Integrand size = 14, antiderivative size = 67 \[ \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx=\frac {3 \operatorname {CosIntegral}\left (b x^n\right ) \sin (a)}{4 n}-\frac {\operatorname {CosIntegral}\left (3 b x^n\right ) \sin (3 a)}{4 n}+\frac {3 \cos (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {\cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \] Output:
3/4*Ci(b*x^n)*sin(a)/n-1/4*Ci(3*b*x^n)*sin(3*a)/n+3/4*cos(a)*Si(b*x^n)/n-1 /4*cos(3*a)*Si(3*b*x^n)/n
Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx=\frac {3 \operatorname {CosIntegral}\left (b x^n\right ) \sin (a)-\operatorname {CosIntegral}\left (3 b x^n\right ) \sin (3 a)+3 \cos (a) \text {Si}\left (b x^n\right )-\cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \] Input:
Integrate[Sin[a + b*x^n]^3/x,x]
Output:
(3*CosIntegral[b*x^n]*Sin[a] - CosIntegral[3*b*x^n]*Sin[3*a] + 3*Cos[a]*Si nIntegral[b*x^n] - Cos[3*a]*SinIntegral[3*b*x^n])/(4*n)
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.
\(\displaystyle \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 3906 |
\(\displaystyle \int \left (\frac {3 \sin \left (a+b x^n\right )}{4 x}-\frac {\sin \left (3 a+3 b x^n\right )}{4 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \sin (a) \operatorname {CosIntegral}\left (b x^n\right )}{4 n}-\frac {\sin (3 a) \operatorname {CosIntegral}\left (3 b x^n\right )}{4 n}+\frac {3 \cos (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {\cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n}\) |
Input:
Int[Sin[a + b*x^n]^3/x,x]
Output:
(3*CosIntegral[b*x^n]*Sin[a])/(4*n) - (CosIntegral[3*b*x^n]*Sin[3*a])/(4*n ) + (3*Cos[a]*SinIntegral[b*x^n])/(4*n) - (Cos[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 = 0.98 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {Si}\left (3 b \,x^{n}\right ) \cos \left (3 a \right )}{4}-\frac {\operatorname {Ci}\left (3 b \,x^{n}\right ) \sin \left (3 a \right )}{4}+\frac {3 \,\operatorname {Si}\left (b \,x^{n}\right ) \cos \left (a \right )}{4}+\frac {3 \,\operatorname {Ci}\left (b \,x^{n}\right ) \sin \left (a \right )}{4}}{n}\) | \(52\) |
default | \(\frac {-\frac {\operatorname {Si}\left (3 b \,x^{n}\right ) \cos \left (3 a \right )}{4}-\frac {\operatorname {Ci}\left (3 b \,x^{n}\right ) \sin \left (3 a \right )}{4}+\frac {3 \,\operatorname {Si}\left (b \,x^{n}\right ) \cos \left (a \right )}{4}+\frac {3 \,\operatorname {Ci}\left (b \,x^{n}\right ) \sin \left (a \right )}{4}}{n}\) | \(52\) |
risch | \(-\frac {i {\mathrm e}^{3 i a} \operatorname {expIntegral}_{1}\left (-3 i b \,x^{n}\right )}{8 n}+\frac {{\mathrm e}^{-3 i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right )}{8 n}-\frac {{\mathrm e}^{-3 i a} \operatorname {Si}\left (3 b \,x^{n}\right )}{4 n}+\frac {i {\mathrm e}^{-3 i a} \operatorname {expIntegral}_{1}\left (-3 i b \,x^{n}\right )}{8 n}-\frac {3 \,{\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right )}{8 n}+\frac {3 \,{\mathrm e}^{-i a} \operatorname {Si}\left (b \,x^{n}\right )}{4 n}-\frac {3 i {\mathrm e}^{-i a} \operatorname {expIntegral}_{1}\left (-i b \,x^{n}\right )}{8 n}+\frac {3 i {\mathrm e}^{i a} \operatorname {expIntegral}_{1}\left (-i b \,x^{n}\right )}{8 n}\) | \(149\) |
