Optimal. Leaf size=292 \[ -\frac {b (9 a-b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 a^2 f (a-b)^3 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\sqrt {b} \left (3 a^2+7 a b-2 b^2\right ) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 a^{3/2} f (a-b)^3 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}+\frac {\tanh (e+f x)}{f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+b) \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3192, 414, 527, 525, 418, 411} \[ \frac {\sqrt {b} \left (3 a^2+7 a b-2 b^2\right ) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 a^{3/2} f (a-b)^3 \sqrt {a+b \sinh ^2(e+f x)} \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}}}-\frac {b (9 a-b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right )}{3 a^2 f (a-b)^3 \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x)}{f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {b (3 a+b) \sinh (e+f x) \cosh (e+f x)}{3 a f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 414
Rule 418
Rule 525
Rule 527
Rule 3192
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {b-3 b x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sinh (e+f x)\right )}{(-a+b) f}\\ &=\frac {b (3 a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {2 (3 a-b) b-b (3 a+b) x^2}{\sqrt {1+x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a (a-b) (-a+b) f}\\ &=\frac {b (3 a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left ((9 a-b) 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 )}{3 a (a-b)^2 (-a+b) f}-\frac {\left (b \left (3 a^2+7 a b-2 b^2\right ) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (a+b x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a (a-b)^2 (-a+b) f}\\ &=\frac {b (3 a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\sqrt {b} \left (3 a^2+7 a b-2 b^2\right ) \cosh (e+f x) E\left (\tan ^{-1}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 a^{3/2} (a-b)^3 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(9 a-b) 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)}}{3 a^2 (a-b)^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 3.46, size = 260, normalized size = 0.89 \[ \frac {-2 i a^2 \left (3 a^2-2 a b-b^2\right ) \left (\frac {2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )+2 i a^2 \left (3 a^2+7 a b-2 b^2\right ) \left (\frac {2 a+b \cosh (2 (e+f x))-b}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+\frac {\tanh (e+f x) \left (24 a^4-24 a^3 b+4 a b \left (6 a^2+5 a b-3 b^2\right ) \cosh (2 (e+f x))+b^2 \left (3 a^2+7 a b-2 b^2\right ) \cosh (4 (e+f x))+41 a^2 b^2-19 a b^3+2 b^4\right )}{\sqrt {2}}}{6 a^2 f (a-b)^3 (2 a+b \cosh (2 (e+f x))-b)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sinh \left (f x + e\right )^{2} + a} \operatorname {sech}\left (f x + e\right )^{2}}{b^{3} \sinh \left (f x + e\right )^{6} + 3 \, a b^{2} \sinh \left (f x + e\right )^{4} + 3 \, a^{2} b \sinh \left (f x + e\right )^{2} + a^{3}}, x\right ) \]
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 [B] time = 0.24, size = 1002, normalized size = 3.43 \[ \text {result too large to display} \]
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 \[ \int \frac {\operatorname {sech}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {cosh}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\operatorname {sech}^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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