Optimal. Leaf size=87 \[ -\frac {a}{d (c+d x)}+\frac {b f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {b \sinh (e+f x)}{d (c+d x)}+\frac {b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, 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 = {3398, 3378,
3384, 3379, 3382} \begin {gather*} -\frac {a}{d (c+d x)}+\frac {b f \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {b \sinh (e+f x)}{d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 3398
Rubi steps
\begin {align*} \int \frac {a+b \sinh (e+f x)}{(c+d x)^2} \, dx &=\int \left (\frac {a}{(c+d x)^2}+\frac {b \sinh (e+f x)}{(c+d x)^2}\right ) \, dx\\ &=-\frac {a}{d (c+d x)}+b \int \frac {\sinh (e+f x)}{(c+d x)^2} \, dx\\ &=-\frac {a}{d (c+d x)}-\frac {b \sinh (e+f x)}{d (c+d x)}+\frac {(b f) \int \frac {\cosh (e+f x)}{c+d x} \, dx}{d}\\ &=-\frac {a}{d (c+d x)}-\frac {b \sinh (e+f x)}{d (c+d x)}+\frac {\left (b f \cosh \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 {\left (b f \sinh \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 {a}{d (c+d x)}+\frac {b f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {b \sinh (e+f x)}{d (c+d x)}+\frac {b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.23, size = 71, normalized size = 0.82 \begin {gather*} \frac {b f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )-\frac {d (a+b \sinh (e+f x))}{c+d x}+b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )}{d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.44, size = 149, normalized size = 1.71
method | result | size |
risch | \(-\frac {a}{d \left (d x +c \right )}+\frac {f b \,{\mathrm e}^{-f x -e}}{2 d \left (d x f +c f \right )}-\frac {f b \,{\mathrm e}^{\frac {c f -d e}{d}} \expIntegral \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d^{2}}-\frac {b f \,{\mathrm e}^{f x +e}}{2 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {b f \,{\mathrm e}^{-\frac {c f -d e}{d}} \expIntegral \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d^{2}}\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 90, normalized size = 1.03 \begin {gather*} \frac {1}{2} \, b {\left (\frac {e^{\left (\frac {c f}{d} - e\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {e^{\left (-\frac {c f}{d} + e\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 178, normalized size = 2.05 \begin {gather*} -\frac {2 \, b d \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, a d - {\left ({\left (b d f x + b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (b d f x + b c f\right )} {\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 ) - {\left ({\left (b d f x + b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (b d f x + b c f\right )} {\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 \, {\left (d^{3} x + c d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 630 vs.
\(2 (90) = 180\).
time = 0.44, size = 630, normalized size = 7.24 \begin {gather*} \frac {1}{2} \, b {\left (\frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - d e f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + c f^{3} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - d f^{2} e^{\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f} + \frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - d e f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + c f^{3} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + d f^{2} e^{\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f}\right )} - \frac {a}{{\left (d x + c\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {sinh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________