Optimal. Leaf size=46 \[ \frac{x}{a+b}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right )}{\sqrt{a} d (a+b)} \]
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Rubi [A] time = 0.0774405, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3660, 3675, 205} \[ \frac{x}{a+b}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right )}{\sqrt{a} d (a+b)} \]
Antiderivative was successfully verified.
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Rule 3660
Rule 3675
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a+b \coth ^2(c+d x)} \, dx &=\frac{x}{a+b}-\frac{b \int \frac{\text{csch}^2(c+d x)}{a+b \coth ^2(c+d x)} \, dx}{a+b}\\ &=\frac{x}{a+b}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\coth (c+d x)\right )}{(a+b) d}\\ &=\frac{x}{a+b}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right )}{\sqrt{a} (a+b) d}\\ \end{align*}
Mathematica [A] time = 0.0575124, size = 47, normalized size = 1.02 \[ \frac{\tanh ^{-1}(\tanh (c+d x))-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{b}}\right )}{\sqrt{a}}}{d (a+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 76, normalized size = 1.7 \begin{align*}{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ) }{d \left ( 2\,b+2\,a \right ) }}+{\frac{b}{d \left ( a+b \right ) }\arctan \left ({b{\rm coth} \left (dx+c\right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) }{d \left ( 2\,b+2\,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 [B] time = 3.2201, size = 1249, normalized size = 27.15 \begin{align*} \left [\frac{2 \, d x + \sqrt{-\frac{b}{a}} \log \left (\frac{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \,{\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} - 6 \, a b + b^{2} + 4 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} -{\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \,{\left ({\left (a^{2} + a b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (a^{2} + a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - a^{2} + a b\right )} \sqrt{-\frac{b}{a}}}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \,{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 2 \,{\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (a + b\right )} \cosh \left (d x + c\right )^{2} - a + b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} -{\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right )}{2 \,{\left (a + b\right )} d}, \frac{d x - \sqrt{\frac{b}{a}} \arctan \left (\frac{{\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) +{\left (a + b\right )} \sinh \left (d x + c\right )^{2} - a + b\right )} \sqrt{\frac{b}{a}}}{2 \, b}\right )}{{\left (a + b\right )} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 27.7931, size = 294, normalized size = 6.39 \begin{align*} \begin{cases} \frac{\tilde{\infty } x}{\coth ^{2}{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{x - \frac{\tanh{\left (c + d x \right )}}{d}}{b} & \text{for}\: a = 0 \\- \frac{d x \tanh ^{2}{\left (c + d x \right )}}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} + \frac{d x}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} - \frac{\tanh{\left (c + d x \right )}}{2 b d \tanh ^{2}{\left (c + d x \right )} - 2 b d} & \text{for}\: a = - b \\\frac{x}{a} & \text{for}\: b = 0 \\\frac{x}{a + b \coth ^{2}{\left (c \right )}} & \text{for}\: d = 0 \\\frac{2 i a \sqrt{b} d x \sqrt{\frac{1}{a}}}{2 i a^{2} \sqrt{b} d \sqrt{\frac{1}{a}} + 2 i a b^{\frac{3}{2}} d \sqrt{\frac{1}{a}}} - \frac{b \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \tanh{\left (c + d x \right )} \right )}}{2 i a^{2} \sqrt{b} d \sqrt{\frac{1}{a}} + 2 i a b^{\frac{3}{2}} d \sqrt{\frac{1}{a}}} + \frac{b \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \tanh{\left (c + d x \right )} \right )}}{2 i a^{2} \sqrt{b} d \sqrt{\frac{1}{a}} + 2 i a b^{\frac{3}{2}} d \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14612, size = 92, normalized size = 2. \begin{align*} -\frac{b \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b}{2 \, \sqrt{a b}}\right )}{\sqrt{a b}{\left (a d + b d\right )}} + \frac{d x + c}{a d + b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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