Optimal. Leaf size=71 \[ -\frac {a^2 \text {Ci}(a+b x)}{2 b^2}+\frac {a \sin (a+b x)}{2 b^2}-\frac {\cos (a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Ci}(a+b x)-\frac {x \sin (a+b x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6504, 6742, 2637, 3296, 2638, 3302} \[ -\frac {a^2 \text {CosIntegral}(a+b x)}{2 b^2}+\frac {a \sin (a+b x)}{2 b^2}-\frac {\cos (a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {CosIntegral}(a+b x)-\frac {x \sin (a+b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 2638
Rule 3296
Rule 3302
Rule 6504
Rule 6742
Rubi steps
\begin {align*} \int x \text {Ci}(a+b x) \, dx &=\frac {1}{2} x^2 \text {Ci}(a+b x)-\frac {1}{2} b \int \frac {x^2 \cos (a+b x)}{a+b x} \, dx\\ &=\frac {1}{2} x^2 \text {Ci}(a+b x)-\frac {1}{2} b \int \left (-\frac {a \cos (a+b x)}{b^2}+\frac {x \cos (a+b x)}{b}+\frac {a^2 \cos (a+b x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {1}{2} x^2 \text {Ci}(a+b x)-\frac {1}{2} \int x \cos (a+b x) \, dx+\frac {a \int \cos (a+b x) \, dx}{2 b}-\frac {a^2 \int \frac {\cos (a+b x)}{a+b x} \, dx}{2 b}\\ &=-\frac {a^2 \text {Ci}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Ci}(a+b x)+\frac {a \sin (a+b x)}{2 b^2}-\frac {x \sin (a+b x)}{2 b}+\frac {\int \sin (a+b x) \, dx}{2 b}\\ &=-\frac {\cos (a+b x)}{2 b^2}-\frac {a^2 \text {Ci}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Ci}(a+b x)+\frac {a \sin (a+b x)}{2 b^2}-\frac {x \sin (a+b x)}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 49, normalized size = 0.69 \[ \frac {\left (b^2 x^2-a^2\right ) \text {Ci}(a+b x)+(a-b x) \sin (a+b x)-\cos (a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Ci}\left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 107, normalized size = 1.51 \[ \frac {1}{2} \, x^{2} \operatorname {Ci}\left (b x + a\right ) - \frac {a^{2} \cos \relax (a)^{2} \operatorname {Ci}\left (b x + a\right ) + a^{2} \cos \relax (a)^{2} \operatorname {Ci}\left (-b x - a\right ) + a^{2} \operatorname {Ci}\left (b x + a\right ) \sin \relax (a)^{2} + a^{2} \operatorname {Ci}\left (-b x - a\right ) \sin \relax (a)^{2} + 2 \, b x \sin \left (b x + a\right ) - 2 \, a \sin \left (b x + a\right ) + 2 \, \cos \left (b x + a\right )}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 60, normalized size = 0.85 \[ \frac {\Ci \left (b x +a \right ) \left (\frac {\left (b x +a \right )^{2}}{2}-a \left (b x +a \right )\right )-\frac {\cos \left (b x +a \right )}{2}-\frac {\left (b x +a \right ) \sin \left (b x +a \right )}{2}+a \sin \left (b x +a \right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Ci}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \frac {x^2\,\mathrm {cosint}\left (a+b\,x\right )}{2}-\frac {\cos \left (a+b\,x\right )-a\,\sin \left (a+b\,x\right )+a^2\,\mathrm {cosint}\left (a+b\,x\right )+b\,x\,\sin \left (a+b\,x\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {Ci}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________