Optimal. Leaf size=174 \[ \frac{i a (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} \text{EllipticF}\left (i e+i f x,\frac{b}{a}\right )}{3 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{b \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{2 i (2 a-b) \sqrt{a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac{b}{a}\right .\right )}{3 f \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.182364, antiderivative size = 174, normalized size of antiderivative = 1., 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 3180
Rule 3172
Rule 3178
Rule 3177
Rule 3183
Rule 3182
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*}
Mathematica [A] time = 0.641269, size = 169, normalized size = 0.97 \[ \frac{2 i \sqrt{2} a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )+b \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)-4 i \sqrt{2} a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac{b}{a}\right .\right )}{6 f \sqrt{4 a+2 b \cosh (2 (e+f x))-2 b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 416, normalized size = 2.4 \begin{align*}{\frac{1}{3\,\cosh \left ( fx+e \right ) f} \left ( \sqrt{-{\frac{b}{a}}}{b}^{2}\sinh \left ( fx+e \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( \sqrt{-{\frac{b}{a}}}ab-\sqrt{-{\frac{b}{a}}}{b}^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}\sinh \left ( fx+e \right ) +3\,{a}^{2}\sqrt{{\frac{b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) -5\,a\sqrt{{\frac{b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) b+2\,\sqrt{{\frac{b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{\it EllipticF} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){b}^{2}+4\,\sqrt{{\frac{b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{\it EllipticE} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) ab-2\,\sqrt{{\frac{b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{\it EllipticE} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){b}^{2} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \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 (b \sinh \left (f x + e\right )^{2} + a\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 (b \sinh \left (f x + e\right )^{2} + a\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 (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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