Optimal. Leaf size=74 \[ -\frac {1}{2} a d^2 \sin (c) \text {Ci}(d x)-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{2 x^2}-\frac {a d \cos (c+d x)}{2 x}+b \sin (c) \text {Ci}(d x)+b \cos (c) \text {Si}(d x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3339, 3297, 3303, 3299, 3302} \[ -\frac {1}{2} a d^2 \sin (c) \text {CosIntegral}(d x)-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{2 x^2}-\frac {a d \cos (c+d x)}{2 x}+b \sin (c) \text {CosIntegral}(d x)+b \cos (c) \text {Si}(d x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3339
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \sin (c+d x)}{x^3} \, dx &=\int \left (\frac {a \sin (c+d x)}{x^3}+\frac {b \sin (c+d x)}{x}\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x^3} \, dx+b \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a \sin (c+d x)}{2 x^2}+\frac {1}{2} (a d) \int \frac {\cos (c+d x)}{x^2} \, dx+(b \cos (c)) \int \frac {\sin (d x)}{x} \, dx+(b \sin (c)) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{2 x}+b \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{2 x^2}+b \cos (c) \text {Si}(d x)-\frac {1}{2} \left (a d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{2 x}+b \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{2 x^2}+b \cos (c) \text {Si}(d x)-\frac {1}{2} \left (a d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx-\frac {1}{2} \left (a d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{2 x}+b \text {Ci}(d x) \sin (c)-\frac {1}{2} a d^2 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{2 x^2}+b \cos (c) \text {Si}(d x)-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 82, normalized size = 1.11 \[ -\frac {1}{2} a d^2 (\sin (c) \text {Ci}(d x)+\cos (c) \text {Si}(d x))-\frac {a \cos (d x) (d x \cos (c)+\sin (c))}{2 x^2}+\frac {a \sin (d x) (d x \sin (c)-\cos (c))}{2 x^2}+b \sin (c) \text {Ci}(d x)+b \cos (c) \text {Si}(d x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 85, normalized size = 1.15 \[ -\frac {2 \, {\left (a d^{2} - 2 \, b\right )} x^{2} \cos \relax (c) \operatorname {Si}\left (d x\right ) + 2 \, a d x \cos \left (d x + c\right ) + 2 \, a \sin \left (d x + c\right ) + {\left ({\left (a d^{2} - 2 \, b\right )} x^{2} \operatorname {Ci}\left (d x\right ) + {\left (a d^{2} - 2 \, b\right )} x^{2} \operatorname {Ci}\left (-d x\right )\right )} \sin \relax (c)}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.32, size = 766, normalized size = 10.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 73, normalized size = 0.99 \[ d^{2} \left (\frac {b \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right )}{d^{2}}+a \left (-\frac {\sin \left (d x +c \right )}{2 x^{2} d^{2}}-\frac {\cos \left (d x +c \right )}{2 x d}-\frac {\Si \left (d x \right ) \cos \relax (c )}{2}-\frac {\Ci \left (d x \right ) \sin \relax (c )}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 1.03, size = 122, normalized size = 1.65 \[ -\frac {2 \, b d x \cos \left (d x + c\right ) + {\left ({\left (a {\left (-i \, \Gamma \left (-2, i \, d x\right ) + i \, \Gamma \left (-2, -i \, d x\right )\right )} \cos \relax (c) - a {\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{4} + {\left (b {\left (2 i \, \Gamma \left (-2, i \, d x\right ) - 2 i \, \Gamma \left (-2, -i \, d x\right )\right )} \cos \relax (c) + 2 \, b {\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{2}\right )} x^{2} + 2 \, b \sin \left (d x + c\right )}{2 \, d^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^2+a\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{2}\right ) \sin {\left (c + d x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________