Optimal. Leaf size=123 \[ -\frac {\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{8 \sqrt {b}}-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right ) \]
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Rubi [A] time = 0.24, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3670, 477, 582, 523, 217, 206, 377} \[ -\frac {\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{8 \sqrt {b}}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 377
Rule 477
Rule 523
Rule 582
Rule 3670
Rubi steps
\begin {align*} \int \coth ^2(x) \left (a+b \coth ^2(x)\right )^{3/2} \, dx &=\operatorname {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^2 \left (-a (4 a+3 b)-b (5 a+4 b) x^2\right )}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}-\frac {\operatorname {Subst}\left (\int \frac {-a b (5 a+4 b)-b \left (3 a^2+12 a b+8 b^2\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )}{8 b}\\ &=-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}+(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )-\frac {1}{8} \left (3 a^2+12 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}+(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{8} \left (3 a^2+12 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )\\ &=-\frac {\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{8 \sqrt {b}}+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{8} (5 a+4 b) \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{4} b \coth ^3(x) \sqrt {a+b \coth ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.98, size = 219, normalized size = 1.78 \[ \frac {\sinh (x) \sqrt {\text {csch}^2(x) ((a+b) \cosh (2 x)-a+b)} \left (\sqrt {b} \left (8 \sqrt {2} (a+b)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {(a+b) \cosh (2 x)-a+b}}\right )-\sqrt {a+b} \coth (x) \text {csch}(x) \sqrt {(a+b) \cosh (2 x)-a+b} \left (5 a+2 b \text {csch}^2(x)+6 b\right )\right )-\sqrt {2} \sqrt {a+b} \left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \cosh (x)}{\sqrt {(a+b) \cosh (2 x)-a+b}}\right )\right )}{8 \sqrt {2} \sqrt {b} \sqrt {a+b} \sqrt {(a+b) \cosh (2 x)-a+b}} \]
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 [B] time = 0.09, size = 633, normalized size = 5.15 \[ \text {result too large to display} \]
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 \[ \int {\left (b \coth \relax (x)^{2} + a\right )}^{\frac {3}{2}} \coth \relax (x)^{2}\,{d x} \]
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 \[ \int {\mathrm {coth}\relax (x)}^2\,{\left (b\,{\mathrm {coth}\relax (x)}^2+a\right )}^{3/2} \,d x \]
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 \[ \int \left (a + b \coth ^{2}{\relax (x )}\right )^{\frac {3}{2}} \coth ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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