Optimal. Leaf size=55 \[ \frac{2}{3} i \text{EllipticF}\left (\frac{\pi }{2}+i x,-1\right )-2 i E\left (\left .i x+\frac{\pi }{2}\right |-1\right )+\frac{1}{3} \sinh (x) \cosh (x) \sqrt{\cosh ^2(x)+1} \]
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Rubi [A] time = 0.0612031, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3180, 3172, 3177, 3182} \[ \frac{2}{3} i F\left (\left .i x+\frac{\pi }{2}\right |-1\right )-2 i E\left (\left .i x+\frac{\pi }{2}\right |-1\right )+\frac{1}{3} \sinh (x) \cosh (x) \sqrt{\cosh ^2(x)+1} \]
Antiderivative was successfully verified.
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Rule 3180
Rule 3172
Rule 3177
Rule 3182
Rubi steps
\begin{align*} \int \left (1+\cosh ^2(x)\right )^{3/2} \, dx &=\frac{1}{3} \cosh (x) \sqrt{1+\cosh ^2(x)} \sinh (x)+\frac{1}{3} \int \frac{4+6 \cosh ^2(x)}{\sqrt{1+\cosh ^2(x)}} \, dx\\ &=\frac{1}{3} \cosh (x) \sqrt{1+\cosh ^2(x)} \sinh (x)-\frac{2}{3} \int \frac{1}{\sqrt{1+\cosh ^2(x)}} \, dx+2 \int \sqrt{1+\cosh ^2(x)} \, dx\\ &=-2 i E\left (\left .\frac{\pi }{2}+i x\right |-1\right )+\frac{2}{3} i F\left (\left .\frac{\pi }{2}+i x\right |-1\right )+\frac{1}{3} \cosh (x) \sqrt{1+\cosh ^2(x)} \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.049945, size = 51, normalized size = 0.93 \[ \frac{4 i \text{EllipticF}\left (i x,\frac{1}{2}\right )-24 i E\left (i x\left |\frac{1}{2}\right .\right )+\sinh (2 x) \sqrt{\cosh (2 x)+3}}{6 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.237, size = 99, normalized size = 1.8 \begin{align*} -{\frac{1}{3\,\sinh \left ( x \right ) }\sqrt{ \left ( 1+ \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}} \left ( - \left ( \cosh \left ( x \right ) \right ) ^{5}+10\,i\sqrt{1+ \left ( \cosh \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( i\cosh \left ( x \right ) ,i \right ) -6\,i\sqrt{1+ \left ( \cosh \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}{\it EllipticE} \left ( i\cosh \left ( x \right ) ,i \right ) +\cosh \left ( x \right ) \right ){\frac{1}{\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{4}-1}}}{\frac{1}{\sqrt{1+ \left ( \cosh \left ( x \right ) \right ) ^{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{\left (\cosh \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\cosh \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (\cosh \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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