Optimal. Leaf size=299 \[ \frac{(9 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \text{EllipticF}\left (\tan ^{-1}(\sinh (e+f x)),1-\frac{b}{a}\right )}{15 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2+7 a b-2 b^2\right ) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 b f}-\frac{\left (3 a^2+7 a b-2 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{15 b f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{b \sinh (e+f x) \cosh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}+\frac{2 (3 a-b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f} \]
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Rubi [A] time = 0.290854, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3192, 416, 528, 531, 418, 492, 411} \[ \frac{\left (3 a^2+7 a b-2 b^2\right ) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 b f}-\frac{\left (3 a^2+7 a b-2 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{15 b f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{b \sinh (e+f x) \cosh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}+\frac{2 (3 a-b) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f}+\frac{(9 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{15 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]
Antiderivative was successfully verified.
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Rule 3192
Rule 416
Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \cosh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx &=\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \sqrt{1+x^2} \left (a+b x^2\right )^{3/2} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{b \cosh ^3(e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x^2} \left (a (5 a-b)+2 (3 a-b) b x^2\right )}{\sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{5 f}\\ &=\frac{2 (3 a-b) \cosh (e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f}+\frac{b \cosh ^3(e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{a (9 a-b) b+b \left (3 a^2+7 a b-2 b^2\right ) x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{15 b f}\\ &=\frac{2 (3 a-b) \cosh (e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f}+\frac{b \cosh ^3(e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}+\frac{\left (a (9 a-b) \sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{15 f}+\frac{\left (\left (3 a^2+7 a b-2 b^2\right ) \sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{15 f}\\ &=\frac{2 (3 a-b) \cosh (e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f}+\frac{b \cosh ^3(e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}+\frac{(9 a-b) F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2+7 a b-2 b^2\right ) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{15 b f}-\frac{\left (\left (3 a^2+7 a b-2 b^2\right ) \sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{15 b f}\\ &=\frac{2 (3 a-b) \cosh (e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f}+\frac{b \cosh ^3(e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{5 f}-\frac{\left (3 a^2+7 a b-2 b^2\right ) E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 b f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{(9 a-b) F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{15 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2+7 a b-2 b^2\right ) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{15 b f}\\ \end{align*}
Mathematica [C] time = 1.32657, size = 213, normalized size = 0.71 \[ \frac{16 i a \left (3 a^2-2 a b-b^2\right ) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )+\sqrt{2} b \sinh (2 (e+f x)) \left (48 a^2+4 b (9 a-2 b) \cosh (2 (e+f x))-28 a b+3 b^2 \cosh (4 (e+f x))+5 b^2\right )-16 i a \left (3 a^2+7 a b-2 b^2\right ) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac{b}{a}\right .\right )}{240 b f \sqrt{2 a+b \cosh (2 (e+f x))-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.12, size = 535, normalized size = 1.8 \begin{align*} \text{result too large to display} \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}} \cosh \left (f x + e\right )^{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 \cosh \left (f x + e\right )^{2} \sinh \left (f x + e\right )^{2} + a \cosh \left (f x + e\right )^{2}\right )} \sqrt{b \sinh \left (f x + e\right )^{2} + a}, 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}} \cosh \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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