Optimal. Leaf size=64 \[ \frac {a \log (c+d x)}{d}+\frac {b \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d}+\frac {b \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d} \]
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Rubi [A] time = 0.12, 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 = {3317, 3303, 3298, 3301} \[ \frac {a \log (c+d x)}{d}+\frac {b \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d}+\frac {b \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 3317
Rubi steps
\begin {align*} \int \frac {a+b \cosh (e+f x)}{c+d x} \, dx &=\int \left (\frac {a}{c+d x}+\frac {b \cosh (e+f x)}{c+d x}\right ) \, dx\\ &=\frac {a \log (c+d x)}{d}+b \int \frac {\cosh (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 {\cosh \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 {\sinh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx\\ &=\frac {b \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d}+\frac {a \log (c+d x)}{d}+\frac {b \sinh \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 = 0.15, size = 57, normalized size = 0.89 \[ \frac {a \log (c+d x)+b \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \cosh \left (e-\frac {c f}{d}\right )+b \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 111, normalized size = 1.73 \[ \frac {{\left (b {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + b {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + 2 \, a \log \left (d x + c\right ) - {\left (b {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - b {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 69, normalized size = 1.08 \[ \frac {b {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f}{d} - e\right )} + b {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f}{d} + e\right )} + 2 \, a \log \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 94, normalized size = 1.47 \[ \frac {a \ln \left (d x +c \right )}{d}-\frac {b \,{\mathrm e}^{\frac {c f -d e}{d}} \Ei \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d}-\frac {b \,{\mathrm e}^{-\frac {c f -d e}{d}} \Ei \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 70, normalized size = 1.09 \[ -\frac {1}{2} \, b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac {a \log \left (d x + c\right )}{d} \]
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 \[ \int \frac {a+b\,\mathrm {cosh}\left (e+f\,x\right )}{c+d\,x} \,d x \]
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 \[ \int \frac {a + b \cosh {\left (e + f x \right )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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