Optimal. Leaf size=59 \[ \frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^2 b d}-\frac {b \log (\sinh (c+d x))}{a^2 d}-\frac {\text {csch}(c+d x)}{a d} \]
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Rubi [A] time = 0.12, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2837, 12, 894} \[ \frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^2 b d}-\frac {b \log (\sinh (c+d x))}{a^2 d}-\frac {\text {csch}(c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b^2 \left (-b^2-x^2\right )}{x^2 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b^3 d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {-b^2-x^2}{x^2 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {b^2}{a x^2}+\frac {b^2}{a^2 x}+\frac {-a^2-b^2}{a^2 (a+x)}\right ) \, dx,x,b \sinh (c+d x)\right )}{b d}\\ &=-\frac {\text {csch}(c+d x)}{a d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^2 b d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 52, normalized size = 0.88 \[ \frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))-a b \text {csch}(c+d x)+b^2 (-\log (\sinh (c+d x)))}{a^2 b d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 299, normalized size = 5.07 \[ -\frac {a^{2} d x \cosh \left (d x + c\right )^{2} + a^{2} d x \sinh \left (d x + c\right )^{2} - a^{2} d x + 2 \, a b \cosh \left (d x + c\right ) - {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{2} + 2 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} \sinh \left (d x + c\right )^{2} - b^{2}\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (a^{2} d x \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right )}{a^{2} b d \cosh \left (d x + c\right )^{2} + 2 \, a^{2} b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} b d \sinh \left (d x + c\right )^{2} - a^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 113, normalized size = 1.92 \[ -\frac {\frac {d x}{b} + \frac {b \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2}} + \frac {b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{2}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - b \right |}\right )}{a^{2} b} + \frac {2 \, e^{\left (d x + c\right )}}{a {\left (e^{\left (d x + c\right )} + 1\right )} {\left (e^{\left (d x + c\right )} - 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 172, normalized size = 2.92 \[ \frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d b}+\frac {\ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d b}+\frac {b \ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d \,a^{2}}-\frac {1}{2 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 131, normalized size = 2.22 \[ \frac {d x + c}{b d} + \frac {2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} - \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 356, normalized size = 6.03 \[ \frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {x}{b}+\frac {\ln \left (8\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^5-16\,a^2\,b^3-4\,a^4\,b+16\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{b\,d}+\frac {b\,\ln \left (8\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^5-16\,a^2\,b^3-4\,a^4\,b+16\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^2\,d}-\frac {b\,\ln \left (4\,a^6+16\,b^6+32\,a^2\,b^4+20\,a^4\,b^2-4\,a^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-16\,b^6\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-32\,a^2\,b^4\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-20\,a^4\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{a^2\,d} \]
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 \[ \int \frac {\cosh {\left (c + d x \right )} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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