Optimal. Leaf size=66 \[ \frac{2 x \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{f}-\frac{4 \sqrt{a+i a \sinh (e+f x)}}{f^2} \]
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Rubi [A] time = 0.0753615, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3319, 3296, 2638} \[ \frac{2 x \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{f}-\frac{4 \sqrt{a+i a \sinh (e+f x)}}{f^2} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x \sqrt{a+i a \sinh (e+f x)} \, dx &=\left (\text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int x \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx\\ &=\frac{2 x \sqrt{a+i a \sinh (e+f x)} \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{f}-\frac{\left (2 \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \cosh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{f}\\ &=-\frac{4 \sqrt{a+i a \sinh (e+f x)}}{f^2}+\frac{2 x \sqrt{a+i a \sinh (e+f x)} \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.16385, size = 87, normalized size = 1.32 \[ \frac{2 \sqrt{a+i a \sinh (e+f x)} \left ((f x-2 i) \sinh \left (\frac{1}{2} (e+f x)\right )+(-2+i f x) \cosh \left (\frac{1}{2} (e+f x)\right )\right )}{f^2 \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 105, normalized size = 1.6 \begin{align*}{\frac{i\sqrt{2} \left ( ixf+fx{{\rm e}^{fx+e}}+2\,i-2\,{{\rm e}^{fx+e}} \right ) \left ({{\rm e}^{fx+e}}-i \right ) }{ \left ( i{{\rm e}^{2\,fx+2\,e}}-i+2\,{{\rm e}^{fx+e}} \right ){f}^{2}}\sqrt{a \left ( i{{\rm e}^{2\,fx+2\,e}}-i+2\,{{\rm e}^{fx+e}} \right ){{\rm e}^{-fx-e}}}} \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 \sqrt{i \, a \sinh \left (f x + e\right ) + a} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{a \left (i \sinh{\left (e + f x \right )} + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{i \, a \sinh \left (f x + e\right ) + a} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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