Integrand size = 20, antiderivative size = 519 \[ \int x^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\frac {a x^4}{4}-\frac {i b e^{i c} f^3 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{-i c} f^3 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{i c} f^2 (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{-i c} f^2 (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{i c} f (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{-i c} f (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{i c} (f+g x)^4 \left (-i d (f+g x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {i b e^{-i c} (f+g x)^4 \left (i d (f+g x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},i d (f+g x)^n\right )}{2 g^4 n} \]
1/4*a*x^4-1/2*I*b*exp(I*c)*f^3*(g*x+f)*GAMMA(1/n,-I*d*(g*x+f)^n)/g^4/n/((- I*d*(g*x+f)^n)^(1/n))+1/2*I*b*f^3*(g*x+f)*GAMMA(1/n,I*d*(g*x+f)^n)/exp(I*c )/g^4/n/((I*d*(g*x+f)^n)^(1/n))+3/2*I*b*exp(I*c)*f^2*(g*x+f)^2*GAMMA(2/n,- I*d*(g*x+f)^n)/g^4/n/((-I*d*(g*x+f)^n)^(2/n))-3/2*I*b*f^2*(g*x+f)^2*GAMMA( 2/n,I*d*(g*x+f)^n)/exp(I*c)/g^4/n/((I*d*(g*x+f)^n)^(2/n))-3/2*I*b*exp(I*c) *f*(g*x+f)^3*GAMMA(3/n,-I*d*(g*x+f)^n)/g^4/n/((-I*d*(g*x+f)^n)^(3/n))+3/2* I*b*f*(g*x+f)^3*GAMMA(3/n,I*d*(g*x+f)^n)/exp(I*c)/g^4/n/((I*d*(g*x+f)^n)^( 3/n))+1/2*I*b*exp(I*c)*(g*x+f)^4*GAMMA(4/n,-I*d*(g*x+f)^n)/g^4/n/((-I*d*(g *x+f)^n)^(4/n))-1/2*I*b*(g*x+f)^4*GAMMA(4/n,I*d*(g*x+f)^n)/exp(I*c)/g^4/n/ ((I*d*(g*x+f)^n)^(4/n))
Time = 0.62 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.79 \[ \int x^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\frac {a x^4}{4}-\frac {i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-4/n} \left (f^3 \left (-i d (f+g x)^n\right )^{3/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )-(f+g x) \left (3 f^2 \left (-i d (f+g x)^n\right )^{2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )-(f+g x) \left (3 f \left (-i d (f+g x)^n\right )^{\frac {1}{n}} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )-(f+g x) \Gamma \left (\frac {4}{n},-i d (f+g x)^n\right )\right )\right )\right )}{2 g^4 n}+\frac {i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-4/n} \left (f^3 \left (i d (f+g x)^n\right )^{3/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )-(f+g x) \left (3 f^2 \left (i d (f+g x)^n\right )^{2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )-(f+g x) \left (3 f \left (i d (f+g x)^n\right )^{\frac {1}{n}} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )-(f+g x) \Gamma \left (\frac {4}{n},i d (f+g x)^n\right )\right )\right )\right )}{2 g^4 n} \]
(a*x^4)/4 - ((I/2)*b*E^(I*c)*(f + g*x)*(f^3*((-I)*d*(f + g*x)^n)^(3/n)*Gam ma[n^(-1), (-I)*d*(f + g*x)^n] - (f + g*x)*(3*f^2*((-I)*d*(f + g*x)^n)^(2/ n)*Gamma[2/n, (-I)*d*(f + g*x)^n] - (f + g*x)*(3*f*((-I)*d*(f + g*x)^n)^n^ (-1)*Gamma[3/n, (-I)*d*(f + g*x)^n] - (f + g*x)*Gamma[4/n, (-I)*d*(f + g*x )^n]))))/(g^4*n*((-I)*d*(f + g*x)^n)^(4/n)) + ((I/2)*b*(f + g*x)*(f^3*(I*d *(f + g*x)^n)^(3/n)*Gamma[n^(-1), I*d*(f + g*x)^n] - (f + g*x)*(3*f^2*(I*d *(f + g*x)^n)^(2/n)*Gamma[2/n, I*d*(f + g*x)^n] - (f + g*x)*(3*f*(I*d*(f + g*x)^n)^n^(-1)*Gamma[3/n, I*d*(f + g*x)^n] - (f + g*x)*Gamma[4/n, I*d*(f + g*x)^n]))))/(E^(I*c)*g^4*n*(I*d*(f + g*x)^n)^(4/n))
Time = 0.80 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2010, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a x^3+b x^3 \sin \left (c+d (f+g x)^n\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a x^4}{4}-\frac {i b e^{i c} f^3 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{-i c} f^3 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{i c} f^2 (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{-i c} f^2 (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{i c} (f+g x)^4 \left (-i d (f+g x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{i c} f (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{-i c} f (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^4 n}-\frac {i b e^{-i c} (f+g x)^4 \left (i d (f+g x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},i d (f+g x)^n\right )}{2 g^4 n}\) |
(a*x^4)/4 - ((I/2)*b*E^(I*c)*f^3*(f + g*x)*Gamma[n^(-1), (-I)*d*(f + g*x)^ n])/(g^4*n*((-I)*d*(f + g*x)^n)^n^(-1)) + ((I/2)*b*f^3*(f + g*x)*Gamma[n^( -1), I*d*(f + g*x)^n])/(E^(I*c)*g^4*n*(I*d*(f + g*x)^n)^n^(-1)) + (((3*I)/ 2)*b*E^(I*c)*f^2*(f + g*x)^2*Gamma[2/n, (-I)*d*(f + g*x)^n])/(g^4*n*((-I)* d*(f + g*x)^n)^(2/n)) - (((3*I)/2)*b*f^2*(f + g*x)^2*Gamma[2/n, I*d*(f + g *x)^n])/(E^(I*c)*g^4*n*(I*d*(f + g*x)^n)^(2/n)) - (((3*I)/2)*b*E^(I*c)*f*( f + g*x)^3*Gamma[3/n, (-I)*d*(f + g*x)^n])/(g^4*n*((-I)*d*(f + g*x)^n)^(3/ n)) + (((3*I)/2)*b*f*(f + g*x)^3*Gamma[3/n, I*d*(f + g*x)^n])/(E^(I*c)*g^4 *n*(I*d*(f + g*x)^n)^(3/n)) + ((I/2)*b*E^(I*c)*(f + g*x)^4*Gamma[4/n, (-I) *d*(f + g*x)^n])/(g^4*n*((-I)*d*(f + g*x)^n)^(4/n)) - ((I/2)*b*(f + g*x)^4 *Gamma[4/n, I*d*(f + g*x)^n])/(E^(I*c)*g^4*n*(I*d*(f + g*x)^n)^(4/n))
3.3.66.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
\[\int x^{3} \left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )d x\]
\[ \int x^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int { {\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x^{3} \,d x } \]
\[ \int x^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int x^{3} \left (a + b \sin {\left (c + d \left (f + g x\right )^{n} \right )}\right )\, dx \]
\[ \int x^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int { {\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x^{3} \,d x } \]
\[ \int x^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int { {\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int x^3\,\left (a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )\right ) \,d x \]