Optimal. Leaf size=46 \[ \frac {b \log \left (b+a \cosh ^2(c+d x)\right )}{2 a (a+b) d}+\frac {\log (\sinh (c+d x))}{(a+b) d} \]
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Rubi [A]
time = 0.06, antiderivative size = 46, 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, 78}
\begin {gather*} \frac {\log (\sinh (c+d x))}{d (a+b)}+\frac {b \log \left (a \cosh ^2(c+d x)+b\right )}{2 a d (a+b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 4223
Rubi steps
\begin {align*} \int \frac {\coth (c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^3}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {x}{(1-x) (b+a x)} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {1}{(-a-b) (-1+x)}-\frac {b}{(a+b) (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {b \log \left (b+a \cosh ^2(c+d x)\right )}{2 a (a+b) d}+\frac {\log (\sinh (c+d x))}{(a+b) d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 42, normalized size = 0.91 \begin {gather*} \frac {2 a \log (\sinh (c+d x))+b \log \left (a+b+a \sinh ^2(c+d x)\right )}{2 a^2 d+2 a b 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. \(124\) vs.
\(2(44)=88\).
time = 2.20, size = 125, normalized size = 2.72
method | result | size |
risch | \(\frac {x}{a}-\frac {2 x}{a +b}-\frac {2 c}{d \left (a +b \right )}-\frac {2 b x}{a \left (a +b \right )}-\frac {2 b c}{a d \left (a +b \right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (a +b \right )}+\frac {b \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a d \left (a +b \right )}\) | \(119\) |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a +b}+\frac {b \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2 a \left (a +b \right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}}{d}\) | \(125\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a +b}+\frac {b \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2 a \left (a +b \right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}}{d}\) | \(125\) |
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. 100 vs.
\(2 (44) = 88\).
time = 0.28, size = 100, normalized size = 2.17 \begin {gather*} \frac {b \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^{2} + a b\right )} d} + \frac {d x + c}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a + b\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a + b\right )} 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. 115 vs.
\(2 (44) = 88\).
time = 0.42, size = 115, normalized size = 2.50 \begin {gather*} -\frac {2 \, {\left (a + b\right )} d x - b \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 \, a \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{2 \, {\left (a^{2} + a b\right )} d} \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 )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, 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 [B]
time = 1.82, size = 228, normalized size = 4.96 \begin {gather*} \frac {\ln \left (8\,a\,b^5-b^6-24\,a^2\,b^4+32\,a^3\,b^3-16\,a^4\,b^2+b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-8\,a\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+24\,a^2\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-32\,a^3\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+16\,a^4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a\,d+b\,d}-\frac {x}{a}+\frac {b\,\ln \left (2\,a^2-a\,b+4\,a^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,a^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+6\,a\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-a\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )}{2\,d\,a^2+2\,b\,d\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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