Optimal. Leaf size=125 \[ \frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b^2 x^2 \cot \left (c+d x^2\right )}{2 d}+\frac {b^2 \log \left (\sin \left (c+d x^2\right )\right )}{2 d^2}+\frac {i a b \text {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i a b \text {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{d^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4290, 4275,
4268, 2317, 2438, 4269, 3556} \begin {gather*} \frac {a^2 x^4}{4}+\frac {i a b \text {Li}_2\left (-e^{i \left (d x^2+c\right )}\right )}{d^2}-\frac {i a b \text {Li}_2\left (e^{i \left (d x^2+c\right )}\right )}{d^2}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {b^2 \log \left (\sin \left (c+d x^2\right )\right )}{2 d^2}-\frac {b^2 x^2 \cot \left (c+d x^2\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 3556
Rule 4268
Rule 4269
Rule 4275
Rule 4290
Rubi steps
\begin {align*} \int x^3 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx &=\frac {1}{2} \text {Subst}\left (\int x (a+b \csc (c+d x))^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a^2 x+2 a b x \csc (c+d x)+b^2 x \csc ^2(c+d x)\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^4}{4}+(a b) \text {Subst}\left (\int x \csc (c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \text {Subst}\left (\int x \csc ^2(c+d x) \, dx,x,x^2\right )\\ &=\frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b^2 x^2 \cot \left (c+d x^2\right )}{2 d}-\frac {(a b) \text {Subst}\left (\int \log \left (1-e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d}+\frac {(a b) \text {Subst}\left (\int \log \left (1+e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{d}+\frac {b^2 \text {Subst}\left (\int \cot (c+d x) \, dx,x,x^2\right )}{2 d}\\ &=\frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b^2 x^2 \cot \left (c+d x^2\right )}{2 d}+\frac {b^2 \log \left (\sin \left (c+d x^2\right )\right )}{2 d^2}+\frac {(i a b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {(i a b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{d^2}\\ &=\frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b^2 x^2 \cot \left (c+d x^2\right )}{2 d}+\frac {b^2 \log \left (\sin \left (c+d x^2\right )\right )}{2 d^2}+\frac {i a b \text {Li}_2\left (-e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i a b \text {Li}_2\left (e^{i \left (c+d x^2\right )}\right )}{d^2}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(268\) vs. \(2(125)=250\).
time = 5.41, size = 268, normalized size = 2.14 \begin {gather*} \frac {2 b^2 d x^2 \cot (c)+d x^2 \left (a^2 d x^2-2 b^2 \cot (c)\right )-2 b^2 \left (d x^2 \cot (c)-\log \left (\sin \left (c+d x^2\right )\right )\right )+4 a b \left (2 \text {ArcTan}(\tan (c)) \tanh ^{-1}\left (\cos (c)-\sin (c) \tan \left (\frac {d x^2}{2}\right )\right )+\frac {\left (\left (d x^2+\text {ArcTan}(\tan (c))\right ) \left (\log \left (1-e^{i \left (d x^2+\text {ArcTan}(\tan (c))\right )}\right )-\log \left (1+e^{i \left (d x^2+\text {ArcTan}(\tan (c))\right )}\right )\right )+i \text {PolyLog}\left (2,-e^{i \left (d x^2+\text {ArcTan}(\tan (c))\right )}\right )-i \text {PolyLog}\left (2,e^{i \left (d x^2+\text {ArcTan}(\tan (c))\right )}\right )\right ) \sec (c)}{\sqrt {\sec ^2(c)}}\right )+b^2 d x^2 \csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} \left (c+d x^2\right )\right ) \sin \left (\frac {d x^2}{2}\right )+b^2 d x^2 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} \left (c+d x^2\right )\right ) \sin \left (\frac {d x^2}{2}\right )}{4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \csc \left (d \,x^{2}+c \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 604 vs. \(2 (107) = 214\).
time = 0.38, size = 604, normalized size = 4.83 \begin {gather*} \frac {1}{4} \, a^{2} x^{4} - \frac {4 \, b^{2} d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + 4 i \, b^{2} d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) - 2 \, {\left (2 \, a b d x^{2} - b^{2} - {\left (2 \, a b d x^{2} - b^{2}\right )} \cos \left (2 \, d x^{2} + 2 \, c\right ) + {\left (-2 i \, a b d x^{2} + i \, b^{2}\right )} \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} \arctan \left (\sin \left (d x^{2} + c\right ), \cos \left (d x^{2} + c\right ) + 1\right ) - 2 \, {\left (b^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + i \, b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) - b^{2}\right )} \arctan \left (\sin \left (d x^{2} + c\right ), \cos \left (d x^{2} + c\right ) - 1\right ) + 4 \, {\left (a b d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + i \, a b d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) - a b d x^{2}\right )} \arctan \left (\sin \left (d x^{2} + c\right ), -\cos \left (d x^{2} + c\right ) + 1\right ) - 4 \, {\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 (i \, d x^{2} + i \, c\right )}\right ) + 4 \, {\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 (i \, d x^{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 (d x^{2} + c\right )^{2} + \sin \left (d x^{2} + c\right )^{2} + 2 \, \cos \left (d x^{2} + c\right ) + 1\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 (d x^{2} + c\right )^{2} + \sin \left (d x^{2} + c\right )^{2} - 2 \, \cos \left (d x^{2} + c\right ) + 1\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 451 vs. \(2 (107) = 214\).
time = 3.19, size = 451, normalized size = 3.61 \begin {gather*} \frac {a^{2} d^{2} x^{4} \sin \left (d x^{2} + c\right ) - 2 \, b^{2} d x^{2} \cos \left (d x^{2} + c\right ) - 2 i \, a b {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) + 2 i \, a b {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - 2 i \, a b {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) + 2 i \, a b {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) \sin \left (d x^{2} + c\right ) - {\left (2 \, a b d x^{2} - b^{2}\right )} \log \left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) \sin \left (d x^{2} + c\right ) - {\left (2 \, a b d x^{2} - b^{2}\right )} \log \left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) \sin \left (d x^{2} + c\right ) - {\left (2 \, a b c - b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{2} + c\right ) - {\left (2 \, a b c - b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{2} + c\right ) + 2 \, {\left (a b d x^{2} + a b c\right )} \log \left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) \sin \left (d x^{2} + c\right ) + 2 \, {\left (a b d x^{2} + a b c\right )} \log \left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) \sin \left (d x^{2} + c\right )}{4 \, d^{2} \sin \left (d x^{2} + c\right )} \end {gather*}
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 \begin {gather*} \int x^{3} \left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^3\,{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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