Optimal. Leaf size=58 \[ -\frac {2 (a c-b d) \tanh ^{-1}\left (\frac {d-c \tanh \left (\frac {x}{2}\right )}{\sqrt {c^2+d^2}}\right )}{c \sqrt {c^2+d^2}}-\frac {b \tanh ^{-1}(\cosh (x))}{c} \]
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Rubi [A] time = 0.17, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2828, 3001, 3770, 2660, 618, 206} \[ -\frac {2 (a c-b d) \tanh ^{-1}\left (\frac {d-c \tanh \left (\frac {x}{2}\right )}{\sqrt {c^2+d^2}}\right )}{c \sqrt {c^2+d^2}}-\frac {b \tanh ^{-1}(\cosh (x))}{c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 2828
Rule 3001
Rule 3770
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}(x)}{c+d \sinh (x)} \, dx &=-\left (i \int \frac {\text {csch}(x) (i b+i a \sinh (x))}{c+d \sinh (x)} \, dx\right )\\ &=\frac {b \int \text {csch}(x) \, dx}{c}+\frac {(a c-b d) \int \frac {1}{c+d \sinh (x)} \, dx}{c}\\ &=-\frac {b \tanh ^{-1}(\cosh (x))}{c}+\frac {(2 (a c-b d)) \operatorname {Subst}\left (\int \frac {1}{c+2 d x-c x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{c}\\ &=-\frac {b \tanh ^{-1}(\cosh (x))}{c}-\frac {(4 (a c-b d)) \operatorname {Subst}\left (\int \frac {1}{4 \left (c^2+d^2\right )-x^2} \, dx,x,2 d-2 c \tanh \left (\frac {x}{2}\right )\right )}{c}\\ &=-\frac {b \tanh ^{-1}(\cosh (x))}{c}-\frac {2 (a c-b d) \tanh ^{-1}\left (\frac {d-c \tanh \left (\frac {x}{2}\right )}{\sqrt {c^2+d^2}}\right )}{c \sqrt {c^2+d^2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 67, normalized size = 1.16 \[ \frac {\frac {2 (a c-b d) \tan ^{-1}\left (\frac {d-c \tanh \left (\frac {x}{2}\right )}{\sqrt {-c^2-d^2}}\right )}{\sqrt {-c^2-d^2}}+b \log \left (\tanh \left (\frac {x}{2}\right )\right )}{c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 172, normalized size = 2.97 \[ -\frac {{\left (a c - b d\right )} \sqrt {c^{2} + d^{2}} \log \left (\frac {d^{2} \cosh \relax (x)^{2} + d^{2} \sinh \relax (x)^{2} + 2 \, c d \cosh \relax (x) + 2 \, c^{2} + d^{2} + 2 \, {\left (d^{2} \cosh \relax (x) + c d\right )} \sinh \relax (x) + 2 \, \sqrt {c^{2} + d^{2}} {\left (d \cosh \relax (x) + d \sinh \relax (x) + c\right )}}{d \cosh \relax (x)^{2} + d \sinh \relax (x)^{2} + 2 \, c \cosh \relax (x) + 2 \, {\left (d \cosh \relax (x) + c\right )} \sinh \relax (x) - d}\right ) + {\left (b c^{2} + b d^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (b c^{2} + b d^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{c^{3} + c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 90, normalized size = 1.55 \[ -\frac {b \log \left (e^{x} + 1\right )}{c} + \frac {b \log \left ({\left | e^{x} - 1 \right |}\right )}{c} + \frac {{\left (a c - b d\right )} \log \left (\frac {{\left | 2 \, d e^{x} + 2 \, c - 2 \, \sqrt {c^{2} + d^{2}} \right |}}{{\left | 2 \, d e^{x} + 2 \, c + 2 \, \sqrt {c^{2} + d^{2}} \right |}}\right )}{\sqrt {c^{2} + d^{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 