Optimal. Leaf size=39 \[ \frac{x}{a+b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (x)}{\sqrt{a}}\right )}{\sqrt{b} (a+b)} \]
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Rubi [A] time = 0.151323, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {481, 207, 205} \[ \frac{x}{a+b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (x)}{\sqrt{a}}\right )}{\sqrt{b} (a+b)} \]
Antiderivative was successfully verified.
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Rule 481
Rule 207
Rule 205
Rubi steps
\begin{align*} \int \frac{\sinh ^2(x)}{a \cosh ^2(x)+b \sinh ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2}{\left (-1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (x)\right )}{a+b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (x)\right )}{a+b}\\ &=\frac{x}{a+b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (x)}{\sqrt{a}}\right )}{\sqrt{b} (a+b)}\\ \end{align*}
Mathematica [A] time = 0.0946278, size = 34, normalized size = 0.87 \[ \frac{x-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (x)}{\sqrt{a}}\right )}{\sqrt{b}}}{a+b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 414, normalized size = 10.6 \begin{align*} 8\,{\frac{\ln \left ( \tanh \left ( x/2 \right ) +1 \right ) }{8\,a+8\,b}}-8\,{\frac{\ln \left ( \tanh \left ( x/2 \right ) -1 \right ) }{8\,a+8\,b}}+4\,{\frac{{a}^{2}}{ \left ( 4\,a+4\,b \right ) \sqrt{b \left ( a+b \right ) }\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}\arctan \left ({\frac{a\tanh \left ( x/2 \right ) }{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}} \right ) }+4\,{\frac{a}{ \left ( 4\,a+4\,b \right ) \sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}\arctan \left ({\frac{a\tanh \left ( x/2 \right ) }{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}} \right ) }+4\,{\frac{ab}{ \left ( 4\,a+4\,b \right ) \sqrt{b \left ( a+b \right ) }\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}\arctan \left ({\frac{a\tanh \left ( x/2 \right ) }{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}} \right ) }+4\,{\frac{{a}^{2}}{ \left ( 4\,a+4\,b \right ) \sqrt{b \left ( a+b \right ) }\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}{\it Artanh} \left ({\frac{a\tanh \left ( x/2 \right ) }{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}} \right ) }-4\,{\frac{a}{ \left ( 4\,a+4\,b \right ) \sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}{\it Artanh} \left ({\frac{a\tanh \left ( x/2 \right ) }{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}} \right ) }+4\,{\frac{ab}{ \left ( 4\,a+4\,b \right ) \sqrt{b \left ( a+b \right ) }\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}{\it Artanh} \left ({\frac{a\tanh \left ( x/2 \right ) }{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}} \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 [A] time = 1.91539, size = 1007, normalized size = 25.82 \begin{align*} \left [\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} +{\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} + 2 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - 6 \, a b + b^{2} + 4 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} +{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4 \,{\left ({\left (a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a b + b^{2}\right )} \sinh \left (x\right )^{2} + a b - b^{2}\right )} \sqrt{-\frac{a}{b}}}{{\left (a + b\right )} \cosh \left (x\right )^{4} + 4 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} +{\left (a + b\right )} \sinh \left (x\right )^{4} + 2 \,{\left (a - b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \,{\left (a + b\right )} \cosh \left (x\right )^{2} + a - b\right )} \sinh \left (x\right )^{2} + 4 \,{\left ({\left (a + b\right )} \cosh \left (x\right )^{3} +{\left (a - b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + a + b}\right ) + 2 \, x}{2 \,{\left (a + b\right )}}, -\frac{\sqrt{\frac{a}{b}} \arctan \left (\frac{{\left ({\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right )^{2} + a - b\right )} \sqrt{\frac{a}{b}}}{2 \, a}\right ) - x}{a + b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.82976, size = 258, normalized size = 6.62 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: a = 0 \wedge b = 0 \\\frac{x - \frac{\sinh{\left (x \right )}}{\cosh{\left (x \right )}}}{a} & \text{for}\: b = 0 \\\frac{x}{b} & \text{for}\: a = 0 \\- \frac{x \sinh ^{2}{\left (x \right )}}{- 2 b \sinh ^{2}{\left (x \right )} + 2 b \cosh ^{2}{\left (x \right )}} + \frac{x \cosh ^{2}{\left (x \right )}}{- 2 b \sinh ^{2}{\left (x \right )} + 2 b \cosh ^{2}{\left (x \right )}} - \frac{\sinh{\left (x \right )} \cosh{\left (x \right )}}{- 2 b \sinh ^{2}{\left (x \right )} + 2 b \cosh ^{2}{\left (x \right )}} & \text{for}\: a = - b \\\frac{2 i \sqrt{a} b x \sqrt{\frac{1}{b}}}{2 i a^{\frac{3}{2}} b \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{2} \sqrt{\frac{1}{b}}} - \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} \cosh{\left (x \right )} + \sinh{\left (x \right )} \right )}}{2 i a^{\frac{3}{2}} b \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{2} \sqrt{\frac{1}{b}}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} \cosh{\left (x \right )} + \sinh{\left (x \right )} \right )}}{2 i a^{\frac{3}{2}} b \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{2} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15839, size = 62, normalized size = 1.59 \begin{align*} -\frac{a \arctan \left (\frac{a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b}{2 \, \sqrt{a b}}\right )}{\sqrt{a b}{\left (a + b\right )}} + \frac{x}{a + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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