Optimal. Leaf size=108 \[ -\frac {a \text {Ci}(2 a+2 b x)}{2 b^2}-\frac {\text {Si}(2 a+2 b x)}{2 b^2}+\frac {\text {Si}(a+b x) \cos (a+b x)}{b^2}+\frac {a \log (a+b x)}{2 b^2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b^2}+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}-\frac {x}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6519, 6742, 2635, 8, 3312, 3302, 6511, 4406, 12, 3299} \[ -\frac {a \text {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\text {Si}(2 a+2 b x)}{2 b^2}+\frac {\text {Si}(a+b x) \cos (a+b x)}{b^2}+\frac {a \log (a+b x)}{2 b^2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b^2}+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}-\frac {x}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 2635
Rule 3299
Rule 3302
Rule 3312
Rule 4406
Rule 6511
Rule 6519
Rule 6742
Rubi steps
\begin {align*} \int x \cos (a+b x) \text {Si}(a+b x) \, dx &=\frac {x \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\int \sin (a+b x) \text {Si}(a+b x) \, dx}{b}-\int \frac {x \sin ^2(a+b x)}{a+b x} \, dx\\ &=\frac {\cos (a+b x) \text {Si}(a+b x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\int \frac {\cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{b}-\int \left (\frac {\sin ^2(a+b x)}{b}-\frac {a \sin ^2(a+b x)}{b (a+b x)}\right ) \, dx\\ &=\frac {\cos (a+b x) \text {Si}(a+b x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\int \sin ^2(a+b x) \, dx}{b}-\frac {\int \frac {\sin (2 a+2 b x)}{2 (a+b x)} \, dx}{b}+\frac {a \int \frac {\sin ^2(a+b x)}{a+b x} \, dx}{b}\\ &=\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\cos (a+b x) \text {Si}(a+b x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\int 1 \, dx}{2 b}-\frac {\int \frac {\sin (2 a+2 b x)}{a+b x} \, dx}{2 b}+\frac {a \int \left (\frac {1}{2 (a+b x)}-\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac {x}{2 b}+\frac {a \log (a+b x)}{2 b^2}+\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\cos (a+b x) \text {Si}(a+b x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b^2}-\frac {a \int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=-\frac {x}{2 b}-\frac {a \text {Ci}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}+\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\cos (a+b x) \text {Si}(a+b x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 74, normalized size = 0.69 \[ \frac {-2 a \text {Ci}(2 (a+b x))-2 \text {Si}(2 (a+b x))+4 \text {Si}(a+b x) (b x \sin (a+b x)+\cos (a+b x))+2 a \log (a+b x)+\sin (2 (a+b x))-2 b x}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.26, size = 528, normalized size = 4.89 \[ {\left (\frac {x \sin \left (b x + a\right )}{b} + \frac {\cos \left (b x + a\right )}{b^{2}}\right )} \operatorname {Si}\left (b x + a\right ) - \frac {2 \, b x \tan \left (b x\right )^{2} \tan \relax (a)^{2} - 2 \, a \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} + a \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} + a \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} + \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} - \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} + 2 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} + 2 \, b x \tan \left (b x\right )^{2} - 2 \, a \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} + a \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + a \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + 2 \, b x \tan \relax (a)^{2} - 2 \, a \log \left ({\left | b x + a \right |}\right ) \tan \relax (a)^{2} + a \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \relax (a)^{2} + a \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \relax (a)^{2} + \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} - \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + 2 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} + 2 \, \tan \left (b x\right )^{2} \tan \relax (a) + \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \relax (a)^{2} - \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \relax (a)^{2} + 2 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \relax (a)^{2} + 2 \, \tan \left (b x\right ) \tan \relax (a)^{2} + 2 \, b x - 2 \, a \log \left ({\left | b x + a \right |}\right ) + a \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) + a \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - 2 \, \tan \left (b x\right ) - 2 \, \tan \relax (a)}{4 \, {\left (b^{2} \tan \left (b x\right )^{2} \tan \relax (a)^{2} + b^{2} \tan \left (b x\right )^{2} + b^{2} \tan \relax (a)^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 105, normalized size = 0.97 \[ \frac {x \Si \left (b x +a \right ) \sin \left (b x +a \right )}{b}+\frac {\cos \left (b x +a \right ) \Si \left (b x +a \right )}{b^{2}}-\frac {\Si \left (2 b x +2 a \right )}{2 b^{2}}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2 b^{2}}-\frac {x}{2 b}-\frac {a}{2 b^{2}}+\frac {a \ln \left (b x +a \right )}{2 b^{2}}-\frac {a \Ci \left (2 b x +2 a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Si}\left (b x + a\right ) \cos \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {sinint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________