Optimal. Leaf size=87 \[ \frac {a f \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {a f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {a \cosh (e+f x)}{d (c+d x)}-\frac {a}{d (c+d x)} \]
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Rubi [A] time = 0.17, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3317, 3297, 3303, 3298, 3301} \[ \frac {a f \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {a f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {a \cosh (e+f x)}{d (c+d x)}-\frac {a}{d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 3317
Rubi steps
\begin {align*} \int \frac {a+a \cosh (e+f x)}{(c+d x)^2} \, dx &=\int \left (\frac {a}{(c+d x)^2}+\frac {a \cosh (e+f x)}{(c+d x)^2}\right ) \, dx\\ &=-\frac {a}{d (c+d x)}+a \int \frac {\cosh (e+f x)}{(c+d x)^2} \, dx\\ &=-\frac {a}{d (c+d x)}-\frac {a \cosh (e+f x)}{d (c+d x)}+\frac {(a f) \int \frac {\sinh (e+f x)}{c+d x} \, dx}{d}\\ &=-\frac {a}{d (c+d x)}-\frac {a \cosh (e+f x)}{d (c+d x)}+\frac {\left (a f \cosh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{d}+\frac {\left (a f \sinh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{d}\\ &=-\frac {a}{d (c+d x)}-\frac {a \cosh (e+f x)}{d (c+d x)}+\frac {a f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {a f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 68, normalized size = 0.78 \[ \frac {a \left (f \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-\frac {d (\cosh (e+f x)+1)}{c+d x}\right )}{d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 162, normalized size = 1.86 \[ -\frac {2 \, a d \cosh \left (f x + e\right ) + 2 \, a d - {\left ({\left (a d f x + a c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a d f x + a c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (a d f x + a c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a d f x + a c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 683, normalized size = 7.85 \[ -\frac {1}{2} \, a {\left (\frac {{\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (\frac {c f - d e}{d}\right )} - c f^{3} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (\frac {c f - d e}{d}\right )} + d f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (\frac {c f - d e}{d} + 1\right )} - d f^{2} e^{\left (\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c d^{4} f + d^{5} e\right )} f} - \frac {{\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (-\frac {c f - d e}{d}\right )} - c f^{3} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (-\frac {c f - d e}{d}\right )} + d f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e}{d}\right ) e^{\left (-\frac {c f - d e}{d} + 1\right )} + d f^{2} e^{\left (-\frac {{\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c d^{4} f + d^{5} e\right )} f}\right )} - \frac {a}{{\left (d x + c\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 149, normalized size = 1.71 \[ -\frac {a}{d \left (d x +c \right )}-\frac {f a \,{\mathrm e}^{-f x -e}}{2 d \left (d f x +c f \right )}+\frac {f a \,{\mathrm e}^{\frac {c f -d e}{d}} \Ei \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d^{2}}-\frac {a f \,{\mathrm e}^{f x +e}}{2 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {a f \,{\mathrm e}^{-\frac {c f -d e}{d}} \Ei \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 87, normalized size = 1.00 \[ -\frac {1}{2} \, a {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a}{d^{2} x + c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+a\,\mathrm {cosh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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