Optimal. Leaf size=54 \[ \frac{2 b \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a d \sqrt{a^2+b^2}}+\frac{x}{a} \]
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Rubi [A] time = 0.0594489, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3783, 2660, 618, 204} \[ \frac{2 b \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a d \sqrt{a^2+b^2}}+\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 3783
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{a+b \text{csch}(c+d x)} \, dx &=\frac{x}{a}-\frac{\int \frac{1}{1+\frac{a \sinh (c+d x)}{b}} \, dx}{a}\\ &=\frac{x}{a}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 i a x}{b}+x^2} \, dx,x,\tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{a d}\\ &=\frac{x}{a}-\frac{(4 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 \tan \left (\frac{1}{2} (i c+i d x)\right )\right )}{a d}\\ &=\frac{x}{a}+\frac{2 b \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}-\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}\\ \end{align*}
Mathematica [A] time = 0.103844, size = 64, normalized size = 1.19 \[ \frac{-\frac{2 b \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a^2-b^2}}\right )}{d \sqrt{-a^2-b^2}}+\frac{c}{d}+x}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 87, normalized size = 1.6 \begin{align*}{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-2\,{\frac{b}{da\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,b\tanh \left ( 1/2\,dx+c/2 \right ) -2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }-{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60734, size = 473, normalized size = 8.76 \begin{align*} \frac{{\left (a^{2} + b^{2}\right )} d x + \sqrt{a^{2} + b^{2}} b \log \left (\frac{a^{2} \cosh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + a^{2} + 2 \, b^{2} + 2 \,{\left (a^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) + 2 \, \sqrt{a^{2} + b^{2}}{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + b\right )}}{a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) + 2 \,{\left (a \cosh \left (d x + c\right ) + b\right )} \sinh \left (d x + c\right ) - a}\right )}{{\left (a^{3} + a b^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \operatorname{csch}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17666, size = 115, normalized size = 2.13 \begin{align*} -\frac{b \log \left (\frac{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} a d} + \frac{d x + c}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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