Optimal. Leaf size=60 \[ \frac {b \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}+\frac {x^2}{2 a} \]
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Rubi [A] time = 0.10, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5437, 3783, 2660, 618, 204} \[ \frac {b \tanh ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}+\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 3783
Rule 5437
Rubi steps
\begin {align*} \int \frac {x}{a+b \text {csch}\left (c+d x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b \text {csch}(c+d x)} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 a}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\frac {a \sinh (c+d x)}{b}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {x^2}{2 a}+\frac {i \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 d}\\ &=\frac {x^2}{2 a}-\frac {(2 i) \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 d}\\ &=\frac {x^2}{2 a}+\frac {b \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 \sqrt {a^2+b^2} d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 71, normalized size = 1.18 \[ \frac {-\frac {2 b \tan ^{-1}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {-a^2-b^2}}\right )}{d \sqrt {-a^2-b^2}}+\frac {c}{d}+x^2}{2 a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 213, normalized size = 3.55 \[ \frac {{\left (a^{2} + b^{2}\right )} d x^{2} + \sqrt {a^{2} + b^{2}} b \log \left (\frac {a^{2} \cosh \left (d x^{2} + c\right )^{2} + a^{2} \sinh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x^{2} + c\right ) + a b\right )} \sinh \left (d x^{2} + c\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right ) + b\right )}}{a \cosh \left (d x^{2} + c\right )^{2} + a \sinh \left (d x^{2} + c\right )^{2} + 2 \, b \cosh \left (d x^{2} + c\right ) + 2 \, {\left (a \cosh \left (d x^{2} + c\right ) + b\right )} \sinh \left (d x^{2} + c\right ) - a}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 92, normalized size = 1.53 \[ -\frac {b \log \left (\frac {{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, \sqrt {a^{2} + b^{2}} a d} + \frac {d x^{2} + c}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 94, normalized size = 1.57 \[ -\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{2 d a}+\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+1\right )}{2 d a}-\frac {b \arctanh \left (\frac {2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d a \sqrt {a^{2}+b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 92, normalized size = 1.53 \[ -\frac {b \log \left (\frac {a e^{\left (-d x^{2} - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x^{2} - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{2 \, \sqrt {a^{2} + b^{2}} a d} + \frac {d x^{2} + c}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 175, normalized size = 2.92 \[ \frac {x^2}{2\,a}-\frac {\mathrm {atan}\left (\frac {a\,d\,\sqrt {b^2}}{\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}-\frac {b\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}{a^2\,d\,\sqrt {b^2}}+\frac {a^2\,b\,d\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {b^2}\,\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}}{a^6\,d^2+a^4\,b^2\,d^2}\right )\,\sqrt {b^2}}{\sqrt {-a^4\,d^2-a^2\,b^2\,d^2}} \]
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 \[ \int \frac {x}{a + b \operatorname {csch}{\left (c + d x^{2} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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