Optimal. Leaf size=109 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {4139, 446, 85, 152, 156, 63, 208} \[ -\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \text {sech}^2(x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 85
Rule 152
Rule 156
Rule 208
Rule 446
Rule 4139
Rubi steps
\begin {align*} \int \frac {\coth (x)}{\left (a+b \text {sech}^2(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (-1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\text {sech}(x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(-1+x) x (a+b x)^{5/2}} \, dx,x,\text {sech}^2(x)\right )\\ &=-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {a+b-b x}{(-1+x) x (a+b x)^{3/2}} \, dx,x,\text {sech}^2(x)\right )}{2 a (a+b)}\\ &=-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}-\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} (a+b)^2+\frac {1}{2} b (2 a+b) x}{(-1+x) x \sqrt {a+b x}} \, dx,x,\text {sech}^2(x)\right )}{a^2 (a+b)^2}\\ &=-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}-\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\text {sech}^2(x)\right )}{2 a^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+b x}} \, dx,x,\text {sech}^2(x)\right )}{2 (a+b)^2}\\ &=-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}-\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \text {sech}^2(x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {sech}^2(x)}\right )}{a^2 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {sech}^2(x)}\right )}{b (a+b)^2}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {b}{3 a (a+b) \left (a+b \text {sech}^2(x)\right )^{3/2}}-\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \text {sech}^2(x)}}\\ \end {align*}
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Mathematica [B] time = 1.11, size = 242, normalized size = 2.22 \[ \frac {\text {sech}^5(x) \left (-\frac {2 b \cosh (x) \left (7 a^2+a (7 a+4 b) \cosh (2 x)+16 a b+6 b^2\right ) (a \cosh (2 x)+a+2 b)}{3 a^2 (a+b)^2}-\frac {(a \cosh (2 x)+a+2 b)^{5/2} \left (\sqrt {a} \left (a^2-2 a b-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {a \cosh (2 x)+a+2 b}}\right )+(a+b)^2 \left (\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {2 a+2 b} \cosh (x)}{\sqrt {a \cosh (2 x)+a+2 b}}\right )-2 \sqrt {a+b} \log \left (\sqrt {a \cosh (2 x)+a+2 b}+\sqrt {2} \sqrt {a} \cosh (x)\right )\right )\right )}{\sqrt {2} a^{5/2} (a+b)^{5/2}}\right )}{8 \left (a+b \text {sech}^2(x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {\coth \relax (x )}{\left (a +b \mathrm {sech}\relax (x )^{2}\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {\coth \relax (x)}{{\left (b \operatorname {sech}\relax (x)^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {coth}\relax (x)}{{\left (a+\frac {b}{{\mathrm {cosh}\relax (x)}^2}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\coth {\relax (x )}}{\left (a + b \operatorname {sech}^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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