Optimal. Leaf size=49 \[ \frac{2 a}{b^2 d \left (a+b \sqrt{\sinh (c+d x)}\right )}+\frac{2 \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2 d} \]
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Rubi [A] time = 0.0545739, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3223, 190, 43} \[ \frac{2 a}{b^2 d \left (a+b \sqrt{\sinh (c+d x)}\right )}+\frac{2 \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 3223
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{\left (a+b \sqrt{\sinh (c+d x)}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \sqrt{x}\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x}{(a+b x)^2} \, dx,x,\sqrt{\sinh (c+d x)}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx,x,\sqrt{\sinh (c+d x)}\right )}{d}\\ &=\frac{2 \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2 d}+\frac{2 a}{b^2 d \left (a+b \sqrt{\sinh (c+d x)}\right )}\\ \end{align*}
Mathematica [A] time = 0.0609125, size = 42, normalized size = 0.86 \[ \frac{2 \left (\frac{a}{a+b \sqrt{\sinh (c+d x)}}+\log \left (a+b \sqrt{\sinh (c+d x)}\right )\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 144, normalized size = 2.9 \begin{align*} -2\,{\frac{{a}^{2}}{d \left ({b}^{2}\sinh \left ( dx+c \right ) -{a}^{2} \right ){b}^{2}}}+{\frac{\ln \left ({b}^{2}\sinh \left ( dx+c \right ) -{a}^{2} \right ) }{d{b}^{2}}}+{\frac{a}{d{b}^{2}} \left ( b\sqrt{\sinh \left ( dx+c \right ) }-a \right ) ^{-1}}-{\frac{1}{d{b}^{2}}\ln \left ( b\sqrt{\sinh \left ( dx+c \right ) }-a \right ) }+{\frac{a}{d{b}^{2}} \left ( a+b\sqrt{\sinh \left ( dx+c \right ) } \right ) ^{-1}}+{\frac{1}{d{b}^{2}}\ln \left ( a+b\sqrt{\sinh \left ( dx+c \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{{\left (b \sqrt{\sinh \left (d x + c\right )} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77643, size = 1370, normalized size = 27.96 \begin{align*} \frac{b^{2} d x + b^{2} c -{\left (b^{2} d x + b^{2} c\right )} \cosh \left (d x + c\right )^{2} -{\left (b^{2} d x + b^{2} c\right )} \sinh \left (d x + c\right )^{2} + 2 \,{\left (a^{2} d x + a^{2} c - 2 \, a^{2}\right )} \cosh \left (d x + c\right ) +{\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac{b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right ) + 4 \,{\left (a b \cosh \left (d x + c\right ) + a b \sinh \left (d x + c\right )\right )} \sqrt{\sinh \left (d x + c\right )}}{b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )}\right ) +{\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac{2 \,{\left (b^{2} \sinh \left (d x + c\right ) - a^{2}\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \,{\left (a^{2} d x + a^{2} c - 2 \, a^{2} -{\left (b^{2} d x + b^{2} c\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \,{\left (a b \cosh \left (d x + c\right ) + a b \sinh \left (d x + c\right )\right )} \sqrt{\sinh \left (d x + c\right )}}{b^{4} d \cosh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{2} - 2 \, a^{2} b^{2} d \cosh \left (d x + c\right ) - b^{4} d + 2 \,{\left (b^{4} d \cosh \left (d x + c\right ) - a^{2} b^{2} d\right )} \sinh \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.92605, size = 151, normalized size = 3.08 \begin{align*} \begin{cases} \frac{x \cosh{\left (c \right )}}{a^{2}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sinh{\left (c + d x \right )}}{a^{2} d} & \text{for}\: b = 0 \\\frac{x \cosh{\left (c \right )}}{\left (a + b \sqrt{\sinh{\left (c \right )}}\right )^{2}} & \text{for}\: d = 0 \\\frac{2 a \log{\left (\frac{a}{b} + \sqrt{\sinh{\left (c + d x \right )}} \right )}}{a b^{2} d + b^{3} d \sqrt{\sinh{\left (c + d x \right )}}} + \frac{2 a}{a b^{2} d + b^{3} d \sqrt{\sinh{\left (c + d x \right )}}} + \frac{2 b \log{\left (\frac{a}{b} + \sqrt{\sinh{\left (c + d x \right )}} \right )} \sqrt{\sinh{\left (c + d x \right )}}}{a b^{2} d + b^{3} d \sqrt{\sinh{\left (c + d x \right )}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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