Optimal. Leaf size=26 \[ \frac {a x^2}{2}-\frac {b \tanh ^{-1}\left (\cosh \left (c+d x^2\right )\right )}{2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 5437, 3770} \[ \frac {a x^2}{2}-\frac {b \tanh ^{-1}\left (\cosh \left (c+d x^2\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3770
Rule 5437
Rubi steps
\begin {align*} \int x \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx &=\int \left (a x+b x \text {csch}\left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^2}{2}+b \int x \text {csch}\left (c+d x^2\right ) \, dx\\ &=\frac {a x^2}{2}+\frac {1}{2} b \operatorname {Subst}\left (\int \text {csch}(c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}-\frac {b \tanh ^{-1}\left (\cosh \left (c+d x^2\right )\right )}{2 d}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 57, normalized size = 2.19 \[ \frac {a x^2}{2}+\frac {b \log \left (\sinh \left (\frac {c}{2}+\frac {d x^2}{2}\right )\right )}{2 d}-\frac {b \log \left (\cosh \left (\frac {c}{2}+\frac {d x^2}{2}\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 55, normalized size = 2.12 \[ \frac {a d x^{2} - b \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) + b \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 49, normalized size = 1.88 \[ \frac {{\left (d x^{2} + c\right )} a}{2 \, d} - \frac {b \log \left (e^{\left (d x^{2} + c\right )} + 1\right )}{2 \, d} + \frac {b \log \left ({\left | e^{\left (d x^{2} + c\right )} - 1 \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 33, normalized size = 1.27 \[ \frac {a \,x^{2}}{2}+\frac {b \ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a c}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 25, normalized size = 0.96 \[ \frac {1}{2} \, a x^{2} + \frac {b \log \left (\tanh \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 47, normalized size = 1.81 \[ \frac {a\,x^2}{2}-\frac {\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \]
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 \[ \int x \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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