Optimal. Leaf size=63 \[ \frac {3 \sin (b x)}{2 b^4}-\frac {3 x \cos (b x)}{2 b^3}-\frac {3 x^2 \sin (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Si}(b x)+\frac {x^3 \cos (b x)}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6503, 12, 3296, 2637} \[ -\frac {3 x^2 \sin (b x)}{4 b^2}+\frac {3 \sin (b x)}{2 b^4}-\frac {3 x \cos (b x)}{2 b^3}+\frac {1}{4} x^4 \text {Si}(b x)+\frac {x^3 \cos (b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2637
Rule 3296
Rule 6503
Rubi steps
\begin {align*} \int x^3 \text {Si}(b x) \, dx &=\frac {1}{4} x^4 \text {Si}(b x)-\frac {1}{4} b \int \frac {x^3 \sin (b x)}{b} \, dx\\ &=\frac {1}{4} x^4 \text {Si}(b x)-\frac {1}{4} \int x^3 \sin (b x) \, dx\\ &=\frac {x^3 \cos (b x)}{4 b}+\frac {1}{4} x^4 \text {Si}(b x)-\frac {3 \int x^2 \cos (b x) \, dx}{4 b}\\ &=\frac {x^3 \cos (b x)}{4 b}-\frac {3 x^2 \sin (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Si}(b x)+\frac {3 \int x \sin (b x) \, dx}{2 b^2}\\ &=-\frac {3 x \cos (b x)}{2 b^3}+\frac {x^3 \cos (b x)}{4 b}-\frac {3 x^2 \sin (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Si}(b x)+\frac {3 \int \cos (b x) \, dx}{2 b^3}\\ &=-\frac {3 x \cos (b x)}{2 b^3}+\frac {x^3 \cos (b x)}{4 b}+\frac {3 \sin (b x)}{2 b^4}-\frac {3 x^2 \sin (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Si}(b x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 50, normalized size = 0.79 \[ \frac {b^4 x^4 \text {Si}(b x)-3 \left (b^2 x^2-2\right ) \sin (b x)+b x \left (b^2 x^2-6\right ) \cos (b x)}{4 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {Si}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.38, size = 49, normalized size = 0.78 \[ \frac {1}{4} \, x^{4} \operatorname {Si}\left (b x\right ) + \frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right )}{4 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{4 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 56, normalized size = 0.89 \[ \frac {\frac {b^{4} x^{4} \Si \left (b x \right )}{4}+\frac {b^{3} x^{3} \cos \left (b x \right )}{4}-\frac {3 b^{2} x^{2} \sin \left (b x \right )}{4}+\frac {3 \sin \left (b x \right )}{2}-\frac {3 b x \cos \left (b x \right )}{2}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} {\rm Si}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \frac {\sin \left (b\,x\right )\,\left (\frac {6}{b^4}-\frac {3\,x^2}{b^2}\right )}{4}+\frac {x^4\,\mathrm {sinint}\left (b\,x\right )}{4}-\frac {\cos \left (b\,x\right )\,\left (\frac {6\,x}{b^3}-\frac {x^3}{b}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.06, size = 61, normalized size = 0.97 \[ \frac {x^{4} \operatorname {Si}{\left (b x \right )}}{4} + \frac {x^{3} \cos {\left (b x \right )}}{4 b} - \frac {3 x^{2} \sin {\left (b x \right )}}{4 b^{2}} - \frac {3 x \cos {\left (b x \right )}}{2 b^{3}} + \frac {3 \sin {\left (b x \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________