Optimal. Leaf size=46 \[ \frac {2 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{3 b}-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2635, 2641} \[ \frac {2 \sinh (a+b x) \sqrt {\cosh (a+b x)}}{3 b}-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin {align*} \int \cosh ^{\frac {3}{2}}(a+b x) \, dx &=\frac {2 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx\\ &=-\frac {2 i F\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{3 b}+\frac {2 \sqrt {\cosh (a+b x)} \sinh (a+b x)}{3 b}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 81, normalized size = 1.76 \[ \frac {2 \sqrt {\sinh (2 (a+b x))+\cosh (2 (a+b x))+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right )+\sinh (2 (a+b x))}{3 b \sqrt {\cosh (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right )^{\frac {3}{2}}, x\right ) \]
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 \[ \int \cosh \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 174, normalized size = 3.78 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (4 \left (\cosh ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-6 \left (\cosh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticF \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]
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 \[ \int \cosh \left (b x + a\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\mathrm {cosh}\left (a+b\,x\right )}^{3/2} \,d x \]
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 \[ \int \cosh ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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