Optimal. Leaf size=276 \[ -\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \text{EllipticF}\left (\tan ^{-1}(\sinh (e+f x)),1-\frac{b}{a}\right )}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{(2 a-b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}+\frac{(2 a-b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{(2 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]
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Rubi [A] time = 0.281847, antiderivative size = 276, 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 = {3188, 475, 583, 531, 418, 492, 411} \[ -\frac{(2 a-b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}+\frac{(2 a-b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{b \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{(2 a-b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]
Antiderivative was successfully verified.
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Rule 3188
Rule 475
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \text{csch}^4(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx &=\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x^4 \sqrt{1+x^2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{-2 a+b-b x^2}{x^2 \sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 f}\\ &=\frac{(2 a-b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{a b+(2 a-b) b x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=\frac{(2 a-b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{\left (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 )}{3 f}-\frac{\left ((2 a-b) b \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 )}{3 a f}\\ &=\frac{(2 a-b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}-\frac{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)}}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{(2 a-b) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a f}+\frac{\left ((2 a-b) \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 )}{3 a f}\\ &=\frac{(2 a-b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a f}-\frac{\coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 f}+\frac{(2 a-b) 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)}}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{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)}}{3 a f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{(2 a-b) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a f}\\ \end{align*}
Mathematica [C] time = 3.02826, size = 208, normalized size = 0.75 \[ \frac{-4 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )+\frac{\coth (e+f x) \text{csch}^2(e+f x) \left (4 \left (2 a^2-4 a b+b^2\right ) \cosh (2 (e+f x))-(2 a-b) (8 a-b \cosh (4 (e+f x))-3 b)\right )}{2 \sqrt{2}}+2 i a (2 a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac{b}{a}\right .\right )}{6 a f \sqrt{2 a+b \cosh (2 (e+f x))-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.121, size = 436, normalized size = 1.6 \begin{align*}{\frac{1}{3\,a \left ( \sinh \left ( fx+e \right ) \right ) ^{3}\cosh \left ( fx+e \right ) f} \left ( 2\,\sqrt{-{\frac{b}{a}}}ab \left ( \sinh \left ( fx+e \right ) \right ) ^{6}-\sqrt{-{\frac{b}{a}}}{b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{6}+b\sqrt{{\frac{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}{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 ) a \left ( \sinh \left ( fx+e \right ) \right ) ^{3}-\sqrt{{\frac{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}{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} \left ( \sinh \left ( fx+e \right ) \right ) ^{3}-2\,\sqrt{{\frac{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}{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 \left ( \sinh \left ( fx+e \right ) \right ) ^{3}+\sqrt{{\frac{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}{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} \left ( \sinh \left ( fx+e \right ) \right ) ^{3}+2\,\sqrt{-{\frac{b}{a}}}{a}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{4}-\sqrt{-{\frac{b}{a}}}{b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+\sqrt{-{\frac{b}{a}}}{a}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{2}-2\,\sqrt{-{\frac{b}{a}}}ab \left ( \sinh \left ( fx+e \right ) \right ) ^{2}-\sqrt{-{\frac{b}{a}}}{a}^{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 \sqrt{b \sinh \left (f x + e\right )^{2} + a} \operatorname{csch}\left (f x + e\right )^{4}\,{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 (\sqrt{b \sinh \left (f x + e\right )^{2} + a} \operatorname{csch}\left (f x + e\right )^{4}, 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 \sqrt{b \sinh \left (f x + e\right )^{2} + a} \operatorname{csch}\left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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