86, normalized size = 1.48 \[ \frac {b \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{c}+\frac {2 \arctanh \left (\frac {2 c \tanh \left (\frac {x}{2}\right )-2 d}{2 \sqrt {c^{2}+d^{2}}}\right ) a}{\sqrt {c^{2}+d^{2}}}-\frac {2 \arctanh \left (\frac {2 c \tanh \left (\frac {x}{2}\right )-2 d}{2 \sqrt {c^{2}+d^{2}}}\right ) b d}{c \sqrt {c^{2}+d^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 141, normalized size = 2.43 \[ -b {\left (\frac {d \log \left (\frac {d e^{\left (-x\right )} - c - \sqrt {c^{2} + d^{2}}}{d e^{\left (-x\right )} - c + \sqrt {c^{2} + d^{2}}}\right )}{\sqrt {c^{2} + d^{2}} c} + \frac {\log \left (e^{\left (-x\right )} + 1\right )}{c} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{c}\right )} + \frac {a \log \left (\frac {d e^{\left (-x\right )} - c - \sqrt {c^{2} + d^{2}}}{d e^{\left (-x\right )} - c + \sqrt {c^{2} + d^{2}}}\right )}{\sqrt {c^{2} + d^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.37, size = 539, normalized size = 9.29 \[ \frac {b\,\ln \left ({\mathrm {e}}^x-1\right )}{c}-\frac {b\,\ln \left ({\mathrm {e}}^x+1\right )}{c}-\frac {\ln \left (\frac {\left (\frac {32\,\left (a^2\,c^2\,d-2\,a\,b\,c\,d^2-4\,{\mathrm {e}}^x\,b^2\,c^3+2\,b^2\,c^2\,d-3\,{\mathrm {e}}^x\,b^2\,c\,d^2+2\,b^2\,d^3\right )}{d^5}-\frac {\left (a\,c-b\,d\right )\,\left (\frac {32\,c^2\,\left (2\,b\,d^2+4\,a\,c^2\,{\mathrm {e}}^x+a\,d^2\,{\mathrm {e}}^x-2\,a\,c\,d-3\,b\,c\,d\,{\mathrm {e}}^x\right )}{d^5}+\frac {32\,c\,\left (a\,c-b\,d\right )\,\left (-4\,{\mathrm {e}}^x\,c^3+3\,c^2\,d-3\,{\mathrm {e}}^x\,c\,d^2+2\,d^3\right )}{d^5\,\sqrt {c^2+d^2}}\right )}{c\,\sqrt {c^2+d^2}}\right )\,\left (a\,c-b\,d\right )}{c\,\sqrt {c^2+d^2}}+\frac {32\,b\,\left (a\,c-b\,d\right )\,\left (a\,d\,{\mathrm {e}}^x-2\,b\,d+4\,b\,c\,{\mathrm {e}}^x\right )}{d^5}\right )\,\left (a\,c-b\,d\right )\,\sqrt {c^2+d^2}}{c^3+c\,d^2}+\frac {\ln \left (\frac {32\,b\,\left (a\,c-b\,d\right )\,\left (a\,d\,{\mathrm {e}}^x-2\,b\,d+4\,b\,c\,{\mathrm {e}}^x\right )}{d^5}-\frac {\left (\frac {32\,\left (a^2\,c^2\,d-2\,a\,b\,c\,d^2-4\,{\mathrm {e}}^x\,b^2\,c^3+2\,b^2\,c^2\,d-3\,{\mathrm {e}}^x\,b^2\,c\,d^2+2\,b^2\,d^3\right )}{d^5}+\frac {\left (a\,c-b\,d\right )\,\left (\frac {32\,c^2\,\left (2\,b\,d^2+4\,a\,c^2\,{\mathrm {e}}^x+a\,d^2\,{\mathrm {e}}^x-2\,a\,c\,d-3\,b\,c\,d\,{\mathrm {e}}^x\right )}{d^5}-\frac {32\,c\,\left (a\,c-b\,d\right )\,\left (-4\,{\mathrm {e}}^x\,c^3+3\,c^2\,d-3\,{\mathrm {e}}^x\,c\,d^2+2\,d^3\right )}{d^5\,\sqrt {c^2+d^2}}\right )}{c\,\sqrt {c^2+d^2}}\right )\,\left (a\,c-b\,d\right )}{c\,\sqrt {c^2+d^2}}\right )\,\left (a\,c-b\,d\right )\,\sqrt {c^2+d^2}}{c^3+c\,d^2} \]
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 \[ \int \frac {a + b \operatorname {csch}{\relax (x )}}{c + d \sinh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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