Optimal. Leaf size=113 \[ \frac {b \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^2 \coth \left (c+d x^2\right )}{2 a d \left (a^2+b^2\right ) \left (a+b \text {csch}\left (c+d x^2\right )\right )}+\frac {x^2}{2 a^2} \]
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Rubi [A] time = 0.23, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5437, 3785, 3919, 3831, 2660, 618, 204} \[ \frac {b \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^2 \coth \left (c+d x^2\right )}{2 a d \left (a^2+b^2\right ) \left (a+b \text {csch}\left (c+d x^2\right )\right )}+\frac {x^2}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 3785
Rule 3831
Rule 3919
Rule 5437
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx,x,x^2\right )\\ &=-\frac {b^2 \coth \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d x^2\right )\right )}-\frac {\operatorname {Subst}\left (\int \frac {-a^2-b^2+a b \text {csch}(c+d x)}{a+b \text {csch}(c+d x)} \, dx,x,x^2\right )}{2 a \left (a^2+b^2\right )}\\ &=\frac {x^2}{2 a^2}-\frac {b^2 \coth \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d x^2\right )\right )}-\frac {\left (b \left (2 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {\text {csch}(c+d x)}{a+b \text {csch}(c+d x)} \, dx,x,x^2\right )}{2 a^2 \left (a^2+b^2\right )}\\ &=\frac {x^2}{2 a^2}-\frac {b^2 \coth \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d x^2\right )\right )}-\frac {\left (2 a^2+b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a \sinh (c+d x)}{b}} \, dx,x,x^2\right )}{2 a^2 \left (a^2+b^2\right )}\\ &=\frac {x^2}{2 a^2}-\frac {b^2 \coth \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d x^2\right )\right )}+\frac {\left (i \left (2 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {2 i a x}{b}+x^2} \, dx,x,i \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a^2 \left (a^2+b^2\right ) d}\\ &=\frac {x^2}{2 a^2}-\frac {b^2 \coth \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d x^2\right )\right )}-\frac {\left (2 i \left (2 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,-\frac {2 i a}{b}+2 i \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{a^2 \left (a^2+b^2\right ) d}\\ &=\frac {x^2}{2 a^2}+\frac {b \left (2 a^2+b^2\right ) \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^2 \coth \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d x^2\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 161, normalized size = 1.42 \[ \frac {\text {csch}\left (c+d x^2\right ) \left (a \sinh \left (c+d x^2\right )+b\right ) \left (-\frac {a b^2 \coth \left (c+d x^2\right )}{a^2+b^2}+\frac {2 b \left (2 a^2+b^2\right ) \left (a+b \text {csch}\left (c+d x^2\right )\right ) \tan ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\left (c+d x^2\right ) \left (a+b \text {csch}\left (c+d x^2\right )\right )\right )}{2 a^2 d \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 711, normalized size = 6.29 \[ \frac {{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \cosh \left (d x^{2} + c\right )^{2} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sinh \left (d x^{2} + c\right )^{2} - 2 \, a^{3} b^{2} - 2 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} - {\left (2 \, a^{3} b + a b^{3} - {\left (2 \, a^{3} b + a b^{3}\right )} \cosh \left (d x^{2} + c\right )^{2} - {\left (2 \, a^{3} b + a b^{3}\right )} \sinh \left (d x^{2} + c\right )^{2} - 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (d x^{2} + c\right ) - 2 \, {\left (2 \, a^{2} b^{2} + b^{4} + {\left (2 \, a^{3} b + a b^{3}\right )} \cosh \left (d x^{2} + c\right )\right )} \sinh \left (d x^{2} + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (d x^{2} + c\right )^{2} + a^{2} \sinh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x^{2} + c\right ) + a b\right )} \sinh \left (d x^{2} + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right ) + b\right )}}{a \cosh \left (d x^{2} + c\right )^{2} + a \sinh \left (d x^{2} + c\right )^{2} + 2 \, b \cosh \left (d x^{2} + c\right ) + 2 \, {\left (a \cosh \left (d x^{2} + c\right ) + b\right )} \sinh \left (d x^{2} + c\right ) - a}\right ) + 2 \, {\left (a^{2} b^{3} + b^{5} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2}\right )} \cosh \left (d x^{2} + c\right ) + 2 \, {\left (a^{2} b^{3} + b^{5} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \cosh \left (d x^{2} + c\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2}\right )} \sinh \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x^{2} + c\right )^{2} + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d x^{2} + c\right )^{2} + 2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d x^{2} + c\right ) - {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d + 2 \, {\left ({\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x^{2} + c\right ) + {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d x^{2} + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 177, normalized size = 1.57 \[ -\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{4} d + a^{2} b^{2} d\right )} \sqrt {a^{2} + b^{2}}} + \frac {b^{3} e^{\left (d x^{2} + c\right )} - a b^{2}}{{\left (a^{4} d + a^{2} b^{2} d\right )} {\left (a e^{\left (2 \, d x^{2} + 2 \, c\right )} + 2 \, b e^{\left (d x^{2} + c\right )} - a\right )}} + \frac {d x^{2} + c}{2 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 255, normalized size = 2.26 \[ -\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{2}}+\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{2}}+\frac {b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{2}\left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-b \right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2}}{d a \left (\left (\tanh ^{2}\left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right ) b -2 a \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-b \right ) \left (a^{2}+b^{2}\right )}-\frac {2 b \arctanh \left (\frac {2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {b^{3} \arctanh \left (\frac {2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 200, normalized size = 1.77 \[ -\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {a e^{\left (-d x^{2} - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x^{2} - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {b^{3} e^{\left (-d x^{2} - c\right )} + a b^{2}}{{\left (a^{5} + a^{3} b^{2} + 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} e^{\left (-d x^{2} - c\right )} - {\left (a^{5} + a^{3} b^{2}\right )} e^{\left (-2 \, d x^{2} - 2 \, c\right )}\right )} d} + \frac {d x^{2} + c}{2 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.97, size = 290, normalized size = 2.57 \[ \frac {x^2}{2\,a^2}-\frac {\frac {b^2}{d\,\left (a^3+a\,b^2\right )}-\frac {b^3\,{\mathrm {e}}^{d\,x^2+c}}{a\,d\,\left (a^3+a\,b^2\right )}}{2\,b\,{\mathrm {e}}^{d\,x^2+c}-a+a\,{\mathrm {e}}^{2\,d\,x^2+2\,c}}-\frac {b\,\ln \left (\frac {2\,b\,x\,{\mathrm {e}}^{d\,x^2+c}\,\left (2\,a^2+b^2\right )}{a^3\,\left (a^2+b^2\right )}-\frac {2\,b\,x\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{d\,x^2+c}\right )}{a^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{2\,a^2\,d\,{\left (a^2+b^2\right )}^{3/2}}+\frac {b\,\ln \left (\frac {2\,b\,x\,{\mathrm {e}}^{d\,x^2+c}\,\left (2\,a^2+b^2\right )}{a^3\,\left (a^2+b^2\right )}+\frac {2\,b\,x\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{d\,x^2+c}\right )}{a^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{2\,a^2\,d\,{\left (a^2+b^2\right )}^{3/2}} \]
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 \[ \int \frac {x}{\left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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