Optimal. Leaf size=367 \[ -\frac{\left (a^2-11 a b+8 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \text{EllipticF}\left (\tan ^{-1}(\sinh (e+f x)),1-\frac{b}{a}\right )}{35 b f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 (a-2 b) \left (a^2+4 a b-4 b^2\right ) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b^2 f}+\frac{\left (a^2-11 a b+8 b^2\right ) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b f}+\frac{2 (a-2 b) \left (a^2+4 a b-4 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{35 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{b \sinh ^5(e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}+\frac{2 (4 a-3 b) \sinh ^3(e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f} \]
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Rubi [A] time = 0.470834, antiderivative size = 367, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3188, 477, 582, 531, 418, 492, 411} \[ -\frac{2 (a-2 b) \left (a^2+4 a b-4 b^2\right ) \tanh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b^2 f}+\frac{\left (a^2-11 a b+8 b^2\right ) \sinh (e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b f}-\frac{\left (a^2-11 a b+8 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{35 b f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{2 (a-2 b) \left (a^2+4 a b-4 b^2\right ) \text{sech}(e+f x) \sqrt{a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac{b}{a}\right )}{35 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac{b \sinh ^5(e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}+\frac{2 (4 a-3 b) \sinh ^3(e+f x) \cosh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f} \]
Antiderivative was successfully verified.
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Rule 3188
Rule 477
Rule 582
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \sinh ^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 (a+b x^2\right )^{3/2}}{\sqrt{1+x^2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac{b \cosh (e+f x) \sinh ^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (a (7 a-5 b)+2 (4 a-3 b) b x^2\right )}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{7 f}\\ &=\frac{2 (4 a-3 b) \cosh (e+f x) \sinh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f}+\frac{b \cosh (e+f x) \sinh ^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}-\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (6 a (4 a-3 b) b-3 b \left (a^2-11 a b+8 b^2\right ) x^2\right )}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{35 b f}\\ &=\frac{\left (a^2-11 a b+8 b^2\right ) \cosh (e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b f}+\frac{2 (4 a-3 b) \cosh (e+f x) \sinh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f}+\frac{b \cosh (e+f x) \sinh ^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}+\frac{\left (\sqrt{\cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{-3 a b \left (a^2-11 a b+8 b^2\right )-6 b \left (a^3+2 a^2 b-12 a b^2+8 b^3\right ) x^2}{\sqrt{1+x^2} \sqrt{a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{105 b^2 f}\\ &=\frac{\left (a^2-11 a b+8 b^2\right ) \cosh (e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b f}+\frac{2 (4 a-3 b) \cosh (e+f x) \sinh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f}+\frac{b \cosh (e+f x) \sinh ^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}-\frac{\left (a \left (a^2-11 a b+8 b^2\right ) \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 )}{35 b f}-\frac{\left (2 \left (a^3+2 a^2 b-12 a b^2+8 b^3\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 )}{35 b f}\\ &=\frac{\left (a^2-11 a b+8 b^2\right ) \cosh (e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b f}+\frac{2 (4 a-3 b) \cosh (e+f x) \sinh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f}+\frac{b \cosh (e+f x) \sinh ^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}-\frac{\left (a^2-11 a b+8 b^2\right ) 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)}}{35 b f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 \left (a^3+2 a^2 b-12 a b^2+8 b^3\right ) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{35 b^2 f}+\frac{\left (2 \left (a^3+2 a^2 b-12 a b^2+8 b^3\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 )}{35 b^2 f}\\ &=\frac{\left (a^2-11 a b+8 b^2\right ) \cosh (e+f x) \sinh (e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 b f}+\frac{2 (4 a-3 b) \cosh (e+f x) \sinh ^3(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{35 f}+\frac{b \cosh (e+f x) \sinh ^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)}}{7 f}+\frac{2 \left (a^3+2 a^2 b-12 a b^2+8 b^3\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)}}{35 b^2 f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{\left (a^2-11 a b+8 b^2\right ) 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)}}{35 b f \sqrt{\frac{\text{sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac{2 \left (a^3+2 a^2 b-12 a b^2+8 b^3\right ) \sqrt{a+b \sinh ^2(e+f x)} \tanh (e+f x)}{35 b^2 f}\\ \end{align*}
Mathematica [C] time = 2.87267, size = 262, normalized size = 0.71 \[ \frac{-64 i a \left (3 a^2 b+2 a^3-13 a b^2+8 b^3\right ) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )+\sqrt{2} b \sinh (2 (e+f x)) \left (b \left (144 a^2-480 a b+299 b^2\right ) \cosh (2 (e+f x))-496 a^2 b+32 a^3+2 b^2 (26 a-27 b) \cosh (4 (e+f x))+684 a b^2+5 b^3 \cosh (6 (e+f x))-250 b^3\right )+128 i a \left (2 a^2 b+a^3-12 a b^2+8 b^3\right ) \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac{b}{a}\right .\right )}{2240 b^2 f \sqrt{2 a+b \cosh (2 (e+f x))-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 743, normalized size = 2. \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{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \sinh \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 ({\left (b \sinh \left (f x + e\right )^{6} + a \sinh \left (f x + e\right )^{4}\right )} \sqrt{b \sinh \left (f x + e\right )^{2} + a}, 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{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \sinh \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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