Optimal. Leaf size=383 \[ \frac {a x^3}{3}+\frac {i b e^{i c} f^2 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{-i c} f^2 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{i c} f (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )}{g^3 n}+\frac {i b e^{-i c} f (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )}{g^3 n}+\frac {i b e^{i c} (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{-i c} (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^3 n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.24, antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {14, 3514,
3446, 2239, 3504, 2250} \begin {gather*} \frac {i b e^{i c} f^2 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{-i c} f^2 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^3 n}+\frac {i b e^{i c} (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{i c} f (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-i d (f+g x)^n\right )}{g^3 n}+\frac {i b e^{-i c} f (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},i d (f+g x)^n\right )}{g^3 n}-\frac {i b e^{-i c} (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^3 n}+\frac {a x^3}{3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2239
Rule 2250
Rule 3446
Rule 3504
Rule 3514
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx &=\int \left (a x^2+b x^2 \sin \left (c+d (f+g x)^n\right )\right ) \, dx\\ &=\frac {a x^3}{3}+b \int x^2 \sin \left (c+d (f+g x)^n\right ) \, dx\\ &=\frac {a x^3}{3}+\frac {b \text {Subst}\left (\int \left (f^2 \sin \left (c+d x^n\right )-2 f x \sin \left (c+d x^n\right )+x^2 \sin \left (c+d x^n\right )\right ) \, dx,x,f+g x\right )}{g^3}\\ &=\frac {a x^3}{3}+\frac {b \text {Subst}\left (\int x^2 \sin \left (c+d x^n\right ) \, dx,x,f+g x\right )}{g^3}-\frac {(2 b f) \text {Subst}\left (\int x \sin \left (c+d x^n\right ) \, dx,x,f+g x\right )}{g^3}+\frac {\left (b f^2\right ) \text {Subst}\left (\int \sin \left (c+d x^n\right ) \, dx,x,f+g x\right )}{g^3}\\ &=\frac {a x^3}{3}+\frac {(i b) \text {Subst}\left (\int e^{-i c-i d x^n} x^2 \, dx,x,f+g x\right )}{2 g^3}-\frac {(i b) \text {Subst}\left (\int e^{i c+i d x^n} x^2 \, dx,x,f+g x\right )}{2 g^3}-\frac {(i b f) \text {Subst}\left (\int e^{-i c-i d x^n} x \, dx,x,f+g x\right )}{g^3}+\frac {(i b f) \text {Subst}\left (\int e^{i c+i d x^n} x \, dx,x,f+g x\right )}{g^3}+\frac {\left (i b f^2\right ) \text {Subst}\left (\int e^{-i c-i d x^n} \, dx,x,f+g x\right )}{2 g^3}-\frac {\left (i b f^2\right ) \text {Subst}\left (\int e^{i c+i d x^n} \, dx,x,f+g x\right )}{2 g^3}\\ &=\frac {a x^3}{3}+\frac {i b e^{i c} f^2 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{-i c} f^2 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{i c} f (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )}{g^3 n}+\frac {i b e^{-i c} f (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )}{g^3 n}+\frac {i b e^{i c} (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{-i c} (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^3 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 5.69, size = 403, normalized size = 1.05 \begin {gather*} \frac {a x^3}{3}+\frac {i b (f+g x) \left (d^2 (f+g x)^{2 n}\right )^{-3/n} \left (-f^2 \left (-i d (f+g x)^n\right )^{3/n} \left (i d (f+g x)^n\right )^{2/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )-(f+g x) \left (-2 f \left (-i d (f+g x)^n\right )^{3/n} \left (i d (f+g x)^n\right )^{\frac {1}{n}} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )-(f+g x) \left (-\left (-i d (f+g x)^n\right )^{3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )+\left (i d (f+g x)^n\right )^{3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right ) (\cos (c)+i \sin (c))^2\right )+2 f \left (-i d (f+g x)^n\right )^{\frac {1}{n}} \left (i d (f+g x)^n\right )^{3/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right ) (\cos (c)+i \sin (c))^2\right )+f^2 \left (-i d (f+g x)^n\right )^{2/n} \left (i d (f+g x)^n\right )^{3/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right ) (\cos (c)+i \sin (c))^2\right ) (\cos (c)-i \sin (c))}{2 g^3 n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \sin {\left (c + d \left (f + g x\right )^{n} \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\left (a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________