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 = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 21
Rule 3177
Rule 3178
Rule 3184
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*}
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Mathematica [A] time = 0.16, 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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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sinh \left (f x + e\right )^{2} + a}}{b^{2} \sinh \left (f x + e\right )^{4} + 2 \, a b \sinh \left (f x + e\right )^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 253, normalized size = 2.20 \[ -\frac {\sqrt {-\frac {b}{a}}\, b \sinh \left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )\right )-a \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b -\sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b}{a \left (a -b \right ) \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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