Optimal. Leaf size=28 \[ \frac {(a+b) \log (\sinh (c+d x))}{d}-\frac {b \log (\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4138, 446, 72} \[ \frac {(a+b) \log (\sinh (c+d x))}{d}-\frac {b \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 4138
Rubi steps
\begin {align*} \int \coth (c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b+a x^2}{x \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {b+a x}{(1-x) x} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {-a-b}{-1+x}+\frac {b}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b \log (\cosh (c+d x))}{d}+\frac {(a+b) \log (\sinh (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 1.57 \[ \frac {a (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{d}-\frac {b (\log (\cosh (c+d x))-\log (\sinh (c+d x)))}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 69, normalized size = 2.46 \[ -\frac {a d x + b \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - {\left (a + b\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 56, normalized size = 2.00 \[ -\frac {a d x - {\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (-2 \, c\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 26, normalized size = 0.93 \[ \frac {b \ln \left (\tanh \left (d x +c \right )\right )}{d}+\frac {a \ln \left (\sinh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 65, normalized size = 2.32 \[ b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d}\right )} + \frac {a \log \left (\sinh \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 167, normalized size = 5.96 \[ \frac {a\,\ln \left (4\,a^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-4\,a^2-16\,b^2-16\,a\,b+16\,b^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+16\,a\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )}{2\,d}-a\,x-\frac {\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-d^2}}{d\,\sqrt {a^2+4\,a\,b+4\,b^2}}+\frac {2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-d^2}}{d\,\sqrt {a^2+4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2+4\,a\,b+4\,b^2}}{\sqrt {-d^2}} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \coth {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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