Optimal. Leaf size=115 \[ -\frac{b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac{b}{a}\right .\right )}{a f (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.0611369, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3184, 21, 3178, 3177} \[ -\frac{b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i \sqrt{a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac{b}{a}\right .\right )}{a f (a-b) \sqrt{\frac{b \sinh ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3184
Rule 21
Rule 3178
Rule 3177
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx &=-\frac{b \cosh (e+f x) \sinh (e+f x)}{a (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{\int \frac{-a-b \sinh ^2(e+f x)}{\sqrt{a+b \sinh ^2(e+f x)}} \, dx}{a (a-b)}\\ &=-\frac{b \cosh (e+f x) \sinh (e+f x)}{a (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\int \sqrt{a+b \sinh ^2(e+f x)} \, dx}{a (a-b)}\\ &=-\frac{b \cosh (e+f x) \sinh (e+f x)}{a (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\sqrt{a+b \sinh ^2(e+f x)} \int \sqrt{1+\frac{b \sinh ^2(e+f x)}{a}} \, dx}{a (a-b) \sqrt{1+\frac{b \sinh ^2(e+f x)}{a}}}\\ &=-\frac{b \cosh (e+f x) \sinh (e+f x)}{a (a-b) f \sqrt{a+b \sinh ^2(e+f x)}}-\frac{i E\left (i e+i f x\left |\frac{b}{a}\right .\right ) \sqrt{a+b \sinh ^2(e+f x)}}{a (a-b) f \sqrt{1+\frac{b \sinh ^2(e+f x)}{a}}}\\ \end{align*}
Mathematica [A] time = 0.143378, size = 100, normalized size = 0.87 \[ \frac{-\sqrt{2} b \sinh (2 (e+f x))-2 i a \sqrt{\frac{2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac{b}{a}\right .\right )}{2 a f (a-b) \sqrt{2 a+b \cosh (2 (e+f x))-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 252, normalized size = 2.2 \begin{align*}{\frac{1}{a \left ( a-b \right ) \cosh \left ( fx+e \right ) f} \left ( -\sqrt{-{\frac{b}{a}}}b\sinh \left ( fx+e \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+a\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}}{\it EllipticF} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \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}}{\it EllipticF} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) b+\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}}{\it EllipticE} \left ( \sinh \left ( fx+e \right ) \sqrt{-{\frac{b}{a}}},\sqrt{{\frac{a}{b}}} \right ) b \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{1}{{\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}}{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(-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{1}{{\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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