Optimal. Leaf size=275 \[ \frac{(3 a+5 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \text{EllipticF}\left (\tan ^{-1}(\sinh (e+f x)),1-\frac{b}{a}\right )}{3 f (a-b)^3 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{4 a \tanh (e+f x)}{3 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\tanh (e+f x) \text{sech}^2(e+f x)}{3 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\sqrt{a} \sqrt{b} (7 a+b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right )|1-\frac{a}{b}\right )}{3 f (a-b)^3 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}} \]
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Rubi [A] time = 0.275938, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3196, 470, 527, 525, 418, 411} \[ -\frac{4 a \tanh (e+f x)}{3 f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\tanh (e+f x) \text{sech}^2(e+f x)}{3 f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\sqrt{a} \sqrt{b} (7 a+b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right )|1-\frac{a}{b}\right )}{3 f (a-b)^3 \sqrt{a+b \sinh ^2(e+f x)} \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac{(3 a+5 b) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{3 f (a-b)^3 \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]
Antiderivative was successfully verified.
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Rule 3196
Rule 470
Rule 527
Rule 525
Rule 418
Rule 411
Rubi steps
\begin{align*} \int \frac{\tanh ^4(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 \frac{x^4}{\left (1+x^2\right )^{5/2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{\text{sech}^2(e+f x) \tanh (e+f x)}{3 (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{a-3 a x^2}{\left (1+x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b) f}\\ &=-\frac{4 a \tanh (e+f x)}{3 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\text{sech}^2(e+f x) \tanh (e+f x)}{3 (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{a (3 a+b)-4 a b x^2}{\sqrt{1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b) (-a+b) f}\\ &=-\frac{4 a \tanh (e+f x)}{3 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\text{sech}^2(e+f x) \tanh (e+f x)}{3 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\left (a b (7 a+b) \sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 (a-b)^2 (-a+b) f}-\frac{\left (a (3 a+5 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-b)^2 (-a+b) f}\\ &=-\frac{\sqrt{a} \sqrt{b} (7 a+b) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac{\sqrt{b} \sinh (e+f x)}{\sqrt{a}}\right )|1-\frac{a}{b}\right )}{3 (a-b)^3 f \sqrt{\frac{a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(3 a+5 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-b)^3 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{4 a \tanh (e+f x)}{3 (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\text{sech}^2(e+f x) \tanh (e+f x)}{3 (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 2.1843, size = 212, normalized size = 0.77 \[ \frac{8 i a (a-b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )-\frac{\tanh (e+f x) \text{sech}^2(e+f x) \left (4 \left (4 a^2+3 a b+b^2\right ) \cosh (2 (e+f x))+8 a^2+b (7 a+b) \cosh (4 (e+f x))+21 a b-5 b^2\right )}{2 \sqrt{2}}-2 i a (7 a+b) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac{b}{a}\right .\right )}{6 f (a-b)^3 \sqrt{2 a+b \cosh (2 (e+f x))-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.204, size = 354, normalized size = 1.3 \begin{align*} -{\frac{1}{3\, \left ( \cosh \left ( fx+e \right ) \right ) ^{3} \left ( a-b \right ) ^{3}f} \left ( \left ( 7\,\sqrt{-{\frac{b}{a}}}ab+\sqrt{-{\frac{b}{a}}}{b}^{2} \right ) \sinh \left ( fx+e \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( 4\,\sqrt{-{\frac{b}{a}}}{a}^{2}-4\,\sqrt{-{\frac{b}{a}}}ab \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}\sinh \left ( fx+e \right ) + \left ( -\sqrt{-{\frac{b}{a}}}{a}^{2}+2\,\sqrt{-{\frac{b}{a}}}ab-\sqrt{-{\frac{b}{a}}}{b}^{2} \right ) \sinh \left ( fx+e \right ) -\sqrt{{\frac{b \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{a}}+{\frac{a-b}{a}}}\sqrt{ \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \left ( 3\,{\it EllipticF} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){a}^{2}-2\,{\it EllipticF} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) ab-{\it EllipticF} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){b}^{2}+7\,{\it EllipticE} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) ab+{\it EllipticE} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ){b}^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{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 \frac{\tanh \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{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} \tanh \left (f x + e\right )^{4}}{b^{2} \sinh \left (f x + e\right )^{4} + 2 \, a b \sinh \left (f x + e\right )^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{4}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \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{\tanh \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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