Integrand size = 18, antiderivative size = 64 \[ \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx=\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{9 b^2}-\frac {4 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{9 b^2} \]
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Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5481, 2715, 2720} \[ \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx=\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{9 b^2}-\frac {4 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{9 b^2}+\frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b} \]
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Rule 2715
Rule 2720
Rule 5481
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {2 \int \cosh ^{\frac {3}{2}}(a+b x) \, dx}{3 b} \\ & = \frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {4 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{9 b^2}-\frac {2 \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx}{9 b} \\ & = \frac {2 x \cosh ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{9 b^2}-\frac {4 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{9 b^2} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx=\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )+2 \sqrt {\cosh (a+b x)} (3 b x \cosh (a+b x)-2 \sinh (a+b x))}{9 b^2} \]
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\[\int x \sinh \left (b x +a \right ) \sqrt {\cosh \left (b x +a \right )}d x\]
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Exception generated. \[ \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx=\int x \sinh {\left (a + b x \right )} \sqrt {\cosh {\left (a + b x \right )}}\, dx \]
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\[ \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx=\int { x \sqrt {\cosh \left (b x + a\right )} \sinh \left (b x + a\right ) \,d x } \]
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\[ \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx=\int { x \sqrt {\cosh \left (b x + a\right )} \sinh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \sqrt {\cosh (a+b x)} \sinh (a+b x) \, dx=\int x\,\sqrt {\mathrm {cosh}\left (a+b\,x\right )}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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