Optimal. Leaf size=241 \[ -\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} \text{EllipticF}\left (i e+i f x,\frac{b}{a}\right )}{3 b f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}+\frac{(a+b) \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sinh (e+f x) \cosh (e+f x)}{3 f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{i (a+b) \sqrt{a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac{b}{a}\right .\right )}{3 a b f (a-b)^2 \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.325695, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3173, 3172, 3178, 3177, 3183, 3182} \[ \frac{(a+b) \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b)^2 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sinh (e+f x) \cosh (e+f x)}{3 f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{i \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1} F\left (i e+i f x\left |\frac{b}{a}\right .\right )}{3 b f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}+\frac{i (a+b) \sqrt{a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac{b}{a}\right .\right )}{3 a b f (a-b)^2 \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3173
Rule 3172
Rule 3178
Rule 3177
Rule 3183
Rule 3182
Rubi steps
\begin{align*} \int \frac{\sinh ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\cosh (e+f x) \sinh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{\int \frac{a-a \sinh ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx}{3 a (a-b)}\\ &=\frac{\cosh (e+f x) \sinh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\int \frac{2 a^2+a (a+b) \sinh ^2(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx}{3 a^2 (a-b)^2}\\ &=\frac{\cosh (e+f x) \sinh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\int \frac{1}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx}{3 (a-b) b}-\frac{(a+b) \int \sqrt{a+b \sinh ^2(e+f x)} \, dx}{3 a (a-b)^2 b}\\ &=\frac{\cosh (e+f x) \sinh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\left ((a+b) \sqrt{a+b \sinh ^2(e+f x)}\right ) \int \sqrt{1+\frac{b \sinh ^2(e+f x)}{a}} \, dx}{3 a (a-b)^2 b \sqrt{1+\frac{b \sinh ^2(e+f x)}{a}}}+\frac{\sqrt{1+\frac{b \sinh ^2(e+f x)}{a}} \int \frac{1}{\sqrt{1+\frac{b \sinh ^2(e+f x)}{a}}} \, dx}{3 (a-b) b \sqrt{a+b \sinh ^2(e+f x)}}\\ &=\frac{\cosh (e+f x) \sinh (e+f x)}{3 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{i (a+b) E\left (i e+i f x\left |\frac{b}{a}\right .\right ) \sqrt{a+b \sinh ^2(e+f x)}}{3 a (a-b)^2 b f \sqrt{1+\frac{b \sinh ^2(e+f x)}{a}}}-\frac{i F\left (i e+i f x\left |\frac{b}{a}\right .\right ) \sqrt{1+\frac{b \sinh ^2(e+f x)}{a}}}{3 (a-b) b f \sqrt{a+b \sinh ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.40341, size = 187, normalized size = 0.78 \[ \frac{-2 i a^2 (a-b) \left (\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} \text{EllipticF}\left (i (e+f x),\frac{b}{a}\right )+\sqrt{2} b \sinh (2 (e+f x)) \left (4 a^2+b (a+b) \cosh (2 (e+f x))-a b-b^2\right )+2 i a^2 (a+b) \left (\frac{2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac{b}{a}\right .\right )}{6 a b f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.114, size = 598, normalized size = 2.5 \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{\sinh \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} \sinh \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{\sinh \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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