Optimal. Leaf size=25 \[ \frac {a x^2}{2}-\frac {b \cos \left (c+d x^2\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 3379, 2638} \[ \frac {a x^2}{2}-\frac {b \cos \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2638
Rule 3379
Rubi steps
\begin {align*} \int x \left (a+b \sin \left (c+d x^2\right )\right ) \, dx &=\int \left (a x+b x \sin \left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^2}{2}+b \int x \sin \left (c+d x^2\right ) \, dx\\ &=\frac {a x^2}{2}+\frac {1}{2} b \operatorname {Subst}\left (\int \sin (c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}-\frac {b \cos \left (c+d x^2\right )}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 41, normalized size = 1.64 \[ \frac {a x^2}{2}+\frac {b \sin (c) \sin \left (d x^2\right )}{2 d}-\frac {b \cos (c) \cos \left (d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 23, normalized size = 0.92 \[ \frac {a d x^{2} - b \cos \left (d x^{2} + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.71, size = 26, normalized size = 1.04 \[ \frac {{\left (d x^{2} + c\right )} a - b \cos \left (d x^{2} + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 27, normalized size = 1.08 \[ \frac {a \left (d \,x^{2}+c \right )-b \cos \left (d \,x^{2}+c \right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 21, normalized size = 0.84 \[ \frac {1}{2} \, a x^{2} - \frac {b \cos \left (d x^{2} + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.61, size = 21, normalized size = 0.84 \[ \frac {a\,x^2}{2}-\frac {b\,\cos \left (d\,x^2+c\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.24, size = 31, normalized size = 1.24 \[ \begin {cases} \frac {a x^{2}}{2} - \frac {b \cos {\left (c + d x^{2} \right )}}{2 d} & \text {for}\: d \neq 0 \\\frac {x^{2} \left (a + b \sin {\relax (c )}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________