Optimal. Leaf size=138 \[ \frac {\log (\cosh (c+d x))}{(a+b)^3 d}+\frac {\log (\tanh (c+d x))}{a^3 d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 (a+b)^3 d}+\frac {b}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (2 a+b)}{2 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.14, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3751, 457, 84}
\begin {gather*} \frac {\log (\tanh (c+d x))}{a^3 d}+\frac {b (2 a+b)}{2 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 d (a+b)^3}+\frac {b}{4 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\log (\cosh (c+d x))}{d (a+b)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 457
Rule 3751
Rubi steps
\begin {align*} \int \frac {\coth (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{(1-x) x (a+b x)^3} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{(a+b)^3 (-1+x)}+\frac {1}{a^3 x}-\frac {b^2}{a (a+b) (a+b x)^3}-\frac {b^2 (2 a+b)}{a^2 (a+b)^2 (a+b x)^2}-\frac {b^2 \left (3 a^2+3 a b+b^2\right )}{a^3 (a+b)^3 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\log (\cosh (c+d x))}{(a+b)^3 d}+\frac {\log (\tanh (c+d x))}{a^3 d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 (a+b)^3 d}+\frac {b}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (2 a+b)}{2 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 1.25, size = 117, normalized size = 0.85 \begin {gather*} \frac {\frac {4 \log (\cosh (c+d x))}{(a+b)^3}+\frac {4 \log (\tanh (c+d x))+\frac {b \left (-2 \left (3 a^2+3 a b+b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )+\frac {a (a+b) \left (a (5 a+3 b)+2 b (2 a+b) \tanh ^2(c+d x)\right )}{\left (a+b \tanh ^2(c+d x)\right )^2}\right )}{(a+b)^3}}{a^3}}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(272\) vs.
\(2(132)=264\).
time = 3.26, size = 273, normalized size = 1.98
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {b \left (\frac {2 \left (3 a^{2}+5 a b +2 b^{2}\right ) a b \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (3 a^{3}+10 a^{2} b +10 a \,b^{2}+3 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (3 a^{2}+5 a b +2 b^{2}\right ) a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (3 a^{2}+3 a b +b^{2}\right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )}{2}\right )}{\left (a +b \right )^{3} a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{\left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{\left (a +b \right )^{3}}}{d}\) | \(273\) |
default | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {b \left (\frac {2 \left (3 a^{2}+5 a b +2 b^{2}\right ) a b \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (3 a^{3}+10 a^{2} b +10 a \,b^{2}+3 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (3 a^{2}+5 a b +2 b^{2}\right ) a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (3 a^{2}+3 a b +b^{2}\right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )}{2}\right )}{\left (a +b \right )^{3} a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{\left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{\left (a +b \right )^{3}}}{d}\) | \(273\) |
risch | \(\frac {x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {2 x}{a^{3}}-\frac {2 c}{a^{3} d}+\frac {6 b x}{a \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {6 b c}{a d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {6 b^{2} x}{a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {6 b^{2} c}{a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 b^{3} x}{a^{3} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 b^{3} c}{a^{3} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (3 a^{2} {\mathrm e}^{4 d x +4 c}+4 a b \,{\mathrm e}^{4 d x +4 c}+b^{2} {\mathrm e}^{4 d x +4 c}+6 a^{2} {\mathrm e}^{2 d x +2 c}-2 a b \,{\mathrm e}^{2 d x +2 c}-2 b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{2}+4 a b +b^{2}\right ) b^{2} {\mathrm e}^{2 d x +2 c}}{\left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} d \left (a +b \right )^{3} a^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{3} d}-\frac {3 b \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 a d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {3 b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 a^{2} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {b^{3} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 a^{3} d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}\) | \(607\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 498 vs.
\(2 (132) = 264\).
time = 0.33, size = 498, normalized size = 3.61 \begin {gather*} -\frac {{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} d} + \frac {d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {2 \, {\left ({\left (3 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{2} b^{2} - a b^{3} - b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5} + 4 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{7} + 7 \, a^{6} b + 6 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 7 \, a^{3} b^{4} + 3 \, a^{2} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4800 vs.
\(2 (132) = 264\).
time = 0.67, size = 4800, normalized size = 34.78 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {\coth {\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 295 vs.
\(2 (132) = 264\).
time = 0.59, size = 295, normalized size = 2.14 \begin {gather*} -\frac {\frac {{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}} + \frac {2 \, {\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{3}} - \frac {4 \, {\left ({\left (3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )} + {\left (3 \, a^{2} b^{2} + a b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} + \frac {2 \, {\left (3 \, a^{3} b^{2} - a^{2} b^{3} - a b^{4}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{a + b}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2} {\left (a + b\right )}^{2} a^{3}}}{2 \, d} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {coth}\left (c+d\,x\right )}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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