Optimal. Leaf size=97 \[ \frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b^3 \sqrt {b \coth (c+d x)}}{d}-\frac {2 b (b \coth (c+d x))^{5/2}}{5 d} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 3476, 329, 212, 206, 203} \[ -\frac {2 b^3 \sqrt {b \coth (c+d x)}}{d}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b (b \coth (c+d x))^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 3473
Rule 3476
Rubi steps
\begin {align*} \int (b \coth (c+d x))^{7/2} \, dx &=-\frac {2 b (b \coth (c+d x))^{5/2}}{5 d}+b^2 \int (b \coth (c+d x))^{3/2} \, dx\\ &=-\frac {2 b^3 \sqrt {b \coth (c+d x)}}{d}-\frac {2 b (b \coth (c+d x))^{5/2}}{5 d}+b^4 \int \frac {1}{\sqrt {b \coth (c+d x)}} \, dx\\ &=-\frac {2 b^3 \sqrt {b \coth (c+d x)}}{d}-\frac {2 b (b \coth (c+d x))^{5/2}}{5 d}-\frac {b^5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-b^2+x^2\right )} \, dx,x,b \coth (c+d x)\right )}{d}\\ &=-\frac {2 b^3 \sqrt {b \coth (c+d x)}}{d}-\frac {2 b (b \coth (c+d x))^{5/2}}{5 d}-\frac {\left (2 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{-b^2+x^4} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}\\ &=-\frac {2 b^3 \sqrt {b \coth (c+d x)}}{d}-\frac {2 b (b \coth (c+d x))^{5/2}}{5 d}+\frac {b^4 \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}+\frac {b^4 \operatorname {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \coth (c+d x)}\right )}{d}\\ &=\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}+\frac {b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b \coth (c+d x)}}{\sqrt {b}}\right )}{d}-\frac {2 b^3 \sqrt {b \coth (c+d x)}}{d}-\frac {2 b (b \coth (c+d x))^{5/2}}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 83, normalized size = 0.86 \[ \frac {b^3 \sqrt {b \coth (c+d x)} \left (-2 \coth ^{\frac {5}{2}}(c+d x)-10 \sqrt {\coth (c+d x)}+5 \tan ^{-1}\left (\sqrt {\coth (c+d x)}\right )+5 \tanh ^{-1}\left (\sqrt {\coth (c+d x)}\right )\right )}{5 d \sqrt {\coth (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1574, normalized size = 16.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 379, normalized size = 3.91 \[ -\frac {10 \, b^{\frac {7}{2}} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt {b}}\right ) \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) + 5 \, b^{\frac {7}{2}} \log \left ({\left | -\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) - \frac {16 \, {\left (5 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{4} b^{4} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) - 10 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{3} b^{\frac {9}{2}} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) + 20 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{2} b^{5} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) - 10 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )} b^{\frac {11}{2}} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) + 3 \, b^{6} \mathrm {sgn}\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )\right )}}{{\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} - \sqrt {b}\right )}^{5}}}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 80, normalized size = 0.82 \[ \frac {b^{\frac {7}{2}} \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{d}+\frac {b^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )}{d}-\frac {2 b \left (b \coth \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d}-\frac {2 b^{3} \sqrt {b \coth \left (d x +c \right )}}{d} \]
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 \[ \int \left (b \coth \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.58, size = 83, normalized size = 0.86 \[ \frac {b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{d}-\frac {2\,b^3\,\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}}{d}-\frac {2\,b\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{5/2}}{5\,d}-\frac {b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,\mathrm {coth}\left (c+d\,x\right )}\,1{}\mathrm {i}}{\sqrt {b}}\right )\,1{}\mathrm {i}}{d} \]
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 \[ \int \left (b \coth {\left (c + d x \right )}\right )^{\frac {7}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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