Optimal. Leaf size=78 \[ -\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b (b \coth (c+d x))^{3/2}}{3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 3476, 329, 298, 203, 206} \[ -\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b (b \coth (c+d x))^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 3473
Rule 3476
Rubi steps
\begin {align*} \int (b \coth (c+d x))^{5/2} \, dx &=-\frac {2 b (b \coth (c+d x))^{3/2}}{3 d}+b^2 \int \sqrt {b \coth (c+d x)} \, dx\\ &=-\frac {2 b (b \coth (c+d x))^{3/2}}{3 d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac {2 b (b \coth (c+d x))^{3/2}}{3 d}-\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{-b^2+x^4} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}\\ &=-\frac {2 b (b \coth (c+d x))^{3/2}}{3 d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}\\ &=-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b (b \coth (c+d x))^{3/2}}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 68, normalized size = 0.87 \[ -\frac {(b \coth (c+d x))^{5/2} \left (2 \coth ^{\frac {3}{2}}(c+d x)+3 \tan ^{-1}\left (\sqrt {\coth (c+d x)}\right )-3 \tanh ^{-1}\left (\sqrt {\coth (c+d x)}\right )\right )}{3 d \coth ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 988, normalized size = 12.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 224, normalized size = 2.87 \[ \frac {6 \, b^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt {b}}\right ) \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) - 3 \, b^{\frac {5}{2}} \log \left ({\left | -\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) + \frac {8 \, {\left (3 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{2} b^{3} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) + b^{4} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )\right )}}{{\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} - \sqrt {b}\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 63, normalized size = 0.81 \[ -\frac {b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{d}+\frac {b^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{d}-\frac {2 b \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d} \]
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 \[ \int \left (b \coth \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 62, normalized size = 0.79 \[ \frac {b^{5/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{d}-\frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{d}-\frac {2\,b\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{3/2}}{3\,d} \]
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 \[ \int \left (b \coth {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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