Optimal. Leaf size=108 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 108, 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 = {3670, 446, 85, 152, 156, 63, 208} \[ \frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 85
Rule 152
Rule 156
Rule 208
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{\left (a+b \coth ^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,\coth (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1-x) x (a+b x)^{5/2}} \, dx,x,\coth ^2(x)\right )\\ &=\frac {b}{3 a (a+b) \left (a+b \coth ^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,\coth ^2(x)\right )}{2 a (a+b)}\\ &=\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^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,\coth ^2(x)\right )}{a^2 (a+b)^2}\\ &=\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{2 a^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )}{2 (a+b)^2}\\ &=\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \coth ^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 \coth ^2(x)}\right )}{b (a+b)^2}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\frac {b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 73, normalized size = 0.68 \[ \frac {(a+b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b \coth ^2(x)}{a}+1\right )-a \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b \coth ^2(x)+a}{a+b}\right )}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}} \]
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 = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x )}{\left (a +b \left (\coth ^{2}\relax (x )\right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {\tanh \relax (x)}{{\left (b \coth \relax (x)^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\mathrm {tanh}\relax (x)}{{\left (b\,{\mathrm {coth}\relax (x)}^2+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\tanh {\relax (x )}}{\left (a + b \coth ^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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