Optimal. Leaf size=351 \[ \frac{(3 a-4 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \text{EllipticF}\left (\tan ^{-1}(\sinh (e+f x)),1-\frac{b}{a}\right )}{3 a^3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{(7 a-8 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)}-\frac{(7 a-8 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)}+\frac{(3 a-4 b) \coth (e+f x)}{3 a^2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(7 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.4119, antiderivative size = 351, 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 = {3196, 469, 579, 583, 531, 418, 492, 411} \[ \frac{(7 a-8 b) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)}-\frac{(7 a-8 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 f (a-b)}+\frac{(3 a-4 b) \coth (e+f x)}{3 a^2 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(3 a-4 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{(7 a-8 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 a^3 f (a-b) \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3196
Rule 469
Rule 579
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\coth ^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{\sqrt{1+x^2}}{x^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\coth (e+f x)}{3 a 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{-4-3 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 f}\\ &=\frac{\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(3 a-4 b) \coth (e+f x)}{3 a^2 (a-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{-7 a+8 b+(-3 a+4 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) f}\\ &=\frac{\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(7 a-8 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{a (3 a-4 b)+(7 a-8 b) b x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^3 (a-b) f}\\ &=\frac{\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(7 a-8 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}+\frac{\left ((3 a-4 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) f}+\frac{\left ((7 a-8 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^3 (a-b) f}\\ &=\frac{\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(7 a-8 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}+\frac{(3 a-4 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) f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{(7 a-8 b) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b) f}-\frac{\left ((7 a-8 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^3 (a-b) f}\\ &=\frac{\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{(7 a-8 b) \coth (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}-\frac{(7 a-8 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^3 (a-b) f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{(3 a-4 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) f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{(7 a-8 b) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b) f}\\ \end{align*}
Mathematica [C] time = 2.6774, size = 226, normalized size = 0.64 \[ \frac{8 i a^2 (a-b) \left (\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )-\frac{\coth (e+f x) \left (4 b \left (11 a^2-19 a b+8 b^2\right ) \cosh (2 (e+f x))-68 a^2 b+24 a^3+b^2 (7 a-8 b) \cosh (4 (e+f x))+69 a b^2-24 b^3\right )}{\sqrt{2}}-2 i a^2 (7 a-8 b) \left (\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac{b}{a}\right .\right )}{6 a^3 f (a-b) (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.24, size = 640, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \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.
[In]
[Out]
________________________________________________________________________________________
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 )^{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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \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.
[In]
[Out]