Integrand size = 18, antiderivative size = 114 \[ \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx=\frac {3 b \cos (a) \operatorname {CosIntegral}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \operatorname {CosIntegral}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \sin \left (a+b x^n\right )}{4 n}+\frac {x^{-n} \sin \left (3 a+3 b x^n\right )}{4 n}-\frac {3 b \sin (a) \text {Si}\left (b x^n\right )}{4 n}+\frac {3 b \sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \] Output:
3/4*b*cos(a)*Ci(b*x^n)/n-3/4*b*cos(3*a)*Ci(3*b*x^n)/n-3/4*sin(a+b*x^n)/n/( x^n)+1/4*sin(3*a+3*b*x^n)/n/(x^n)-3/4*b*sin(a)*Si(b*x^n)/n+3/4*b*sin(3*a)* Si(3*b*x^n)/n
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx=\frac {x^{-n} \left (3 b x^n \cos (a) \operatorname {CosIntegral}\left (b x^n\right )-3 b x^n \cos (3 a) \operatorname {CosIntegral}\left (3 b x^n\right )-3 \sin \left (a+b x^n\right )+\sin \left (3 \left (a+b x^n\right )\right )-3 b x^n \sin (a) \text {Si}\left (b x^n\right )+3 b x^n \sin (3 a) \text {Si}\left (3 b x^n\right )\right )}{4 n} \] Input:
Integrate[x^(-1 - n)*Sin[a + b*x^n]^3,x]
Output:
(3*b*x^n*Cos[a]*CosIntegral[b*x^n] - 3*b*x^n*Cos[3*a]*CosIntegral[3*b*x^n] - 3*Sin[a + b*x^n] + Sin[3*(a + b*x^n)] - 3*b*x^n*Sin[a]*SinIntegral[b*x^ n] + 3*b*x^n*Sin[3*a]*SinIntegral[3*b*x^n])/(4*n*x^n)
Time = 0.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3906, 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.
| \(\displaystyle \int x^{-n-1} \sin ^3\left (a+b x^n\right ) \, dx\) |
| \(\Big \downarrow \) 3906 |
| \(\displaystyle \int \left (\frac {3}{4} x^{-n-1} \sin \left (a+b x^n\right )-\frac {1}{4} x^{-n-1} \sin \left (3 a+3 b x^n\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b \cos (a) \operatorname {CosIntegral}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \operatorname {CosIntegral}\left (3 b x^n\right )}{4 n}-\frac {3 b \sin (a) \text {Si}\left (b x^n\right )}{4 n}+\frac {3 b \sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \sin \left (a+b x^n\right )}{4 n}+\frac {x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{4 n}\) |
Input:
Int[x^(-1 - n)*Sin[a + b*x^n]^3,x]
Output:
(3*b*Cos[a]*CosIntegral[b*x^n])/(4*n) - (3*b*Cos[3*a]*CosIntegral[3*b*x^n] )/(4*n) - (3*Sin[a + b*x^n])/(4*n*x^n) + Sin[3*(a + b*x^n)]/(4*n*x^n) - (3 *b*Sin[a]*SinIntegral[b*x^n])/(4*n) + (3*b*Sin[3*a]*SinIntegral[3*b*x^n])/ (4*n)
Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x _Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 5.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.87
| method | result | size |
| default | \(\frac {3 b \left (-\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{b}-\operatorname {Si}\left (b \,x^{n}\right ) \sin \left (a \right )+\operatorname {Ci}\left (b \,x^{n}\right ) \cos \left (a \right )\right )}{4 n}-\frac {3 b \left (-\frac {\sin \left (3 a +3 b \,x^{n}\right ) x^{-n}}{3 b}-\operatorname {Si}\left (3 b \,x^{n}\right ) \sin \left (3 a \right )+\operatorname {Ci}\left (3 b \,x^{n}\right ) \cos \left (3 a \right )\right )}{4 n}\) | \(99\) |
| risch | \(-\frac {\left (3 i b \,{\mathrm e}^{-3 i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{n}-3 i b \,{\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{n}-6 i b \,{\mathrm e}^{-3 i a} \operatorname {Si}\left (3 b \,x^{n}\right ) x^{n}+6 i b \,{\mathrm e}^{-i a} \operatorname {Si}\left (b \,x^{n}\right ) x^{n}-3 b \,{\mathrm e}^{3 i a} \operatorname {expIntegral}_{1}\left (-3 i b \,x^{n}\right ) x^{n}-3 b \,{\mathrm e}^{-3 i a} \operatorname {expIntegral}_{1}\left (-3 i b \,x^{n}\right ) x^{n}+3 b \,{\mathrm e}^{-i a} \operatorname {expIntegral}_{1}\left (-i b \,x^{n}\right ) x^{n}+3 b \,{\mathrm e}^{i a} \operatorname {expIntegral}_{1}\left (-i b \,x^{n}\right ) x^{n}+6 \sin \left (a +b \,x^{n}\right )-2 \sin \left (3 a +3 b \,x^{n}\right )\right ) x^{-n}}{8 n}\) | \(190\) |
