}\left (\int x \, dx,x,x^2\right )-\frac {b^2 \operatorname {Subst}\left (\int \tan (c+d x) \, dx,x,x^2\right )}{2 d}\\ &=\frac {a^2 x^4}{4}+\frac {1}{2} i a b x^4-\frac {b^2 x^4}{4}-\frac {a b x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 \log \left (\cos \left (c+d x^2\right )\right )}{2 d^2}+\frac {b^2 x^2 \tan \left (c+d x^2\right )}{2 d}+\frac {(a b) \operatorname {Subst}\left (\int \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,x^2\right )}{d}\\ &=\frac {a^2 x^4}{4}+\frac {1}{2} i a b x^4-\frac {b^2 x^4}{4}-\frac {a b x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 \log \left (\cos \left (c+d x^2\right )\right )}{2 d^2}+\frac {b^2 x^2 \tan \left (c+d x^2\right )}{2 d}-\frac {(i a b) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \left (c+d x^2\right )}\right )}{2 d^2}\\ &=\frac {a^2 x^4}{4}+\frac {1}{2} i a b x^4-\frac {b^2 x^4}{4}-\frac {a b x^2 \log \left (1+e^{2 i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 \log \left (\cos \left (c+d x^2\right )\right )}{2 d^2}+\frac {i a b \text {Li}_2\left (-e^{2 i \left (c+d x^2\right )}\right )}{2 d^2}+\frac {b^2 x^2 \tan \left (c+d x^2\right )}{2 d}\\ \end {align*}
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Mathematica [B] time = 6.55, size = 295, normalized size = 2.34 \[ \frac {1}{4} x^4 \sec (c) \left (a^2 \cos (c)+2 a b \sin (c)-b^2 \cos (c)\right )-\frac {a b \csc (c) \sec (c) \left (d^2 x^4 e^{-i \tan ^{-1}(\cot (c))}-\frac {\cot (c) \left (i \text {Li}_2\left (e^{2 i \left (d x^2-\tan ^{-1}(\cot (c))\right )}\right )+i d x^2 \left (-2 \tan ^{-1}(\cot (c))-\pi \right )-2 \left (d x^2-\tan ^{-1}(\cot (c))\right ) \log \left (1-e^{2 i \left (d x^2-\tan ^{-1}(\cot (c))\right )}\right )-2 \tan ^{-1}(\cot (c)) \log \left (\sin \left (d x^2-\tan ^{-1}(\cot (c))\right )\right )-\pi \log \left (1+e^{-2 i d x^2}\right )+\pi \log \left (\cos \left (d x^2\right )\right )\right )}{\sqrt {\cot ^2(c)+1}}\right )}{2 d^2 \sqrt {\csc ^2(c) \left (\sin ^2(c)+\cos ^2(c)\right )}}+\frac {b^2 \sec (c) \left (d x^2 \sin (c)+\cos (c) \log \left (\cos (c) \cos \left (d x^2\right )-\sin (c) \sin \left (d x^2\right )\right )\right )}{2 d^2 \left (\sin ^2(c)+\cos ^2(c)\right )}+\frac {b^2 x^2 \sec (c) \sin \left (d x^2\right ) \sec \left (c+d x^2\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 199, normalized size = 1.58 \[ \frac {{\left (a^{2} - b^{2}\right )} d^{2} x^{4} + 2 \, b^{2} d x^{2} \tan \left (d x^{2} + c\right ) - i \, a b {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (d x^{2} + c\right ) - 1\right )}}{\tan \left (d x^{2} + c\right )^{2} + 1} + 1\right ) + i \, a b {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (d x^{2} + c\right ) - 1\right )}}{\tan \left (d x^{2} + c\right )^{2} + 1} + 1\right ) - {\left (2 \, a b d x^{2} - b^{2}\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (d x^{2} + c\right ) - 1\right )}}{\tan \left (d x^{2} + c\right )^{2} + 1}\right ) - {\left (2 \, a b d x^{2} - b^{2}\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (d x^{2} + c\right ) - 1\right )}}{\tan \left (d x^{2} + c\right )^{2} + 1}\right )}{4 \, d^{2}} \]
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 \[ \int {\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.92, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a +b \tan \left (d \,x^{2}+c \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.67, size = 398, normalized size = 3.16 \[ \frac {1}{4} \, a^{2} x^{4} + \frac {{\left (2 \, a b + i \, b^{2}\right )} d^{2} x^{4} - {\left (4 \, a b d x^{2} - 2 \, b^{2} + 2 \, {\left (2 \, a b d x^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (4 i \, a b d x^{2} - 2 i \, b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \arctan \left (\sin \left (2 \, d x^{2} + 2 \, c\right ), \cos \left (2 \, d x^{2} + 2 \, c\right ) + 1\right ) + {\left ({\left (2 \, a b + i \, b^{2}\right )} d^{2} x^{4} - 4 \, b^{2} d x^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + 2 \, {\left (a b \cos \left (2 \, d x^{2} + 2 \, c\right ) + i \, a b \sin \left (2 \, d x^{2} + 2 \, c\right ) + a b\right )} {\rm Li}_2\left (-e^{\left (2 i \, d x^{2} + 2 i \, c\right )}\right ) - {\left (-2 i \, a b d x^{2} + i \, b^{2} + {\left (-2 i \, a b d x^{2} + i \, b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (2 \, a b d x^{2} - b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \log \left (\cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x^{2} + 2 \, c\right ) + 1\right ) - {\left ({\left (-2 i \, a b + b^{2}\right )} d^{2} x^{4} + 4 i \, b^{2} d x^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )}{-4 i \, d^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + 4 \, d^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) - 4 i \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2 \,d x \]
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 \[ \int x^{3} \left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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