Integrand size = 10, antiderivative size = 187 \[ \int \sin ^3\left (a+b x^n\right ) \, dx=\frac {3 i e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{8 n}-\frac {3 i e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{8 n}-\frac {i 3^{-1/n} e^{3 i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 i b x^n\right )}{8 n}+\frac {i 3^{-1/n} e^{-3 i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 i b x^n\right )}{8 n} \] Output:
3/8*I*exp(I*a)*x*GAMMA(1/n,-I*b*x^n)/n/((-I*b*x^n)^(1/n))-3/8*I*x*GAMMA(1/ n,I*b*x^n)/exp(I*a)/n/((I*b*x^n)^(1/n))-1/8*I*exp(3*I*a)*x*GAMMA(1/n,-3*I* b*x^n)/(3^(1/n))/n/((-I*b*x^n)^(1/n))+1/8*I*x*GAMMA(1/n,3*I*b*x^n)/(3^(1/n ))/exp(3*I*a)/n/((I*b*x^n)^(1/n))
Time = 0.42 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \sin ^3\left (a+b x^n\right ) \, dx=\frac {i 3^{-1/n} e^{-3 i a} x \left (b^2 x^{2 n}\right )^{-1/n} \left (3^{1+\frac {1}{n}} e^{4 i a} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-i b x^n\right )-3^{1+\frac {1}{n}} e^{2 i a} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},i b x^n\right )-e^{6 i a} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-3 i b x^n\right )+\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},3 i b x^n\right )\right )}{8 n} \] Input:
Integrate[Sin[a + b*x^n]^3,x]
Output:
((I/8)*x*(3^(1 + n^(-1))*E^((4*I)*a)*(I*b*x^n)^n^(-1)*Gamma[n^(-1), (-I)*b *x^n] - 3^(1 + n^(-1))*E^((2*I)*a)*((-I)*b*x^n)^n^(-1)*Gamma[n^(-1), I*b*x ^n] - E^((6*I)*a)*(I*b*x^n)^n^(-1)*Gamma[n^(-1), (-3*I)*b*x^n] + ((-I)*b*x ^n)^n^(-1)*Gamma[n^(-1), (3*I)*b*x^n]))/(3^n^(-1)*E^((3*I)*a)*n*(b^2*x^(2* n))^n^(-1))
Time = 0.31 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3848, 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.
\(\displaystyle \int \sin ^3\left (a+b x^n\right ) \, dx\) |
\(\Big \downarrow \) 3848 |
\(\displaystyle \int \left (\frac {3}{4} \sin \left (a+b x^n\right )-\frac {1}{4} \sin \left (3 a+3 b x^n\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 i e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{8 n}-\frac {i e^{3 i a} 3^{-1/n} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 i b x^n\right )}{8 n}-\frac {3 i e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{8 n}+\frac {i e^{-3 i a} 3^{-1/n} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 i b x^n\right )}{8 n}\) |
Input:
Int[Sin[a + b*x^n]^3,x]
Output:
(((3*I)/8)*E^(I*a)*x*Gamma[n^(-1), (-I)*b*x^n])/(n*((-I)*b*x^n)^n^(-1)) - (((3*I)/8)*x*Gamma[n^(-1), I*b*x^n])/(E^(I*a)*n*(I*b*x^n)^n^(-1)) - ((I/8) *E^((3*I)*a)*x*Gamma[n^(-1), (-3*I)*b*x^n])/(3^n^(-1)*n*((-I)*b*x^n)^n^(-1 )) + ((I/8)*x*Gamma[n^(-1), (3*I)*b*x^n])/(3^n^(-1)*E^((3*I)*a)*n*(I*b*x^n )^n^(-1))
Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_), x_Sy mbol] :> Int[ExpandTrigReduce[(a + b*Sin[c + d*(e + f*x)^n])^p, x], x] /; F reeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]
\[\int \sin \left (a +b \,x^{n}\right )^{3}d x\]
Input:
int(sin(a+b*x^n)^3,x)
Output:
int(sin(a+b*x^n)^3,x)
\[ \int \sin ^3\left (a+b x^n\right ) \, dx=\int { \sin \left (b x^{n} + a\right )^{3} \,d x } \] Input:
integrate(sin(a+b*x^n)^3,x, algorithm="fricas")
Output:
integral(-(cos(b*x^n + a)^2 - 1)*sin(b*x^n + a), x)
\[ \int \sin ^3\left (a+b x^n\right ) \, dx=\int \sin ^{3}{\left (a + b x^{n} \right )}\, dx \] Input:
integrate(sin(a+b*x**n)**3,x)
Output:
Integral(sin(a + b*x**n)**3, x)
\[ \int \sin ^3\left (a+b x^n\right ) \, dx=\int { \sin \left (b x^{n} + a\right )^{3} \,d x } \] Input:
integrate(sin(a+b*x^n)^3,x, algorithm="maxima")
Output:
integrate(sin(b*x^n + a)^3, x)
\[ \int \sin ^3\left (a+b x^n\right ) \, dx=\int { \sin \left (b x^{n} + a\right )^{3} \,d x } \] Input:
integrate(sin(a+b*x^n)^3,x, algorithm="giac")
Output:
integrate(sin(b*x^n + a)^3, x)
Timed out. \[ \int \sin ^3\left (a+b x^n\right ) \, dx=\int {\sin \left (a+b\,x^n\right )}^3 \,d x \] Input:
int(sin(a + b*x^n)^3,x)
Output:
int(sin(a + b*x^n)^3, x)
\[ \int \sin ^3\left (a+b x^n\right ) \, dx=\int \sin \left (x^{n} b +a \right )^{3}d x \] Input:
int(sin(a+b*x^n)^3,x)
Output:
int(sin(x**n*b + a)**3,x)