Integrand size = 20, antiderivative size = 73 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx=\frac {\operatorname {CosIntegral}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n}+\frac {\cos (a) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \text {Si}\left (b x^n\right )}{n} \] Output:
Ci(b*x^n)*csc(a+b*x^n)*sin(a)*(c*sin(a+b*x^n)^3)^(1/3)/n+cos(a)*csc(a+b*x^ n)*(c*sin(a+b*x^n)^3)^(1/3)*Si(b*x^n)/n
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx=\frac {\csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\operatorname {CosIntegral}\left (b x^n\right ) \sin (a)+\cos (a) \text {Si}\left (b x^n\right )\right )}{n} \] Input:
Integrate[(c*Sin[a + b*x^n]^3)^(1/3)/x,x]
Output:
(Csc[a + b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3)*(CosIntegral[b*x^n]*Sin[a] + Co s[a]*SinIntegral[b*x^n]))/n
Time = 0.42 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7271, 3858, 3856, 3857}
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.
\(\displaystyle \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \int \frac {\sin \left (b x^n+a\right )}{x}dx\) |
\(\Big \downarrow \) 3858 |
\(\displaystyle \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\sin (a) \int \frac {\cos \left (b x^n\right )}{x}dx+\cos (a) \int \frac {\sin \left (b x^n\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3856 |
\(\displaystyle \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\sin (a) \int \frac {\cos \left (b x^n\right )}{x}dx+\frac {\cos (a) \text {Si}\left (b x^n\right )}{n}\right )\) |
\(\Big \downarrow \) 3857 |
\(\displaystyle \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\frac {\sin (a) \operatorname {CosIntegral}\left (b x^n\right )}{n}+\frac {\cos (a) \text {Si}\left (b x^n\right )}{n}\right )\) |
Input:
Int[(c*Sin[a + b*x^n]^3)^(1/3)/x,x]
Output:
Csc[a + b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3)*((CosIntegral[b*x^n]*Sin[a])/n + (Cos[a]*SinIntegral[b*x^n])/n)
Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] / ; FreeQ[{d, n}, x]
Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] / ; FreeQ[{d, n}, x]
Int[Sin[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[Sin[c] Int[Cos[d* x^n]/x, x], x] + Simp[Cos[c] Int[Sin[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]
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 = 2.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.79
method | result | size |
risch | \(-\frac {\left (i c \,{\mathrm e}^{-3 i \left (a +b \,x^{n}\right )} \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3}\right )^{\frac {1}{3}} \left (i {\mathrm e}^{i b \,x^{n}} \pi \,\operatorname {csgn}\left (b \,x^{n}\right )-2 i {\mathrm e}^{i b \,x^{n}} \operatorname {Si}\left (b \,x^{n}\right )+\operatorname {expIntegral}_{1}\left (-i b \,x^{n}\right ) {\mathrm e}^{i \left (b \,x^{n}+2 a \right )}-{\mathrm e}^{i b \,x^{n}} \operatorname {expIntegral}_{1}\left (-i b \,x^{n}\right )\right )}{2 n \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )}\) | \(131\) |
Input:
int((c*sin(a+b*x^n)^3)^(1/3)/x,x,method=_RETURNVERBOSE)
Output:
-1/2*(I*c*exp(-3*I*(a+b*x^n))*(exp(2*I*(a+b*x^n))-1)^3)^(1/3)*(I*exp(I*b*x ^n)*Pi*csgn(b*x^n)-2*I*exp(I*b*x^n)*Si(b*x^n)+Ei(1,-I*b*x^n)*exp(I*(b*x^n+ 2*a))-exp(I*b*x^n)*Ei(1,-I*b*x^n))/n/(exp(2*I*(a+b*x^n))-1)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx=\frac {{\left (-i \, {\rm Ei}\left (i \, b x^{n}\right ) e^{\left (i \, a\right )} + i \, {\rm Ei}\left (-i \, b x^{n}\right ) e^{\left (-i \, a\right )}\right )} \left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {1}{3}}}{2 \, n \sin \left (b x^{n} + a\right )} \] Input:
integrate((c*sin(a+b*x^n)^3)^(1/3)/x,x, algorithm="fricas")
Output:
1/2*(-I*Ei(I*b*x^n)*e^(I*a) + I*Ei(-I*b*x^n)*e^(-I*a))*(-(c*cos(b*x^n + a) ^2 - c)*sin(b*x^n + a))^(1/3)/(n*sin(b*x^n + a))
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx=\int \frac {\sqrt [3]{c \sin ^{3}{\left (a + b x^{n} \right )}}}{x}\, dx \] Input:
integrate((c*sin(a+b*x**n)**3)**(1/3)/x,x)
Output:
Integral((c*sin(a + b*x**n)**3)**(1/3)/x, x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx=\frac {{\left ({\left ({\left (\sqrt {3} + i\right )} {\rm Ei}\left (i \, b x^{n}\right ) - {\left (\sqrt {3} + i\right )} {\rm Ei}\left (-i \, b x^{n}\right ) - {\left (\sqrt {3} - i\right )} {\rm Ei}\left (i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\left (\sqrt {3} - i\right )} {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (a\right ) - {\left ({\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (i \, b x^{n}\right ) + {\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-i \, b x^{n}\right ) + {\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (a\right )\right )} c^{\frac {1}{3}}}{8 \, n} \] Input:
integrate((c*sin(a+b*x^n)^3)^(1/3)/x,x, algorithm="maxima")
Output:
1/8*(((sqrt(3) + I)*Ei(I*b*x^n) - (sqrt(3) + I)*Ei(-I*b*x^n) - (sqrt(3) - I)*Ei(I*b*e^(n*conjugate(log(x)))) + (sqrt(3) - I)*Ei(-I*b*e^(n*conjugate( log(x)))))*cos(a) - ((-I*sqrt(3) + 1)*Ei(I*b*x^n) + (-I*sqrt(3) + 1)*Ei(-I *b*x^n) + (I*sqrt(3) + 1)*Ei(I*b*e^(n*conjugate(log(x)))) + (I*sqrt(3) + 1 )*Ei(-I*b*e^(n*conjugate(log(x)))))*sin(a))*c^(1/3)/n
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx=\int { \frac {\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}}}{x} \,d x } \] Input:
integrate((c*sin(a+b*x^n)^3)^(1/3)/x,x, algorithm="giac")
Output:
integrate((c*sin(b*x^n + a)^3)^(1/3)/x, x)
Timed out. \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx=\int \frac {{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{1/3}}{x} \,d x \] Input:
int((c*sin(a + b*x^n)^3)^(1/3)/x,x)
Output:
int((c*sin(a + b*x^n)^3)^(1/3)/x, x)
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x} \, dx=c^{\frac {1}{3}} \left (\int \frac {\sin \left (x^{n} b +a \right )}{x}d x \right ) \] Input:
int((c*sin(a+b*x^n)^3)^(1/3)/x,x)
Output:
c**(1/3)*int(sin(x**n*b + a)/x,x)