Optimal. Leaf size=38 \[ a x-b d \cos (c) \text {Ci}\left (\frac {d}{x}\right )+b d \sin (c) \text {Si}\left (\frac {d}{x}\right )+b x \sin \left (c+\frac {d}{x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3361, 3297, 3303, 3299, 3302} \[ a x-b d \cos (c) \text {CosIntegral}\left (\frac {d}{x}\right )+b d \sin (c) \text {Si}\left (\frac {d}{x}\right )+b x \sin \left (c+\frac {d}{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3361
Rubi steps
\begin {align*} \int \left (a+b \sin \left (c+\frac {d}{x}\right )\right ) \, dx &=a x+b \int \sin \left (c+\frac {d}{x}\right ) \, dx\\ &=a x-b \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=a x+b x \sin \left (c+\frac {d}{x}\right )-(b d) \operatorname {Subst}\left (\int \frac {\cos (c+d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a x+b x \sin \left (c+\frac {d}{x}\right )-(b d \cos (c)) \operatorname {Subst}\left (\int \frac {\cos (d x)}{x} \, dx,x,\frac {1}{x}\right )+(b d \sin (c)) \operatorname {Subst}\left (\int \frac {\sin (d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a x-b d \cos (c) \text {Ci}\left (\frac {d}{x}\right )+b x \sin \left (c+\frac {d}{x}\right )+b d \sin (c) \text {Si}\left (\frac {d}{x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 50, normalized size = 1.32 \[ a x-b d \left (\cos (c) \text {Ci}\left (\frac {d}{x}\right )-\sin (c) \text {Si}\left (\frac {d}{x}\right )\right )+b x \sin (c) \cos \left (\frac {d}{x}\right )+b x \cos (c) \sin \left (\frac {d}{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 52, normalized size = 1.37 \[ b d \sin \relax (c) \operatorname {Si}\left (\frac {d}{x}\right ) + b x \sin \left (\frac {c x + d}{x}\right ) + a x - \frac {1}{2} \, {\left (b d \operatorname {Ci}\left (\frac {d}{x}\right ) + b d \operatorname {Ci}\left (-\frac {d}{x}\right )\right )} \cos \relax (c) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.55, size = 137, normalized size = 3.61 \[ a x - \frac {{\left (c d^{2} \cos \relax (c) \operatorname {Ci}\left (-c + \frac {c x + d}{x}\right ) + c d^{2} \sin \relax (c) \operatorname {Si}\left (c - \frac {c x + d}{x}\right ) - \frac {{\left (c x + d\right )} d^{2} \cos \relax (c) \operatorname {Ci}\left (-c + \frac {c x + d}{x}\right )}{x} - \frac {{\left (c x + d\right )} d^{2} \sin \relax (c) \operatorname {Si}\left (c - \frac {c x + d}{x}\right )}{x} + d^{2} \sin \left (\frac {c x + d}{x}\right )\right )} b}{{\left (c - \frac {c x + d}{x}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 43, normalized size = 1.13 \[ a x -b d \left (-\frac {\sin \left (c +\frac {d}{x}\right ) x}{d}-\Si \left (\frac {d}{x}\right ) \sin \relax (c )+\Ci \left (\frac {d}{x}\right ) \cos \relax (c )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.36, size = 65, normalized size = 1.71 \[ -\frac {1}{2} \, {\left ({\left ({\left ({\rm Ei}\left (\frac {i \, d}{x}\right ) + {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \cos \relax (c) - {\left (-i \, {\rm Ei}\left (\frac {i \, d}{x}\right ) + i \, {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \sin \relax (c)\right )} d - 2 \, x \sin \left (\frac {c x + d}{x}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int a+b\,\sin \left (c+\frac {d}{x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + \frac {d}{x} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________