Optimal. Leaf size=28 \[ \frac {(b c-a d) \log (c+d \coth (x))}{d^2}-\frac {b \coth (x)}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4344, 43} \[ \frac {(b c-a d) \log (c+d \coth (x))}{d^2}-\frac {b \coth (x)}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 4344
Rubi steps
\begin {align*} \int \frac {(a+b \coth (x)) \text {csch}^2(x)}{c+d \coth (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b x}{c+d x} \, dx,x,\coth (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx,x,\coth (x)\right )\\ &=-\frac {b \coth (x)}{d}+\frac {(b c-a d) \log (c+d \coth (x))}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 56, normalized size = 2.00 \[ \frac {\sinh (x) (a+b \coth (x)) (-(b c-a d) (\log (\sinh (x))-\log (c \sinh (x)+d \cosh (x)))-b d \coth (x))}{d^2 (a \sinh (x)+b \cosh (x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 174, normalized size = 6.21 \[ -\frac {2 \, b d - {\left ({\left (b c - a d\right )} \cosh \relax (x)^{2} + 2 \, {\left (b c - a d\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b c - a d\right )} \sinh \relax (x)^{2} - b c + a d\right )} \log \left (\frac {2 \, {\left (d \cosh \relax (x) + c \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + {\left ({\left (b c - a d\right )} \cosh \relax (x)^{2} + 2 \, {\left (b c - a d\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b c - a d\right )} \sinh \relax (x)^{2} - b c + a d\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{d^{2} \cosh \relax (x)^{2} + 2 \, d^{2} \cosh \relax (x) \sinh \relax (x) + d^{2} \sinh \relax (x)^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 113, normalized size = 4.04 \[ \frac {{\left (b c^{2} - a c d + b c d - a d^{2}\right )} \log \left ({\left | c e^{\left (2 \, x\right )} + d e^{\left (2 \, x\right )} - c + d \right |}\right )}{c d^{2} + d^{3}} - \frac {{\left (b c - a d\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{d^{2}} + \frac {b c e^{\left (2 \, x\right )} - a d e^{\left (2 \, x\right )} - b c + a d - 2 \, b d}{d^{2} {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 94, normalized size = 3.36 \[ -\frac {b \tanh \left (\frac {x}{2}\right )}{2 d}-\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) d +2 c \tanh \left (\frac {x}{2}\right )+d \right ) a}{d}+\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) d +2 c \tanh \left (\frac {x}{2}\right )+d \right ) c b}{d^{2}}-\frac {b}{2 d \tanh \left (\frac {x}{2}\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) a}{d}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) c b}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 77, normalized size = 2.75 \[ b {\left (\frac {c \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{2}} - \frac {c \log \left (e^{\left (-x\right )} + 1\right )}{d^{2}} - \frac {c \log \left (e^{\left (-x\right )} - 1\right )}{d^{2}} + \frac {2}{d e^{\left (-2 \, x\right )} - d}\right )} - \frac {a \log \left (d \coth \relax (x) + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.16, size = 297, normalized size = 10.61 \[ \frac {2\,\mathrm {atan}\left ({\mathrm {e}}^{2\,x}\,\left (\frac {4\,\left (a\,d\,\sqrt {-d^4}-b\,c\,\sqrt {-d^4}\right )}{d^2\,\sqrt {{\left (a\,d-b\,c\right )}^2}\,\left (c+d\right )\,\left (c-d\right )\,\sqrt {-d^4}}-\frac {4\,c^2\,\sqrt {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{d^4\,\left (c+d\right )\,\left (c-d\right )\,\left (a\,d-b\,c\right )}\right )\,\left (\frac {d^2\,\sqrt {-d^4}}{4}+\frac {c\,d\,\sqrt {-d^4}}{4}\right )-\frac {4\,c\,\left (d^2\,\sqrt {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}-c\,d\,\sqrt {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}\right )\,\left (\frac {d^2\,\sqrt {-d^4}}{4}+\frac {c\,d\,\sqrt {-d^4}}{4}\right )}{d^5\,\left (c+d\right )\,\left (c-d\right )\,\left (a\,d-b\,c\right )}\right )\,\sqrt {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{\sqrt {-d^4}}-\frac {2\,b}{d\,\left ({\mathrm {e}}^{2\,x}-1\right )} \]
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 \[ \int \frac {\left (a + b \coth {\relax (x )}\right ) \operatorname {csch}^{2}{\relax (x )}}{c + d \coth {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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