Optimal. Leaf size=251 \[ -\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 a^2 f (a-b)^2 \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}-\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {i \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} F\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \]
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Rubi [A] time = 0.28, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3184, 3173, 3172, 3178, 3177, 3183, 3182} \[ -\frac {2 b (2 a-b) \sinh (e+f x) \cosh (e+f x)}{3 a^2 f (a-b)^2 \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 a^2 f (a-b)^2 \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}-\frac {b \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {i \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1} F\left (i e+i f x\left |\frac {b}{a}\right .\right )}{3 a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3172
Rule 3173
Rule 3177
Rule 3178
Rule 3182
Rule 3183
Rule 3184
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {-3 a+2 b+b \sinh ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx}{3 a (a-b)}\\ &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\int \frac {-a (3 a-b)-2 (2 a-b) b \sinh ^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx}{3 a^2 (a-b)^2}\\ &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (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 (a-b)}+\frac {(2 (2 a-b)) \int \sqrt {a+b \sinh ^2(e+f x)} \, dx}{3 a^2 (a-b)^2}\\ &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (2 (2 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^2 (a-b)^2 \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 (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\\ &=-\frac {b \cosh (e+f x) \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) b \cosh (e+f x) \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 i (2 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^2 (a-b)^2 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 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.38, size = 190, normalized size = 0.76 \[ \frac {\sqrt {2} b \sinh (2 (e+f x)) \left (-5 a^2+b (b-2 a) \cosh (2 (e+f x))+5 a b-b^2\right )+i a^2 (a-b) \left (\frac {2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a^2 (2 a-b) \left (\frac {2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{3 a^2 f (a-b)^2 (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sinh \left (f x + e\right )^{2} + a}}{b^{3} \sinh \left (f x + e\right )^{6} + 3 \, a b^{2} \sinh \left (f x + e\right )^{4} + 3 \, a^{2} b \sinh \left (f x + e\right )^{2} + a^{3}}, x\right ) \]
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 = 0.35, size = 406, normalized size = 1.62 \[ \frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (-\frac {\sinh \left (f x +e \right ) \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}{3 a b \left (a -b \right ) \left (\sinh ^{2}\left (f x +e \right )+\frac {a}{b}\right )^{2}}-\frac {2 b \left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right ) \left (2 a -b \right )}{3 a^{2} \left (a -b \right )^{2} \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}+\frac {\left (3 a -b \right ) \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )}{\left (3 a^{3}-6 a^{2} b +3 a \,b^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}-\frac {2 b \left (2 a -b \right ) \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (\EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )}{3 a^{2} \left (a -b \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]
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 \[ \int \frac {1}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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