Optimal. Leaf size=83 \[ \frac {b^2}{2 a^2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )}+\frac {b (2 a+b) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 (a+b)^2 d}+\frac {\log (\sinh (c+d x))}{(a+b)^2 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 83, 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 = {4223, 457, 90}
\begin {gather*} \frac {b^2}{2 a^2 d (a+b) \left (a \cosh ^2(c+d x)+b\right )}+\frac {b (2 a+b) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^2 d (a+b)^2}+\frac {\log (\sinh (c+d x))}{d (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rule 4223
Rubi steps
\begin {align*} \int \frac {\coth (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^5}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {x^2}{(1-x) (b+a x)^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {1}{(a+b)^2 (-1+x)}+\frac {b^2}{a (a+b) (b+a x)^2}-\frac {b (2 a+b)}{a (a+b)^2 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {b^2}{2 a^2 (a+b) d \left (b+a \cosh ^2(c+d x)\right )}+\frac {b (2 a+b) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^2 (a+b)^2 d}+\frac {\log (\sinh (c+d x))}{(a+b)^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 115, normalized size = 1.39 \begin {gather*} \frac {(a+b) \left (2 a^2 \log (\sinh (c+d x))+b \left (b+(2 a+b) \log \left (a+b+a \sinh ^2(c+d x)\right )\right )\right )+a \left (2 a^2 \log (\sinh (c+d x))+b (2 a+b) \log \left (a+b+a \sinh ^2(c+d x)\right )\right ) \sinh ^2(c+d x)}{a^2 (a+b)^2 d (a+2 b+a \cosh (2 (c+d x)))} \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. \(205\) vs.
\(2(79)=158\).
time = 2.68, size = 206, normalized size = 2.48
method | result | size |
derivativedivides | \(\frac {\frac {b \left (-\frac {2 a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (2 a +b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{2} \left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}}{d}\) | \(206\) |
default | \(\frac {\frac {b \left (-\frac {2 a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (2 a +b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{2} \left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}}{d}\) | \(206\) |
risch | \(\frac {x}{a^{2}}-\frac {2 x}{a^{2}+2 a b +b^{2}}-\frac {2 c}{d \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 b x}{a \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 b c}{a d \left (a^{2}+2 a b +b^{2}\right )}-\frac {2 b^{2} x}{a^{2} \left (a^{2}+2 a b +b^{2}\right )}-\frac {2 b^{2} c}{a^{2} d \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 b^{2} {\mathrm e}^{2 d x +2 c}}{a^{2} d \left (a +b \right ) \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{a d \left (a^{2}+2 a b +b^{2}\right )}+\frac {b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{2} d \left (a^{2}+2 a b +b^{2}\right )}\) | \(332\) |
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. 209 vs.
\(2 (79) = 158\).
time = 0.28, size = 209, normalized size = 2.52 \begin {gather*} \frac {2 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{4} + a^{3} b + 2 \, {\left (a^{4} + 3 \, a^{3} b + 2 \, a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{4} + a^{3} b\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {{\left (2 \, a b + b^{2}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {d x + c}{a^{2} 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. 1031 vs.
\(2 (79) = 158\).
time = 0.48, size = 1031, normalized size = 12.42 \begin {gather*} -\frac {2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d x \cosh \left (d x + c\right )^{4} + 8 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d x \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d x - 4 \, {\left (a b^{2} + b^{3} - {\left (a^{3} + 4 \, a^{2} b + 5 \, a b^{2} + 2 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d x \cosh \left (d x + c\right )^{2} - a b^{2} - b^{3} + {\left (a^{3} + 4 \, a^{2} b + 5 \, a b^{2} + 2 \, b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (2 \, a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (2 \, a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (2 \, a^{2} b + a b^{2}\right )} \sinh \left (d x + c\right )^{4} + 2 \, a^{2} b + a b^{2} + 2 \, {\left (2 \, a^{2} b + 5 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{2} b + 5 \, a b^{2} + 2 \, b^{3} + 3 \, {\left (2 \, a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (2 \, a^{2} b + 5 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) - 2 \, {\left (a^{3} \cosh \left (d x + c\right )^{4} + 4 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} \sinh \left (d x + c\right )^{4} + a^{3} + 2 \, {\left (a^{3} + 2 \, a^{2} b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} \cosh \left (d x + c\right )^{2} + a^{3} + 2 \, a^{2} b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} \cosh \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left ({\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} d x \cosh \left (d x + c\right )^{3} - {\left (a b^{2} + b^{3} - {\left (a^{3} + 4 \, a^{2} b + 5 \, a b^{2} + 2 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{5} + 4 \, a^{4} b + 5 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{5} + 4 \, a^{4} b + 5 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d + 4 \, {\left ({\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d \cosh \left (d x + c\right )^{3} + {\left (a^{5} + 4 \, a^{4} b + 5 \, a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \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 \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {cosh}\left (c+d\,x\right )}^4\,\mathrm {coth}\left (c+d\,x\right )}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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