Integrand size = 20, antiderivative size = 121 \[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx=-\frac {\cos (2 a) \operatorname {CosIntegral}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {1}{2} \csc ^2\left (a+b x^n\right ) \log (x) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {\csc ^2\left (a+b x^n\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \text {Si}\left (2 b x^n\right )}{2 n} \] Output:
-1/2*cos(2*a)*Ci(2*b*x^n)*csc(a+b*x^n)^2*(c*sin(a+b*x^n)^3)^(2/3)/n+1/2*cs c(a+b*x^n)^2*ln(x)*(c*sin(a+b*x^n)^3)^(2/3)+1/2*csc(a+b*x^n)^2*sin(2*a)*(c *sin(a+b*x^n)^3)^(2/3)*Si(2*b*x^n)/n
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52 \[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx=\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \left (-\cos (2 a) \operatorname {CosIntegral}\left (2 b x^n\right )+n \log (x)+\sin (2 a) \text {Si}\left (2 b x^n\right )\right )}{2 n} \] Input:
Integrate[(c*Sin[a + b*x^n]^3)^(2/3)/x,x]
Output:
(Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3)*(-(Cos[2*a]*CosIntegral[2*b*x ^n]) + n*Log[x] + Sin[2*a]*SinIntegral[2*b*x^n]))/(2*n)
Time = 0.39 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7271, 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.
\(\displaystyle \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int \frac {\sin ^2\left (b x^n+a\right )}{x}dx\) |
\(\Big \downarrow \) 3906 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int \left (\frac {1}{2 x}-\frac {\cos \left (2 b x^n+2 a\right )}{2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \left (-\frac {\cos (2 a) \operatorname {CosIntegral}\left (2 b x^n\right )}{2 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2}\right )\) |
Input:
Int[(c*Sin[a + b*x^n]^3)^(2/3)/x,x]
Output:
Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3)*(-1/2*(Cos[2*a]*CosIntegral[2* b*x^n])/n + Log[x]/2 + (Sin[2*a]*SinIntegral[2*b*x^n])/(2*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]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {\left (i c \,{\mathrm e}^{-3 i \left (a +b \,x^{n}\right )} \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3}\right )^{\frac {2}{3}} \left (-i {\mathrm e}^{2 i b \,x^{n}} \pi \,\operatorname {csgn}\left (b \,x^{n}\right )+2 i {\mathrm e}^{2 i b \,x^{n}} \operatorname {Si}\left (2 b \,x^{n}\right )+2 \ln \left (x \right ) {\mathrm e}^{2 i \left (a +b \,x^{n}\right )} n +{\mathrm e}^{2 i b \,x^{n}} \operatorname {expIntegral}_{1}\left (-2 i b \,x^{n}\right )+\operatorname {expIntegral}_{1}\left (-2 i b \,x^{n}\right ) {\mathrm e}^{2 i \left (b \,x^{n}+2 a \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2} n}\) | \(147\) |
Input:
int((c*sin(a+b*x^n)^3)^(2/3)/x,x,method=_RETURNVERBOSE)
Output:
-1/4*(I*c*exp(-3*I*(a+b*x^n))*(exp(2*I*(a+b*x^n))-1)^3)^(2/3)*(-I*exp(2*I* b*x^n)*Pi*csgn(b*x^n)+2*I*exp(2*I*b*x^n)*Si(2*b*x^n)+2*ln(x)*exp(2*I*(a+b* x^n))*n+exp(2*I*b*x^n)*Ei(1,-2*I*b*x^n)+Ei(1,-2*I*b*x^n)*exp(2*I*(b*x^n+2* a)))/(exp(2*I*(a+b*x^n))-1)^2/n
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx=\frac {{\left ({\rm Ei}\left (2 i \, b x^{n}\right ) e^{\left (2 i \, a\right )} + {\rm Ei}\left (-2 i \, b x^{n}\right ) e^{\left (-2 i \, a\right )} - 2 \, n \log \left (x\right )\right )} \left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {2}{3}}}{4 \, {\left (n \cos \left (b x^{n} + a\right )^{2} - n\right )}} \] Input:
integrate((c*sin(a+b*x^n)^3)^(2/3)/x,x, algorithm="fricas")
Output:
1/4*(Ei(2*I*b*x^n)*e^(2*I*a) + Ei(-2*I*b*x^n)*e^(-2*I*a) - 2*n*log(x))*(-( c*cos(b*x^n + a)^2 - c)*sin(b*x^n + a))^(2/3)/(n*cos(b*x^n + a)^2 - n)
\[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx=\int \frac {\left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}}{x}\, dx \] Input:
integrate((c*sin(a+b*x**n)**3)**(2/3)/x,x)
Output:
Integral((c*sin(a + b*x**n)**3)**(2/3)/x, x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.26 \[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx=\frac {{\left ({\left ({\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (2 i \, b x^{n}\right ) + {\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-2 i \, b x^{n}\right ) + {\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (2 \, a\right ) - 4 \, n \log \left (x\right ) - {\left ({\left (\sqrt {3} - i\right )} {\rm Ei}\left (2 i \, b x^{n}\right ) - {\left (\sqrt {3} - i\right )} {\rm Ei}\left (-2 i \, b x^{n}\right ) - {\left (\sqrt {3} + i\right )} {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\left (\sqrt {3} + i\right )} {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (2 \, a\right )\right )} c^{\frac {2}{3}}}{16 \, n} \] Input:
integrate((c*sin(a+b*x^n)^3)^(2/3)/x,x, algorithm="maxima")
Output:
1/16*(((I*sqrt(3) + 1)*Ei(2*I*b*x^n) + (I*sqrt(3) + 1)*Ei(-2*I*b*x^n) + (- I*sqrt(3) + 1)*Ei(2*I*b*e^(n*conjugate(log(x)))) + (-I*sqrt(3) + 1)*Ei(-2* I*b*e^(n*conjugate(log(x)))))*cos(2*a) - 4*n*log(x) - ((sqrt(3) - I)*Ei(2* I*b*x^n) - (sqrt(3) - I)*Ei(-2*I*b*x^n) - (sqrt(3) + I)*Ei(2*I*b*e^(n*conj ugate(log(x)))) + (sqrt(3) + I)*Ei(-2*I*b*e^(n*conjugate(log(x)))))*sin(2* a))*c^(2/3)/n
\[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx=\int { \frac {\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}}}{x} \,d x } \] Input:
integrate((c*sin(a+b*x^n)^3)^(2/3)/x,x, algorithm="giac")
Output:
integrate((c*sin(b*x^n + a)^3)^(2/3)/x, x)
Timed out. \[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx=\int \frac {{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3}}{x} \,d x \] Input:
int((c*sin(a + b*x^n)^3)^(2/3)/x,x)
Output:
int((c*sin(a + b*x^n)^3)^(2/3)/x, x)
\[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx=c^{\frac {2}{3}} \left (\int \frac {\sin \left (x^{n} b +a \right )^{2}}{x}d x \right ) \] Input:
int((c*sin(a+b*x^n)^3)^(2/3)/x,x)
Output:
c**(2/3)*int(sin(x**n*b + a)**2/x,x)