Optimal. Leaf size=64 \[ \frac {a \log (c+d x)}{d}+\frac {b \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d}+\frac {b \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d} \]
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Rubi [A]
time = 0.10, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3398, 3384,
3379, 3382} \begin {gather*} \frac {a \log (c+d x)}{d}+\frac {b \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d}+\frac {b \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3398
Rubi steps
\begin {align*} \int \frac {a+b \sinh (e+f x)}{c+d x} \, dx &=\int \left (\frac {a}{c+d x}+\frac {b \sinh (e+f x)}{c+d x}\right ) \, dx\\ &=\frac {a \log (c+d x)}{d}+b \int \frac {\sinh (e+f x)}{c+d x} \, dx\\ &=\frac {a \log (c+d x)}{d}+\left (b \cosh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx+\left (b \sinh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx\\ &=\frac {a \log (c+d x)}{d}+\frac {b \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d}+\frac {b \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 57, normalized size = 0.89 \begin {gather*} \frac {a \log (c+d x)+b \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+b \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 94, normalized size = 1.47
method | result | size |
risch | \(\frac {a \ln \left (d x +c \right )}{d}+\frac {b \,{\mathrm e}^{\frac {c f -d e}{d}} \expIntegral \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d}-\frac {b \,{\mathrm e}^{-\frac {c f -d e}{d}} \expIntegral \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 73, normalized size = 1.14 \begin {gather*} \frac {1}{2} \, b {\left (\frac {e^{\left (\frac {c f}{d} - e\right )} E_{1}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {e^{\left (-\frac {c f}{d} + e\right )} E_{1}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac {a \log \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 122, normalized size = 1.91 \begin {gather*} \frac {{\left (b {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - b {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right ) + 2 \, a \log \left (d x + c\right ) + {\left (b {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + b {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right )}{2 \, 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 {a + b \sinh {\left (e + f x \right )}}{c + d x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 68, normalized size = 1.06 \begin {gather*} \frac {b {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - b {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + 2 \, a \log \left (d x + c\right )}{2 \, d} \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 = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {sinh}\left (e+f\,x\right )}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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