Optimal. Leaf size=43 \[ \frac{2 \sqrt{\sinh (c+d x)}}{b d}-\frac{2 a \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2 d} \]
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Rubi [A] time = 0.050232, antiderivative size = 43, 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} \[ \frac{2 \sqrt{\sinh (c+d x)}}{b d}-\frac{2 a \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 3223
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{a+b \sqrt{\sinh (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \sqrt{x}} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,\sqrt{\sinh (c+d x)}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,\sqrt{\sinh (c+d x)}\right )}{d}\\ &=-\frac{2 a \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2 d}+\frac{2 \sqrt{\sinh (c+d x)}}{b d}\\ \end{align*}
Mathematica [A] time = 0.0274741, size = 41, normalized size = 0.95 \[ \frac{2 \left (\frac{\sqrt{\sinh (c+d x)}}{b}-\frac{a \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 89, normalized size = 2.1 \begin{align*} 2\,{\frac{\sqrt{\sinh \left ( dx+c \right ) }}{bd}}+{\frac{a}{d{b}^{2}}\ln \left ( b\sqrt{\sinh \left ( dx+c \right ) }-a \right ) }-{\frac{a}{d{b}^{2}}\ln \left ( a+b\sqrt{\sinh \left ( dx+c \right ) } \right ) }-{\frac{a\ln \left ({b}^{2}\sinh \left ( dx+c \right ) -{a}^{2} \right ) }{d{b}^{2}}} \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 )}{b \sqrt{\sinh \left (d x + c\right )} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 6.3547, size = 563, normalized size = 13.09 \begin{align*} \frac{a d x + a \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 ) - a \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 \, b \sqrt{\sinh \left (d x + c\right )}}{b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.90048, size = 68, normalized size = 1.58 \begin{align*} \begin{cases} \frac{x \cosh{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sinh{\left (c + d x \right )}}{a d} & \text{for}\: b = 0 \\\frac{x \cosh{\left (c \right )}}{a + b \sqrt{\sinh{\left (c \right )}}} & \text{for}\: d = 0 \\- \frac{2 a \log{\left (\frac{a}{b} + \sqrt{\sinh{\left (c + d x \right )}} \right )}}{b^{2} d} + \frac{2 \sqrt{\sinh{\left (c + d x \right )}}}{b d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23697, size = 92, normalized size = 2.14 \begin{align*} -\frac{2 \, a \log \left ({\left | b \sqrt{\frac{1}{2} \, e^{\left (d x + c\right )} - \frac{1}{2} \, e^{\left (-d x - c\right )}} + a \right |}\right )}{b^{2} d} + \frac{2 \, \sqrt{\frac{1}{2} \, e^{\left (d x + c\right )} - \frac{1}{2} \, e^{\left (-d x - c\right )}}}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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