Optimal. Leaf size=55 \[ \frac {2}{3} i F\left (\left .i x+\frac {\pi }{2}\right |-1\right )-2 i E\left (\left .i x+\frac {\pi }{2}\right |-1\right )+\frac {1}{3} \sinh (x) \cosh (x) \sqrt {\cosh ^2(x)+1} \]
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Rubi [A] time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 3172
Rule 3177
Rule 3180
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*}
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Mathematica [A] time = 0.05, size = 51, normalized size = 0.93 \[ \frac {4 i F\left (i x\left |\frac {1}{2}\right .\right )-24 i E\left (i x\left |\frac {1}{2}\right .\right )+\sinh (2 x) \sqrt {\cosh (2 x)+3}}{6 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cosh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\cosh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 99, normalized size = 1.80 \[ -\frac {\sqrt {\left (1+\cosh ^{2}\relax (x )\right ) \left (\sinh ^{2}\relax (x )\right )}\, \left (-\left (\cosh ^{5}\relax (x )\right )+10 i \sqrt {1+\cosh ^{2}\relax (x )}\, \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \EllipticF \left (i \cosh \relax (x ), i\right )-6 i \sqrt {1+\cosh ^{2}\relax (x )}\, \sqrt {-\left (\sinh ^{2}\relax (x )\right )}\, \EllipticE \left (i \cosh \relax (x ), i\right )+\cosh \relax (x )\right )}{3 \sqrt {\cosh ^{4}\relax (x )-1}\, \sinh \relax (x ) \sqrt {1+\cosh ^{2}\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\cosh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left ({\mathrm {cosh}\relax (x)}^2+1\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\cosh ^{2}{\relax (x )} + 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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