Optimal. Leaf size=123 \[ \frac {1}{3} b \sinh (x) \cosh (x) \sqrt {a+b \sinh ^2(x)}+\frac {i a (a-b) \sqrt {\frac {b \sinh ^2(x)}{a}+1} F\left (i x\left |\frac {b}{a}\right .\right )}{3 \sqrt {a+b \sinh ^2(x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(x)} E\left (i x\left |\frac {b}{a}\right .\right )}{3 \sqrt {\frac {b \sinh ^2(x)}{a}+1}} \]
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Rubi [A] time = 0.16, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ \frac {1}{3} b \sinh (x) \cosh (x) \sqrt {a+b \sinh ^2(x)}+\frac {i a (a-b) \sqrt {\frac {b \sinh ^2(x)}{a}+1} F\left (i x\left |\frac {b}{a}\right .\right )}{3 \sqrt {a+b \sinh ^2(x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(x)} E\left (i x\left |\frac {b}{a}\right .\right )}{3 \sqrt {\frac {b \sinh ^2(x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3172
Rule 3177
Rule 3178
Rule 3180
Rule 3182
Rule 3183
Rubi steps
\begin {align*} \int \left (a+b \sinh ^2(x)\right )^{3/2} \, dx &=\frac {1}{3} b \cosh (x) \sinh (x) \sqrt {a+b \sinh ^2(x)}+\frac {1}{3} \int \frac {a (3 a-b)+2 (2 a-b) b \sinh ^2(x)}{\sqrt {a+b \sinh ^2(x)}} \, dx\\ &=\frac {1}{3} b \cosh (x) \sinh (x) \sqrt {a+b \sinh ^2(x)}-\frac {1}{3} (a (a-b)) \int \frac {1}{\sqrt {a+b \sinh ^2(x)}} \, dx+\frac {1}{3} (2 (2 a-b)) \int \sqrt {a+b \sinh ^2(x)} \, dx\\ &=\frac {1}{3} b \cosh (x) \sinh (x) \sqrt {a+b \sinh ^2(x)}+\frac {\left (2 (2 a-b) \sqrt {a+b \sinh ^2(x)}\right ) \int \sqrt {1+\frac {b \sinh ^2(x)}{a}} \, dx}{3 \sqrt {1+\frac {b \sinh ^2(x)}{a}}}-\frac {\left (a (a-b) \sqrt {1+\frac {b \sinh ^2(x)}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b \sinh ^2(x)}{a}}} \, dx}{3 \sqrt {a+b \sinh ^2(x)}}\\ &=\frac {1}{3} b \cosh (x) \sinh (x) \sqrt {a+b \sinh ^2(x)}-\frac {2 i (2 a-b) E\left (i x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(x)}}{3 \sqrt {1+\frac {b \sinh ^2(x)}{a}}}+\frac {i a (a-b) F\left (i x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(x)}{a}}}{3 \sqrt {a+b \sinh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 132, normalized size = 1.07 \[ \frac {\sqrt {2} b \sinh (2 x) (2 a+b \cosh (2 x)-b)+4 i a (a-b) \sqrt {\frac {2 a+b \cosh (2 x)-b}{a}} F\left (i x\left |\frac {b}{a}\right .\right )-8 i a (2 a-b) \sqrt {\frac {2 a+b \cosh (2 x)-b}{a}} E\left (i x\left |\frac {b}{a}\right .\right )}{12 \sqrt {2 a+b \cosh (2 x)-b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sinh \relax (x)^{2} + a\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (b \sinh \relax (x)^{2} + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 329, normalized size = 2.67 \[ \frac {\sqrt {-\frac {b}{a}}\, b^{2} \sinh \relax (x ) \left (\cosh ^{4}\relax (x )\right )+\left (\sqrt {-\frac {b}{a}}\, a b -\sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{2}\relax (x )\right ) \sinh \relax (x )+3 a^{2} \sqrt {\frac {b \left (\cosh ^{2}\relax (x )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \relax (x ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-5 a b \sqrt {\frac {b \left (\cosh ^{2}\relax (x )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \relax (x ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 \sqrt {\frac {b \left (\cosh ^{2}\relax (x )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \relax (x ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+4 a b \sqrt {\frac {b \left (\cosh ^{2}\relax (x )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \relax (x ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 \sqrt {\frac {b \left (\cosh ^{2}\relax (x )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \relax (x ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}}{3 \sqrt {-\frac {b}{a}}\, \cosh \relax (x ) \sqrt {a +b \left (\sinh ^{2}\relax (x )\right )}} \]
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 \[ \int {\left (b \sinh \relax (x)^{2} + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,{\mathrm {sinh}\relax (x)}^2+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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