3.3.67 \(\int x^2 (a+b \sin (c+d (f+g x)^n)) \, dx\) [267]

3.3.67.1 Optimal result

Integrand size = 20, antiderivative size = 383 \[ \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\frac {a x^3}{3}+\frac {i b e^{i c} f^2 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{-i c} f^2 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{i c} f (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )}{g^3 n}+\frac {i b e^{-i c} f (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )}{g^3 n}+\frac {i b e^{i c} (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{-i c} (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^3 n} \]

output

1/3*a*x^3+1/2*I*b*exp(I*c)*f^2*(g*x+f)*GAMMA(1/n,-I*d*(g*x+f)^n)/g^3/n/((- 
I*d*(g*x+f)^n)^(1/n))-1/2*I*b*f^2*(g*x+f)*GAMMA(1/n,I*d*(g*x+f)^n)/exp(I*c 
)/g^3/n/((I*d*(g*x+f)^n)^(1/n))-I*b*exp(I*c)*f*(g*x+f)^2*GAMMA(2/n,-I*d*(g 
*x+f)^n)/g^3/n/((-I*d*(g*x+f)^n)^(2/n))+I*b*f*(g*x+f)^2*GAMMA(2/n,I*d*(g*x 
+f)^n)/exp(I*c)/g^3/n/((I*d*(g*x+f)^n)^(2/n))+1/2*I*b*exp(I*c)*(g*x+f)^3*G 
AMMA(3/n,-I*d*(g*x+f)^n)/g^3/n/((-I*d*(g*x+f)^n)^(3/n))-1/2*I*b*(g*x+f)^3* 
GAMMA(3/n,I*d*(g*x+f)^n)/exp(I*c)/g^3/n/((I*d*(g*x+f)^n)^(3/n))
 

3.3.67.2 Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.82 \[ \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\frac {a x^3}{3}+\frac {i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-3/n} \left (f^2 \left (-i d (f+g x)^n\right )^{2/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )-(f+g x) \left (2 f \left (-i d (f+g x)^n\right )^{\frac {1}{n}} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )-(f+g x) \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )\right )\right )}{2 g^3 n}-\frac {i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-3/n} \left (f^2 \left (i d (f+g x)^n\right )^{2/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )-(f+g x) \left (2 f \left (i d (f+g x)^n\right )^{\frac {1}{n}} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )-(f+g x) \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )\right )\right )}{2 g^3 n} \]

input

Integrate[x^2*(a + b*Sin[c + d*(f + g*x)^n]),x]
 

output

(a*x^3)/3 + ((I/2)*b*E^(I*c)*(f + g*x)*(f^2*((-I)*d*(f + g*x)^n)^(2/n)*Gam 
ma[n^(-1), (-I)*d*(f + g*x)^n] - (f + g*x)*(2*f*((-I)*d*(f + g*x)^n)^n^(-1 
)*Gamma[2/n, (-I)*d*(f + g*x)^n] - (f + g*x)*Gamma[3/n, (-I)*d*(f + g*x)^n 
])))/(g^3*n*((-I)*d*(f + g*x)^n)^(3/n)) - ((I/2)*b*(f + g*x)*(f^2*(I*d*(f 
+ g*x)^n)^(2/n)*Gamma[n^(-1), I*d*(f + g*x)^n] - (f + g*x)*(2*f*(I*d*(f + 
g*x)^n)^n^(-1)*Gamma[2/n, I*d*(f + g*x)^n] - (f + g*x)*Gamma[3/n, I*d*(f + 
 g*x)^n])))/(E^(I*c)*g^3*n*(I*d*(f + g*x)^n)^(3/n))
 

3.3.67.3 Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 383, 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.

\(\displaystyle \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \int \left (a x^2+b x^2 \sin \left (c+d (f+g x)^n\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a x^3}{3}+\frac {i b e^{i c} f^2 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{-i c} f^2 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^3 n}+\frac {i b e^{i c} (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^3 n}-\frac {i b e^{i c} f (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )}{g^3 n}+\frac {i b e^{-i c} f (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )}{g^3 n}-\frac {i b e^{-i c} (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^3 n}\)

input

Int[x^2*(a + b*Sin[c + d*(f + g*x)^n]),x]
 

output

(a*x^3)/3 + ((I/2)*b*E^(I*c)*f^2*(f + g*x)*Gamma[n^(-1), (-I)*d*(f + g*x)^ 
n])/(g^3*n*((-I)*d*(f + g*x)^n)^n^(-1)) - ((I/2)*b*f^2*(f + g*x)*Gamma[n^( 
-1), I*d*(f + g*x)^n])/(E^(I*c)*g^3*n*(I*d*(f + g*x)^n)^n^(-1)) - (I*b*E^( 
I*c)*f*(f + g*x)^2*Gamma[2/n, (-I)*d*(f + g*x)^n])/(g^3*n*((-I)*d*(f + g*x 
)^n)^(2/n)) + (I*b*f*(f + g*x)^2*Gamma[2/n, I*d*(f + g*x)^n])/(E^(I*c)*g^3 
*n*(I*d*(f + g*x)^n)^(2/n)) + ((I/2)*b*E^(I*c)*(f + g*x)^3*Gamma[3/n, (-I) 
*d*(f + g*x)^n])/(g^3*n*((-I)*d*(f + g*x)^n)^(3/n)) - ((I/2)*b*(f + g*x)^3 
*Gamma[3/n, I*d*(f + g*x)^n])/(E^(I*c)*g^3*n*(I*d*(f + g*x)^n)^(3/n))
 

3.3.67.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

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]]
 

3.3.67.4 Maple [F]

\[\int x^{2} \left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )d x\]

input

int(x^2*(a+b*sin(c+d*(g*x+f)^n)),x)
 

output

int(x^2*(a+b*sin(c+d*(g*x+f)^n)),x)
 

3.3.67.5 Fricas [F]

\[ \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int { {\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x^{2} \,d x } \]

input

integrate(x^2*(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="fricas")
 

output

integral(b*x^2*sin((g*x + f)^n*d + c) + a*x^2, x)
 

3.3.67.6 Sympy [F]

\[ \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int x^{2} \left (a + b \sin {\left (c + d \left (f + g x\right )^{n} \right )}\right )\, dx \]

input

integrate(x**2*(a+b*sin(c+d*(g*x+f)**n)),x)
 

output

Integral(x**2*(a + b*sin(c + d*(f + g*x)**n)), x)
 

3.3.67.7 Maxima [F]

\[ \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int { {\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x^{2} \,d x } \]

input

integrate(x^2*(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="maxima")
 

output

1/3*a*x^3 + b*integrate(x^2*sin((g*x + f)^n*d + c), x)
 

3.3.67.8 Giac [F]

\[ \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int { {\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x^{2} \,d x } \]

input

integrate(x^2*(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="giac")
 

output

integrate((b*sin((g*x + f)^n*d + c) + a)*x^2, x)
 

3.3.67.9 Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx=\int x^2\,\left (a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )\right ) \,d x \]

input

int(x^2*(a + b*sin(c + d*(f + g*x)^n)),x)
 

output

int(x^2*(a + b*sin(c + d*(f + g*x)^n)), x)