Input:
int(sin(a+b*x^n)^3/x,x,method=_RETURNVERBOSE)
Output:
1/n*(-1/4*Si(3*b*x^n)*cos(3*a)-1/4*Ci(3*b*x^n)*sin(3*a)+3/4*Si(b*x^n)*cos( a)+3/4*Ci(b*x^n)*sin(a))
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx=-\frac {\operatorname {Ci}\left (3 \, b x^{n}\right ) \sin \left (3 \, a\right ) - 3 \, \operatorname {Ci}\left (b x^{n}\right ) \sin \left (a\right ) + \cos \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) - 3 \, \cos \left (a\right ) \operatorname {Si}\left (b x^{n}\right )}{4 \, n} \] Input:
integrate(sin(a+b*x^n)^3/x,x, algorithm="fricas")
Output:
-1/4*(cos_integral(3*b*x^n)*sin(3*a) - 3*cos_integral(b*x^n)*sin(a) + cos( 3*a)*sin_integral(3*b*x^n) - 3*cos(a)*sin_integral(b*x^n))/n
\[ \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx=\int \frac {\sin ^{3}{\left (a + b x^{n} \right )}}{x}\, dx \] Input:
integrate(sin(a+b*x**n)**3/x,x)
Output:
Integral(sin(a + b*x**n)**3/x, x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.27 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.70 \[ \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx=\frac {{\left (i \, {\rm Ei}\left (3 i \, b x^{n}\right ) - i \, {\rm Ei}\left (-3 i \, b x^{n}\right ) + i \, {\rm Ei}\left (3 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) - i \, {\rm Ei}\left (-3 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (3 \, a\right ) - 3 \, {\left (i \, {\rm Ei}\left (i \, b x^{n}\right ) - i \, {\rm Ei}\left (-i \, b x^{n}\right ) + i \, {\rm Ei}\left (i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) - i \, {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (a\right ) - {\left ({\rm Ei}\left (3 i \, b x^{n}\right ) + {\rm Ei}\left (-3 i \, b x^{n}\right ) + {\rm Ei}\left (3 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\rm Ei}\left (-3 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (3 \, a\right ) + 3 \, {\left ({\rm Ei}\left (i \, b x^{n}\right ) + {\rm Ei}\left (-i \, b x^{n}\right ) + {\rm Ei}\left (i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (a\right )}{16 \, n} \] Input:
integrate(sin(a+b*x^n)^3/x,x, algorithm="maxima")
Output:
1/16*((I*Ei(3*I*b*x^n) - I*Ei(-3*I*b*x^n) + I*Ei(3*I*b*e^(n*conjugate(log( x)))) - I*Ei(-3*I*b*e^(n*conjugate(log(x)))))*cos(3*a) - 3*(I*Ei(I*b*x^n) - I*Ei(-I*b*x^n) + I*Ei(I*b*e^(n*conjugate(log(x)))) - I*Ei(-I*b*e^(n*conj ugate(log(x)))))*cos(a) - (Ei(3*I*b*x^n) + Ei(-3*I*b*x^n) + Ei(3*I*b*e^(n* conjugate(log(x)))) + Ei(-3*I*b*e^(n*conjugate(log(x)))))*sin(3*a) + 3*(Ei (I*b*x^n) + Ei(-I*b*x^n) + Ei(I*b*e^(n*conjugate(log(x)))) + Ei(-I*b*e^(n* conjugate(log(x)))))*sin(a))/n
\[ \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx=\int { \frac {\sin \left (b x^{n} + a\right )^{3}}{x} \,d x } \] Input:
integrate(sin(a+b*x^n)^3/x,x, algorithm="giac")
Output:
integrate(sin(b*x^n + a)^3/x, x)
Timed out. \[ \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx=\int \frac {{\sin \left (a+b\,x^n\right )}^3}{x} \,d x \] Input:
int(sin(a + b*x^n)^3/x,x)
Output:
int(sin(a + b*x^n)^3/x, x)
\[ \int \frac {\sin ^3\left (a+b x^n\right )}{x} \, dx=\int \frac {\sin \left (x^{n} b +a \right )^{3}}{x}d x \] Input:
int(sin(a+b*x^n)^3/x,x)
Output:
int(sin(x**n*b + a)**3/x,x)