Optimal. Leaf size=174 \[ \frac {b \sinh (e+f x) \cosh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {i a (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} F\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.19, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ \frac {b \sinh (e+f x) \cosh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {i a (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} F\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 f \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}} \]
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(e+f x)\right )^{3/2} \, dx &=\frac {b \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {1}{3} \int \frac {a (3 a-b)+2 (2 a-b) b \sinh ^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\\ &=\frac {b \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {1}{3} (a (a-b)) \int \frac {1}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx+\frac {1}{3} (2 (2 a-b)) \int \sqrt {a+b \sinh ^2(e+f x)} \, dx\\ &=\frac {b \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {\left (2 (2 a-b) \sqrt {a+b \sinh ^2(e+f x)}\right ) \int \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}} \, dx}{3 \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}-\frac {\left (a (a-b) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}} \, dx}{3 \sqrt {a+b \sinh ^2(e+f x)}}\\ &=\frac {b \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {2 i (2 a-b) E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}+\frac {i a (a-b) F\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 169, normalized size = 0.97 \[ \frac {b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+2 i \sqrt {2} a (a-b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-4 i \sqrt {2} a (2 a-b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{6 f \sqrt {4 a+2 b \cosh (2 (e+f x))-2 b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 416, normalized size = 2.39 \[ \frac {\sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right ) \left (\cosh ^{4}\left (f x +e \right )\right )+\left (\sqrt {-\frac {b}{a}}\, a b -\sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )+3 a^{2} \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-5 a \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b +2 \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+4 \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -2 \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}}{3 \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]
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 \[ \int {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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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