Optimal. Leaf size=105 \[ -\frac {2 \coth (c+d x)}{d \sqrt {b \coth ^3(c+d x)}}-\frac {\coth ^{\frac {3}{2}}(c+d x) \tan ^{-1}\left (\sqrt {\coth (c+d x)}\right )}{d \sqrt {b \coth ^3(c+d x)}}+\frac {\coth ^{\frac {3}{2}}(c+d x) \tanh ^{-1}\left (\sqrt {\coth (c+d x)}\right )}{d \sqrt {b \coth ^3(c+d x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3658, 3474, 3476, 329, 298, 203, 206} \[ -\frac {2 \coth (c+d x)}{d \sqrt {b \coth ^3(c+d x)}}-\frac {\coth ^{\frac {3}{2}}(c+d x) \tan ^{-1}\left (\sqrt {\coth (c+d x)}\right )}{d \sqrt {b \coth ^3(c+d x)}}+\frac {\coth ^{\frac {3}{2}}(c+d x) \tanh ^{-1}\left (\sqrt {\coth (c+d x)}\right )}{d \sqrt {b \coth ^3(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 3474
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \coth ^3(c+d x)}} \, dx &=\frac {\coth ^{\frac {3}{2}}(c+d x) \int \frac {1}{\coth ^{\frac {3}{2}}(c+d x)} \, dx}{\sqrt {b \coth ^3(c+d x)}}\\ &=-\frac {2 \coth (c+d x)}{d \sqrt {b \coth ^3(c+d x)}}+\frac {\coth ^{\frac {3}{2}}(c+d x) \int \sqrt {\coth (c+d x)} \, dx}{\sqrt {b \coth ^3(c+d x)}}\\ &=-\frac {2 \coth (c+d x)}{d \sqrt {b \coth ^3(c+d x)}}-\frac {\coth ^{\frac {3}{2}}(c+d x) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \sqrt {b \coth ^3(c+d x)}}\\ &=-\frac {2 \coth (c+d x)}{d \sqrt {b \coth ^3(c+d x)}}-\frac {\left (2 \coth ^{\frac {3}{2}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {\coth (c+d x)}\right )}{d \sqrt {b \coth ^3(c+d x)}}\\ &=-\frac {2 \coth (c+d x)}{d \sqrt {b \coth ^3(c+d x)}}+\frac {\coth ^{\frac {3}{2}}(c+d x) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\coth (c+d x)}\right )}{d \sqrt {b \coth ^3(c+d x)}}-\frac {\coth ^{\frac {3}{2}}(c+d x) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\coth (c+d x)}\right )}{d \sqrt {b \coth ^3(c+d x)}}\\ &=-\frac {2 \coth (c+d x)}{d \sqrt {b \coth ^3(c+d x)}}-\frac {\tan ^{-1}\left (\sqrt {\coth (c+d x)}\right ) \coth ^{\frac {3}{2}}(c+d x)}{d \sqrt {b \coth ^3(c+d x)}}+\frac {\tanh ^{-1}\left (\sqrt {\coth (c+d x)}\right ) \coth ^{\frac {3}{2}}(c+d x)}{d \sqrt {b \coth ^3(c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 41, normalized size = 0.39 \[ -\frac {2 \coth (c+d x) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};\coth ^2(c+d x)\right )}{d \sqrt {b \coth ^3(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.21, size = 907, normalized size = 8.64 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 279, normalized size = 2.66 \[ \frac {\frac {2 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt {b}}\right )}{\sqrt {b} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {\log \left ({\left | -\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right )}{\sqrt {b} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {8}{{\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} + \sqrt {b}\right )} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 92, normalized size = 0.88 \[ -\frac {\coth \left (d x +c \right ) \left (2 b^{\frac {5}{2}}-\arctanh \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{2} \sqrt {b \coth \left (d x +c \right )}+\arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right ) b^{2} \sqrt {b \coth \left (d x +c \right )}\right )}{d \sqrt {b \left (\coth ^{3}\left (d x +c \right )\right )}\, b^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{\sqrt {b \coth \left (d x + c\right )^{3}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {b \coth ^{3}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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