Optimal. Leaf size=57 \[ \frac {a x^6}{6}+\frac {b \cos \left (c+d x^2\right )}{d^3}+\frac {b x^2 \sin \left (c+d x^2\right )}{d^2}-\frac {b x^4 \cos \left (c+d x^2\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {14, 3379, 3296, 2638} \[ \frac {a x^6}{6}+\frac {b x^2 \sin \left (c+d x^2\right )}{d^2}+\frac {b \cos \left (c+d x^2\right )}{d^3}-\frac {b x^4 \cos \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2638
Rule 3296
Rule 3379
Rubi steps
\begin {align*} \int x^5 \left (a+b \sin \left (c+d x^2\right )\right ) \, dx &=\int \left (a x^5+b x^5 \sin \left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^6}{6}+b \int x^5 \sin \left (c+d x^2\right ) \, dx\\ &=\frac {a x^6}{6}+\frac {1}{2} b \operatorname {Subst}\left (\int x^2 \sin (c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^6}{6}-\frac {b x^4 \cos \left (c+d x^2\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int x \cos (c+d x) \, dx,x,x^2\right )}{d}\\ &=\frac {a x^6}{6}-\frac {b x^4 \cos \left (c+d x^2\right )}{2 d}+\frac {b x^2 \sin \left (c+d x^2\right )}{d^2}-\frac {b \operatorname {Subst}\left (\int \sin (c+d x) \, dx,x,x^2\right )}{d^2}\\ &=\frac {a x^6}{6}+\frac {b \cos \left (c+d x^2\right )}{d^3}-\frac {b x^4 \cos \left (c+d x^2\right )}{2 d}+\frac {b x^2 \sin \left (c+d x^2\right )}{d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 51, normalized size = 0.89 \[ \frac {a d^3 x^6-3 b \left (d^2 x^4-2\right ) \cos \left (c+d x^2\right )+6 b d x^2 \sin \left (c+d x^2\right )}{6 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.74, size = 51, normalized size = 0.89 \[ \frac {a d^{3} x^{6} + 6 \, b d x^{2} \sin \left (d x^{2} + c\right ) - 3 \, {\left (b d^{2} x^{4} - 2 \, b\right )} \cos \left (d x^{2} + c\right )}{6 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.51, size = 69, normalized size = 1.21 \[ \frac {a d x^{6} + 3 \, {\left (\frac {2 \, x^{2} \sin \left (d x^{2} + c\right )}{d} - \frac {{\left ({\left (d x^{2} + c\right )}^{2} - 2 \, {\left (d x^{2} + c\right )} c + c^{2} - 2\right )} \cos \left (d x^{2} + c\right )}{d^{2}}\right )} b}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 62, normalized size = 1.09 \[ \frac {a \,x^{6}}{6}+b \left (-\frac {x^{4} \cos \left (d \,x^{2}+c \right )}{2 d}+\frac {\frac {x^{2} \sin \left (d \,x^{2}+c \right )}{d}+\frac {\cos \left (d \,x^{2}+c \right )}{d^{2}}}{d}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.65, size = 47, normalized size = 0.82 \[ \frac {1}{6} \, a x^{6} + \frac {{\left (2 \, d x^{2} \sin \left (d x^{2} + c\right ) - {\left (d^{2} x^{4} - 2\right )} \cos \left (d x^{2} + c\right )\right )} b}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.19, size = 53, normalized size = 0.93 \[ \frac {a\,x^6}{6}+\frac {b\,\cos \left (d\,x^2+c\right )-\frac {b\,d^2\,x^4\,\cos \left (d\,x^2+c\right )}{2}+b\,d\,x^2\,\sin \left (d\,x^2+c\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.51, size = 65, normalized size = 1.14 \[ \begin {cases} \frac {a x^{6}}{6} - \frac {b x^{4} \cos {\left (c + d x^{2} \right )}}{2 d} + \frac {b x^{2} \sin {\left (c + d x^{2} \right )}}{d^{2}} + \frac {b \cos {\left (c + d x^{2} \right )}}{d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{6} \left (a + b \sin {\relax (c )}\right )}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________