Optimal. Leaf size=225 \[ -\frac {b \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{2 a d^2 \sqrt {a^2+b^2}}+\frac {b \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{2 a d^2 \sqrt {a^2+b^2}}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{2 a d \sqrt {a^2+b^2}}+\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{2 a d \sqrt {a^2+b^2}}+\frac {x^4}{4 a} \]
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Rubi [A] time = 0.51, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5437, 4191, 3322, 2264, 2190, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a d^2 \sqrt {a^2+b^2}}+\frac {b \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}\right )}{2 a d^2 \sqrt {a^2+b^2}}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{2 a d \sqrt {a^2+b^2}}+\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{2 a d \sqrt {a^2+b^2}}+\frac {x^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3322
Rule 4191
Rule 5437
Rubi steps
\begin {align*} \int \frac {x^3}{a+b \text {csch}\left (c+d x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{a+b \text {csch}(c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {x}{a}-\frac {b x}{a (b+a \sinh (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac {x^4}{4 a}-\frac {b \operatorname {Subst}\left (\int \frac {x}{b+a \sinh (c+d x)} \, dx,x,x^2\right )}{2 a}\\ &=\frac {x^4}{4 a}-\frac {b \operatorname {Subst}\left (\int \frac {e^{c+d x} x}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^2\right )}{a}\\ &=\frac {x^4}{4 a}-\frac {b \operatorname {Subst}\left (\int \frac {e^{c+d x} x}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{\sqrt {a^2+b^2}}+\frac {b \operatorname {Subst}\left (\int \frac {e^{c+d x} x}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{\sqrt {a^2+b^2}}\\ &=\frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a \sqrt {a^2+b^2} d}-\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a \sqrt {a^2+b^2} d}\\ &=\frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\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^{c+d x^2}\right )}{2 a \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^{c+d x^2}\right )}{2 a \sqrt {a^2+b^2} d^2}\\ &=\frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d}-\frac {b \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d^2}+\frac {b \text {Li}_2\left (-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a \sqrt {a^2+b^2} d^2}\\ \end {align*}
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Mathematica [C] time = 3.21, size = 1166, normalized size = 5.18 \[ \frac {\text {csch}\left (d x^2+c\right ) \left (x^4+\frac {2 i b \pi \tanh ^{-1}\left (\frac {b \tanh \left (\frac {1}{2} \left (d x^2+c\right )\right )-a}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d^2}+\frac {2 b \left (2 \left (c+i \cos ^{-1}\left (-\frac {i b}{a}\right )\right ) \tan ^{-1}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )}{\sqrt {-a^2-b^2}}\right )+\left (-2 i d x^2-2 i c+\pi \right ) \tanh ^{-1}\left (\frac {(b-i a) \tan \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )}{\sqrt {-a^2-b^2}}\right )-\left (\cos ^{-1}\left (-\frac {i b}{a}\right )-2 \tan ^{-1}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {(a+i b) \left (a-i b+\sqrt {-a^2-b^2}\right ) \left (i \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )+1\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac {i b}{a}\right )+2 \tan ^{-1}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {i (a+i b) \left (-a+i b+\sqrt {-a^2-b^2}\right ) \left (\cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )+i\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )\right )}\right )+\left (\cos ^{-1}\left (-\frac {i b}{a}\right )+2 \tan ^{-1}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )}{\sqrt {-a^2-b^2}}\right )-2 i \tanh ^{-1}\left (\frac {(b-i a) \tan \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (-\frac {(-1)^{3/4} \sqrt {-a^2-b^2} e^{-\frac {d x^2}{2}-\frac {c}{2}}}{\sqrt {2} \sqrt {-i a} \sqrt {b+a \sinh \left (d x^2+c\right )}}\right )+\left (\cos ^{-1}\left (-\frac {i b}{a}\right )-2 \tan ^{-1}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )}{\sqrt {-a^2-b^2}}\right )+2 i \tanh ^{-1}\left (\frac {(b-i a) \tan \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {-a^2-b^2} e^{\frac {1}{2} \left (d x^2+c\right )}}{\sqrt {2} \sqrt {-i a} \sqrt {b+a \sinh \left (d x^2+c\right )}}\right )+i \left (\text {Li}_2\left (\frac {\left (i b+\sqrt {-a^2-b^2}\right ) \left (a+i b-i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )\right )}\right )-\text {Li}_2\left (\frac {\left (b+i \sqrt {-a^2-b^2}\right ) \left (i a-b+\sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i d x^2+2 i c+\pi \right )\right )\right )}\right )\right )\right )}{\sqrt {-a^2-b^2} d^2}\right ) \left (b+a \sinh \left (d x^2+c\right )\right )}{4 a \left (a+b \text {csch}\left (d x^2+c\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 505, normalized size = 2.24 \[ \frac {{\left (a^{2} + b^{2}\right )} d^{2} x^{4} - 2 \, a b c \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) + 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) + 2 \, a b c \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) - 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) - 2 \, a b \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a} + 1\right ) + 2 \, a b \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a} + 1\right ) - 2 \, {\left (a b d x^{2} + a b c\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a}\right ) + 2 \, {\left (a b d x^{2} + a b c\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - a}{a}\right )}{4 \, {\left (a^{3} + a b^{2}\right )} d^{2}} \]
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 \[ \int \frac {x^{3}}{b \operatorname {csch}\left (d x^{2} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )}\, dx \]
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 \[ \frac {x^{4}}{4 \, a} - 2 \, b \int \frac {x^{3} e^{\left (d x^{2} + c\right )}}{a^{2} e^{\left (2 \, d x^{2} + 2 \, c\right )} + 2 \, a b e^{\left (d x^{2} + c\right )} - a^{2}}\,{d x} \]
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 \[ \int \frac {x^3}{a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}} \,d x \]
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 \[ \int \frac {x^{3}}{a + b \operatorname {csch}{\left (c + d x^{2} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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