Optimal. Leaf size=84 \[ \frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b \text {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{2 d^2}-\frac {i b \text {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{2 d^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {14, 4290, 4268,
2317, 2438} \begin {gather*} \frac {a x^4}{4}+\frac {i b \text {Li}_2\left (-e^{i \left (d x^2+c\right )}\right )}{2 d^2}-\frac {i b \text {Li}_2\left (e^{i \left (d x^2+c\right )}\right )}{2 d^2}-\frac {b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2317
Rule 2438
Rule 4268
Rule 4290
Rubi steps
\begin {align*} \int x^3 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx &=\int \left (a x^3+b x^3 \csc \left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^4}{4}+b \int x^3 \csc \left (c+d x^2\right ) \, dx\\ &=\frac {a x^4}{4}+\frac {1}{2} b \text {Subst}\left (\int x \csc (c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b \text {Subst}\left (\int \log \left (1-e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{2 d}+\frac {b \text {Subst}\left (\int \log \left (1+e^{i (c+d x)}\right ) \, dx,x,x^2\right )}{2 d}\\ &=\frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{2 d^2}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{2 d^2}\\ &=\frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b \text {Li}_2\left (-e^{i \left (c+d x^2\right )}\right )}{2 d^2}-\frac {i b \text {Li}_2\left (e^{i \left (c+d x^2\right )}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 118, normalized size = 1.40 \begin {gather*} \frac {a x^4}{4}+\frac {b \left (\left (c+d x^2\right ) \left (\log \left (1-e^{i \left (c+d x^2\right )}\right )-\log \left (1+e^{i \left (c+d x^2\right )}\right )\right )-c \log \left (\tan \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )+i \left (\text {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )-\text {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )\right )\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \csc \left (d \,x^{2}+c \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 288 vs. \(2 (66) = 132\).
time = 3.72, size = 288, normalized size = 3.43 \begin {gather*} \frac {a d^{2} x^{4} - b d x^{2} \log \left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) - b d x^{2} \log \left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) - b c \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 ) - b c \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 ) - i \, b {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + i \, b {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) - i \, b {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + i \, b {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) + {\left (b d x^{2} + b c\right )} \log \left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) + {\left (b d x^{2} + b c\right )} \log \left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right )}{4 \, d^{2}} \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 )\, 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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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