Optimal. Leaf size=385 \[ \frac{(3 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \text{EllipticF}\left (\tan ^{-1}(\sinh (e+f x)),1-\frac{b}{a}\right )}{3 a^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{8 (a-2 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}-\frac{8 (a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac{(3 a-8 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{8 (a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.561806, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3196, 468, 579, 583, 531, 418, 492, 411} \[ \frac{8 (a-2 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}-\frac{8 (a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac{(3 a-8 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(3 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{8 (a-2 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3196
Rule 468
Rule 579
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\coth ^4(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{\left (1+x^2\right )^{3/2}}{x^4 \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 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-2 b)+(2 a-5 b) x^2}{x^4 \sqrt{1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a b f}\\ &=-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b 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 (3 a-8 b) (a-b)+6 (a-3 b) (a-b) x^2}{x^4 \sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^2 (a-b) b f}\\ &=-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(3 a-8 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 b f}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{24 (a-2 b) (a-b) b+3 (3 a-8 b) (a-b) b x^2}{x^2 \sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{9 a^3 (a-b) b f}\\ &=-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{8 (a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac{(3 a-8 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{-3 a (3 a-8 b) (a-b) b-24 (a-2 b) (a-b) b^2 x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{9 a^4 (a-b) b f}\\ &=-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{8 (a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac{(3 a-8 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 b f}+\frac{\left ((3 a-8 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^3 f}+\frac{\left (8 (a-2 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^4 f}\\ &=-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{8 (a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac{(3 a-8 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 b f}+\frac{(3 a-8 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^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{8 (a-2 b) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^4 f}-\frac{\left (8 (a-2 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^4 f}\\ &=-\frac{(a-b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{2 (a-3 b) \coth (e+f x) \text{csch}^2(e+f x)}{3 a^2 b f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{8 (a-2 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^4 f}+\frac{(3 a-8 b) \coth (e+f x) \text{csch}^2(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 b f}-\frac{8 (a-2 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^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{(3 a-8 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^4 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{8 (a-2 b) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^4 f}\\ \end{align*}
Mathematica [C] time = 2.78841, size = 247, normalized size = 0.64 \[ -\frac{i \left (2 a^2 b \left (\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} \left ((8 b-5 a) \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )+8 (a-2 b) E\left (i (e+f x)\left |\frac{b}{a}\right .\right )\right )+\frac{i b \coth (e+f x) \text{csch}^2(e+f x) \left (-2 \left (-38 a^2 b+8 a^3+63 a b^2-30 b^3\right ) \cosh (2 (e+f x))-b \left (13 a^2-36 a b+24 b^2\right ) \cosh (4 (e+f x))-63 a^2 b+8 a^3-2 a b^2 \cosh (6 (e+f x))+92 a b^2+4 b^3 \cosh (6 (e+f x))-40 b^3\right )}{\sqrt{2}}\right )}{6 a^4 b f (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.226, size = 924, normalized size = 2.4 \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{\coth \left (f x + e\right )^{4}}{{\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} \coth \left (f x + e\right )^{4}}{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{\coth \left (f x + e\right )^{4}}{{\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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