Integrand size = 18, antiderivative size = 88 \[ \int \frac {a+b \sin (e+f x)}{(c+d x)^2} \, dx=-\frac {a}{d (c+d x)}+\frac {b f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {b \sin (e+f x)}{d (c+d x)}-\frac {b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3398, 3378, 3384, 3380, 3383} \[ \int \frac {a+b \sin (e+f x)}{(c+d x)^2} \, dx=-\frac {a}{d (c+d x)}+\frac {b f \operatorname {CosIntegral}\left (x f+\frac {c f}{d}\right ) \cos \left (e-\frac {c f}{d}\right )}{d^2}-\frac {b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {b \sin (e+f x)}{d (c+d x)} \]
[In]
[Out]
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{(c+d x)^2}+\frac {b \sin (e+f x)}{(c+d x)^2}\right ) \, dx \\ & = -\frac {a}{d (c+d x)}+b \int \frac {\sin (e+f x)}{(c+d x)^2} \, dx \\ & = -\frac {a}{d (c+d x)}-\frac {b \sin (e+f x)}{d (c+d x)}+\frac {(b f) \int \frac {\cos (e+f x)}{c+d x} \, dx}{d} \\ & = -\frac {a}{d (c+d x)}-\frac {b \sin (e+f x)}{d (c+d x)}+\frac {\left (b f \cos \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cos \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{d}-\frac {\left (b f \sin \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sin \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx}{d} \\ & = -\frac {a}{d (c+d x)}+\frac {b f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {b \sin (e+f x)}{d (c+d x)}-\frac {b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d^2} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \sin (e+f x)}{(c+d x)^2} \, dx=\frac {b f \cos \left (e-\frac {c f}{d}\right ) \operatorname {CosIntegral}\left (f \left (\frac {c}{d}+x\right )\right )-\frac {d (a+b \sin (e+f x))}{c+d x}-b f \sin \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )}{d^2} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.39
method | result | size |
parts | \(-\frac {a}{d \left (d x +c \right )}+b f \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )\) | \(122\) |
derivativedivides | \(\frac {-\frac {f^{2} a}{\left (c f -d e +d \left (f x +e \right )\right ) d}+f^{2} b \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )}{f}\) | \(141\) |
default | \(\frac {-\frac {f^{2} a}{\left (c f -d e +d \left (f x +e \right )\right ) d}+f^{2} b \left (-\frac {\sin \left (f x +e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}+\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}}{d}\right )}{f}\) | \(141\) |
risch | \(-\frac {a}{d \left (d x +c \right )}-\frac {f b \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{2 d^{2}}-\frac {f b \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-i f x -i e -\frac {i c f -i d e}{d}\right )}{2 d^{2}}-\frac {b \left (-2 d x f -2 c f \right ) \sin \left (f x +e \right )}{2 d \left (d x +c \right ) \left (-d x f -c f \right )}\) | \(154\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \sin (e+f x)}{(c+d x)^2} \, dx=\frac {{\left (b d f x + b c f\right )} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) - b d \sin \left (f x + e\right ) + {\left (b d f x + b c f\right )} \sin \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right ) - a d}{d^{3} x + c d^{2}} \]
[In]
[Out]
\[ \int \frac {a+b \sin (e+f x)}{(c+d x)^2} \, dx=\int \frac {a + b \sin {\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.23 \[ \int \frac {a+b \sin (e+f x)}{(c+d x)^2} \, dx=-\frac {\frac {2 \, a f^{2}}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f} - \frac {{\left (f^{2} {\left (-i \, E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac {d e - c f}{d}\right ) + f^{2} {\left (E_{2}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{2}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )\right )} b}{{\left (f x + e\right )} d^{2} - d^{2} e + c d f}}{2 \, f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (89) = 178\).
Time = 0.33 (sec) , antiderivative size = 533, normalized size of antiderivative = 6.06 \[ \int \frac {a+b \sin (e+f x)}{(c+d x)^2} \, dx=\frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) - d e f^{2} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) + c f^{3} \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) + {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} \sin \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) - d e f^{2} \sin \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) + c f^{3} \sin \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) + d f^{2} \sin \left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )\right )} b d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f} - \frac {a}{{\left (d x + c\right )} d} \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \sin (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+b\,\sin \left (e+f\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]
[In]
[Out]