Input:
int(x^(-1-n)*sin(a+b*x^n)^3,x,method=_RETURNVERBOSE)
Output:
3/4/n*b*(-sin(a+b*x^n)/b/(x^n)-Si(b*x^n)*sin(a)+Ci(b*x^n)*cos(a))-3/4/n*b* (-1/3*sin(3*a+3*b*x^n)/b/(x^n)-Si(3*b*x^n)*sin(3*a)+Ci(3*b*x^n)*cos(3*a))
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx=-\frac {3 \, b x^{n} \cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x^{n}\right ) - 3 \, b x^{n} \cos \left (a\right ) \operatorname {Ci}\left (b x^{n}\right ) - 3 \, b x^{n} \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) + 3 \, b x^{n} \sin \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) - 4 \, {\left (\cos \left (b x^{n} + a\right )^{2} - 1\right )} \sin \left (b x^{n} + a\right )}{4 \, n x^{n}} \] Input:
integrate(x^(-1-n)*sin(a+b*x^n)^3,x, algorithm="fricas")
Output:
-1/4*(3*b*x^n*cos(3*a)*cos_integral(3*b*x^n) - 3*b*x^n*cos(a)*cos_integral (b*x^n) - 3*b*x^n*sin(3*a)*sin_integral(3*b*x^n) + 3*b*x^n*sin(a)*sin_inte gral(b*x^n) - 4*(cos(b*x^n + a)^2 - 1)*sin(b*x^n + a))/(n*x^n)
\[ \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx=\int x^{- n - 1} \sin ^{3}{\left (a + b x^{n} \right )}\, dx \] Input:
integrate(x**(-1-n)*sin(a+b*x**n)**3,x)
Output:
Integral(x**(-n - 1)*sin(a + b*x**n)**3, x)
\[ \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \sin \left (b x^{n} + a\right )^{3} \,d x } \] Input:
integrate(x^(-1-n)*sin(a+b*x^n)^3,x, algorithm="maxima")
Output:
integrate(x^(-n - 1)*sin(b*x^n + a)^3, x)
\[ \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \sin \left (b x^{n} + a\right )^{3} \,d x } \] Input:
integrate(x^(-1-n)*sin(a+b*x^n)^3,x, algorithm="giac")
Output:
integrate(x^(-n - 1)*sin(b*x^n + a)^3, x)
Timed out. \[ \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx=\int \frac {{\sin \left (a+b\,x^n\right )}^3}{x^{n+1}} \,d x \] Input:
int(sin(a + b*x^n)^3/x^(n + 1),x)
Output:
int(sin(a + b*x^n)^3/x^(n + 1), x)
\[ \int x^{-1-n} \sin ^3\left (a+b x^n\right ) \, dx =\text {Too large to display} \] Input:
int(x^(-1-n)*sin(a+b*x^n)^3,x)
Output:
(2*( - 3*cos(x**n*b + a)*sin(x**n*b + a)*tan((x**n*b + a)/2)**6 - 9*cos(x* *n*b + a)*sin(x**n*b + a)*tan((x**n*b + a)/2)**4 - 9*cos(x**n*b + a)*sin(x **n*b + a)*tan((x**n*b + a)/2)**2 - 3*cos(x**n*b + a)*sin(x**n*b + a) + 20 *x**n*int(tan((x**n*b + a)/2)**3/(x**n*tan((x**n*b + a)/2)**6*x + 3*x**n*t an((x**n*b + a)/2)**4*x + 3*x**n*tan((x**n*b + a)/2)**2*x + x**n*x),x)*tan ((x**n*b + a)/2)**6*n + 60*x**n*int(tan((x**n*b + a)/2)**3/(x**n*tan((x**n *b + a)/2)**6*x + 3*x**n*tan((x**n*b + a)/2)**4*x + 3*x**n*tan((x**n*b + a )/2)**2*x + x**n*x),x)*tan((x**n*b + a)/2)**4*n + 60*x**n*int(tan((x**n*b + a)/2)**3/(x**n*tan((x**n*b + a)/2)**6*x + 3*x**n*tan((x**n*b + a)/2)**4* x + 3*x**n*tan((x**n*b + a)/2)**2*x + x**n*x),x)*tan((x**n*b + a)/2)**2*n + 20*x**n*int(tan((x**n*b + a)/2)**3/(x**n*tan((x**n*b + a)/2)**6*x + 3*x* *n*tan((x**n*b + a)/2)**4*x + 3*x**n*tan((x**n*b + a)/2)**2*x + x**n*x),x) *n - sin(x**n*b + a)**3*tan((x**n*b + a)/2)**6 - 3*sin(x**n*b + a)**3*tan( (x**n*b + a)/2)**4 - 3*sin(x**n*b + a)**3*tan((x**n*b + a)/2)**2 - sin(x** n*b + a)**3 - 3*sin(x**n*b + a)*tan((x**n*b + a)/2)**6 - 9*sin(x**n*b + a) *tan((x**n*b + a)/2)**4 - 9*sin(x**n*b + a)*tan((x**n*b + a)/2)**2 - 3*sin (x**n*b + a) + 20*tan((x**n*b + a)/2)**3 + 12*tan((x**n*b + a)/2)))/(5*x** n*n*(tan((x**n*b + a)/2)**6 + 3*tan((x**n*b + a)/2)**4 + 3*tan((x**n*b + a )/2)**2 + 1))