Optimal. Leaf size=50 \[ \frac{x}{b}-\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d} \]
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Rubi [A] time = 0.0817953, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3191, 391, 206, 208} \[ \frac{x}{b}-\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d} \]
Antiderivative was successfully verified.
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Rule 3191
Rule 391
Rule 206
Rule 208
Rubi steps
\begin{align*} \int \frac{\cosh ^2(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{b d}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{b d}\\ &=\frac{x}{b}-\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d}\\ \end{align*}
Mathematica [A] time = 0.0816948, size = 50, normalized size = 1. \[ \frac{-\frac{\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a}}+c+d x}{b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 577, normalized size = 11.5 \begin{align*} \text{result too large to display} \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 = 1.62478, size = 1088, normalized size = 21.76 \begin{align*} \left [\frac{2 \, d x + \sqrt{\frac{a - b}{a}} \log \left (\frac{b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} + 2 \,{\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \,{\left (b^{2} \cosh \left (d x + c\right )^{3} +{\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \,{\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + 2 \, a^{2} - a b\right )} \sqrt{\frac{a - b}{a}}}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \,{\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (b \cosh \left (d x + c\right )^{3} +{\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right )}{2 \, b d}, \frac{d x + \sqrt{-\frac{a - b}{a}} \arctan \left (-\frac{{\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sqrt{-\frac{a - b}{a}}}{2 \,{\left (a - b\right )}}\right )}{b d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28074, size = 93, normalized size = 1.86 \begin{align*} -\frac{{\left (a - b\right )} \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} b d} + \frac{d x + c}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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