Optimal. Leaf size=385 \[ -\frac{2 b (3 a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \text{EllipticF}\left (\tan ^{-1}(\sinh (e+f x)),1-\frac{b}{a}\right )}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2-13 a b+8 b^2\right ) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 b (3 a-2 b) \coth (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{b \coth (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.456218, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3188, 472, 579, 583, 531, 418, 492, 411} \[ \frac{\left (3 a^2-13 a b+8 b^2\right ) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 b (3 a-2 b) \coth (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}-\frac{2 b (3 a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b)^2 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{b \coth (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3188
Rule 472
Rule 579
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{3 a-4 b-3 b x^2}{x^2 \sqrt{1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a (a-b) f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{3 a^2-13 a b+8 b^2-2 (3 a-2 b) b x^2}{x^2 \sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^2 (a-b)^2 f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{2 a (3 a-2 b) b-b \left (3 a^2-13 a b+8 b^2\right ) x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^3 (a-b)^2 f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{\left (2 (3 a-2 b) 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 a^2 (a-b)^2 f}+\frac{\left (b \left (3 a^2-13 a b+8 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 )}{3 a^3 (a-b)^2 f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{2 (3 a-2 b) 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^3 (a-b)^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2-13 a b+8 b^2\right ) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b)^2 f}-\frac{\left (\left (3 a^2-13 a b+8 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 )}{3 a^3 (a-b)^2 f}\\ &=-\frac{b \coth (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (3 a-2 b) b \coth (e+f x)}{3 a^2 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left (3 a^2-13 a b+8 b^2\right ) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b)^2 f}-\frac{\left (3 a^2-13 a b+8 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)}}{3 a^3 (a-b)^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 (3 a-2 b) 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^3 (a-b)^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\left (3 a^2-13 a b+8 b^2\right ) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b)^2 f}\\ \end{align*}
Mathematica [C] time = 2.3105, size = 234, normalized size = 0.61 \[ \frac{i \left (4 a^2 \left (\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} \left (\left (3 a^2-7 a b+4 b^2\right ) \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )+\left (-3 a^2+13 a b-8 b^2\right ) E\left (i (e+f x)\left |\frac{b}{a}\right .\right )\right )+2 i \sqrt{2} \left (-2 a b^2 (a-b) \sinh (2 (e+f x))-b^2 (7 a-5 b) \sinh (2 (e+f x)) (2 a+b \cosh (2 (e+f x))-b)+3 (a-b)^2 \coth (e+f x) (2 a+b \cosh (2 (e+f x))-b)^2\right )\right )}{12 a^3 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.128, size = 747, normalized size = 1.9 \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 \frac{\operatorname{csch}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{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 (\frac{\sqrt{b \sinh \left (f x + e\right )^{2} + a} \operatorname{csch}\left (f x + e\right )^{2}}{b^{3} \sinh \left (f x + e\right )^{6} + 3 \, a b^{2} \sinh \left (f x + e\right )^{4} + 3 \, a^{2} b \sinh \left (f x + e\right )^{2} + a^{3}}, 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 \frac{\operatorname{csch}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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