Optimal. Leaf size=519 \[ -\frac {b \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b^2 \log \left (a \sinh \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2+b^2\right )}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d \sqrt {a^2+b^2}}+\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a d \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^2\right )+b\right )}+\frac {b^3 \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 d^2 \left (a^2+b^2\right )^{3/2}}-\frac {b^3 \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 d^2 \left (a^2+b^2\right )^{3/2}}+\frac {b^3 x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{2 a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^3 x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{2 a^2 d \left (a^2+b^2\right )^{3/2}}+\frac {x^4}{4 a^2} \]
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Rubi [A] time = 1.08, antiderivative size = 519, 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 = {5437, 4191, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac {b \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b^3 \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 d^2 \left (a^2+b^2\right )^{3/2}}+\frac {b \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}\right )}{a^2 d^2 \sqrt {a^2+b^2}}-\frac {b^3 \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}\right )}{2 a^2 d^2 \left (a^2+b^2\right )^{3/2}}+\frac {b^2 \log \left (a \sinh \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2+b^2\right )}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d \sqrt {a^2+b^2}}+\frac {b^3 x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{2 a^2 d \left (a^2+b^2\right )^{3/2}}+\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {b^3 x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{2 a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a d \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^2\right )+b\right )}+\frac {x^4}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3322
Rule 3324
Rule 4191
Rule 5437
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(a+b \text {csch}(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 \sinh (c+d x))^2}-\frac {2 b x}{a^2 (b+a \sinh (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 \sinh (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {x}{(b+a \sinh (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {x^4}{4 a^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}-\frac {(2 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^2}+\frac {b^3 \operatorname {Subst}\left (\int \frac {x}{b+a \sinh (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 {\cosh (c+d x)}{b+a \sinh (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 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}+\frac {b^3 \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^2 \left (a^2+b^2\right )}-\frac {(2 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 )}{a \sqrt {a^2+b^2}}+\frac {(2 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 )}{a \sqrt {a^2+b^2}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \sinh \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 {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \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 )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 \log \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}+\frac {b^3 \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 )}{a \left (a^2+b^2\right )^{3/2}}-\frac {b^3 \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 )}{a \left (a^2+b^2\right )^{3/2}}+\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 )}{a^2 \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 )}{a^2 \sqrt {a^2+b^2} d}\\ &=\frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 \log \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \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^{c+d x^2}\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^{c+d x^2}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^3 \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^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 \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^2 \left (a^2+b^2\right )^{3/2} d}\\ &=\frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 \log \left (b+a \sinh \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^{c+d x^2}}{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^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \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^{c+d x^2}\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^{c+d x^2}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d^2}\\ &=\frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 \log \left (b+a \sinh \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^{c+d x^2}}{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^{c+d x^2}}{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^{c+d x^2}}{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^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}\\ \end {align*}
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Mathematica [A] time = 6.44, size = 747, normalized size = 1.44 \[ \frac {\text {csch}^2\left (c+d x^2\right ) \left (a \sinh \left (c+d x^2\right )+b\right ) \left (-\frac {2 b \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^2\right )+b\right ) \left (-\sqrt {-a^2-b^2} \left (2 a^2+b^2\right ) \text {Li}_2\left (\frac {a e^{d x^2+c}}{\sqrt {a^2+b^2}-b}\right )+\sqrt {-a^2-b^2} \left (2 a^2+b^2\right ) \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )-b \sqrt {-\left (a^2+b^2\right )^2} \left (c+d x^2\right )-2 a^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )+2 a^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )-b^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )+b^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )+b \sqrt {-\left (a^2+b^2\right )^2} \log \left (a \left (e^{2 \left (c+d x^2\right )}-1\right )+2 b e^{c+d x^2}\right )-4 a^2 c \sqrt {a^2+b^2} \tan ^{-1}\left (\frac {b-a e^{-c-d x^2}}{\sqrt {-a^2-b^2}}\right )-2 b^2 c \sqrt {a^2+b^2} \tan ^{-1}\left (\frac {b-a e^{-c-d x^2}}{\sqrt {-a^2-b^2}}\right )+2 b^2 \sqrt {a^2+b^2} \tan ^{-1}\left (\frac {b-a e^{-c-d x^2}}{\sqrt {-a^2-b^2}}\right )+2 b^2 \sqrt {a^2+b^2} \tan ^{-1}\left (\frac {a e^{c+d x^2}+b}{\sqrt {-a^2-b^2}}\right )\right )}{\left (-\left (a^2+b^2\right )^2\right )^{3/2}}-\frac {2 a b^2 d x^2 \cosh \left (c+d x^2\right )}{a^2+b^2}+\left (d x^2-c\right ) \left (c+d x^2\right ) \left (a \sinh \left (c+d x^2\right )+b\right )\right )}{4 a^2 d^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.48, size = 2383, normalized size = 4.59 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{3}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -4 \, a^{2} b d \int \frac {x^{3} e^{\left (d x^{2} + c\right )}}{a^{5} d e^{\left (2 \, d x^{2} + 2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, d x^{2} + 2 \, c\right )} + 2 \, a^{4} b d e^{\left (d x^{2} + c\right )} + 2 \, a^{2} b^{3} d e^{\left (d x^{2} + c\right )} - a^{5} d - a^{3} b^{2} d}\,{d x} - 2 \, b^{3} d \int \frac {x^{3} e^{\left (d x^{2} + c\right )}}{a^{5} d e^{\left (2 \, d x^{2} + 2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, d x^{2} + 2 \, c\right )} + 2 \, a^{4} b d e^{\left (d x^{2} + c\right )} + 2 \, a^{2} b^{3} d e^{\left (d x^{2} + c\right )} - a^{5} d - a^{3} b^{2} d}\,{d x} + \frac {1}{2} \, a b^{2} {\left (\frac {b \log \left (\frac {a e^{\left (d x^{2} + c\right )} + b - \sqrt {a^{2} + b^{2}}}{a e^{\left (d x^{2} + c\right )} + b + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{5} + a^{3} b^{2}\right )} \sqrt {a^{2} + b^{2}} d^{2}} - \frac {2 \, {\left (d x^{2} + c\right )}}{{\left (a^{5} + a^{3} b^{2}\right )} d^{2}} + \frac {\log \left (a e^{\left (2 \, d x^{2} + 2 \, c\right )} + 2 \, b e^{\left (d x^{2} + c\right )} - a\right )}{{\left (a^{5} + a^{3} b^{2}\right )} d^{2}}\right )} - \frac {b^{3} \log \left (\frac {a e^{\left (d x^{2} + c\right )} + b - \sqrt {a^{2} + b^{2}}}{a e^{\left (d x^{2} + c\right )} + b + \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d^{2}} - \frac {{\left (a^{3} d e^{\left (2 \, c\right )} + a b^{2} d e^{\left (2 \, c\right )}\right )} x^{4} e^{\left (2 \, d x^{2}\right )} - 4 \, a b^{2} x^{2} - {\left (a^{3} d + a b^{2} d\right )} x^{4} + 2 \, {\left (2 \, b^{3} x^{2} e^{c} + {\left (a^{2} b d e^{c} + b^{3} d e^{c}\right )} x^{4}\right )} e^{\left (d x^{2}\right )}}{4 \, {\left (a^{5} d + a^{3} b^{2} d - {\left (a^{5} d e^{\left (2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{2}\right )} - 2 \, {\left (a^{4} b d e^{c} + a^{2} b^{3} d e^{c}\right )} e^{\left (d x^{2}\right )}\right )}} \]
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 \[ \int \frac {x^3}{{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2} \,d x \]
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 \[ \int \frac {x^{3}}{\left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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