Optimal. Leaf size=290 \[ \frac {(a-2 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f (a-b)}-\frac {(a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f (a-b)}-\frac {b \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{a^2 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(a-2 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{a^2 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b \coth (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \]
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Rubi [A] time = 0.31, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3188, 472, 583, 531, 418, 492, 411} \[ \frac {(a-2 b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f (a-b)}-\frac {(a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f (a-b)}-\frac {b \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{a^2 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(a-2 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{a^2 f (a-b) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b \coth (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 472
Rule 492
Rule 531
Rule 583
Rule 3188
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(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 {1}{x^2 \sqrt {1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {b \coth (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {a-2 b-b x^2}{x^2 \sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{a (a-b) f}\\ &=-\frac {b \coth (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 (a-b) f}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {a b-(a-2 b) b x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{a^2 (a-b) f}\\ &=-\frac {b \coth (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 (a-b) f}-\frac {\left (b \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 )}{a (a-b) f}+\frac {\left ((a-2 b) b \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 )}{a^2 (a-b) f}\\ &=-\frac {b \coth (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 (a-b) f}-\frac {b 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)}}{a^2 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{a^2 (a-b) f}-\frac {\left ((a-2 b) \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 )}{a^2 (a-b) f}\\ &=-\frac {b \coth (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 (a-b) f}-\frac {(a-2 b) 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)}}{a^2 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b 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)}}{a^2 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{a^2 (a-b) f}\\ \end {align*}
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Mathematica [C] time = 1.29, size = 185, normalized size = 0.64 \[ \frac {-\left (\coth (e+f x) \left (2 a^2+b (a-2 b) \cosh (2 (e+f x))-3 a b+2 b^2\right )\right )+i \sqrt {2} a (a-b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )-i \sqrt {2} a (a-2 b) \sqrt {\frac {2 a+b \cosh (2 (e+f x))-b}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{a^2 f (a-b) \sqrt {4 a+2 b \cosh (2 (e+f x))-2 b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 3.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sinh \left (f x + e\right )^{2} + a} \operatorname {csch}\left (f x + e\right )^{2}}{b^{2} \sinh \left (f x + e\right )^{4} + 2 \, a b \sinh \left (f x + e\right )^{2} + a^{2}}, x\right ) \]
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 = 0.17, size = 284, normalized size = 0.98 \[ -\frac {\sinh \left (f x +e \right ) \sqrt {\frac {b \left (\cosh ^{2}\left (f x +e \right )\right )}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (2 a \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 b \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a +2 b \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )+\left (\sqrt {-\frac {b}{a}}\, a b -2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{4}\left (f x +e \right )\right )+\left (\sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b +2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{a^{2} \left (a -b \right ) \sqrt {-\frac {b}{a}}\, \sinh \left (f x +e \right ) \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f} \]
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 \[ \int \frac {\operatorname {csch}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\mathrm {sinh}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {csch}^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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