Integrand size = 20, antiderivative size = 98 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx=-\frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{2 x^2}+\frac {1}{2} b \cos (a) \operatorname {CosIntegral}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}-\frac {1}{2} b \csc \left (a+b x^2\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \text {Si}\left (b x^2\right ) \] Output:
-1/2*(c*sin(b*x^2+a)^3)^(1/3)/x^2+1/2*b*cos(a)*Ci(b*x^2)*csc(b*x^2+a)*(c*s in(b*x^2+a)^3)^(1/3)-1/2*b*csc(b*x^2+a)*sin(a)*(c*sin(b*x^2+a)^3)^(1/3)*Si (b*x^2)
Time = 0.35 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx=-\frac {\csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (-b x^2 \cos (a) \operatorname {CosIntegral}\left (b x^2\right )+\sin \left (a+b x^2\right )+b x^2 \sin (a) \text {Si}\left (b x^2\right )\right )}{2 x^2} \] Input:
Integrate[(c*Sin[a + b*x^2]^3)^(1/3)/x^3,x]
Output:
-1/2*(Csc[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3)*(-(b*x^2*Cos[a]*CosIntegra l[b*x^2]) + Sin[a + b*x^2] + b*x^2*Sin[a]*SinIntegral[b*x^2]))/x^2
Time = 0.65 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {7271, 3860, 3042, 3778, 3042, 3784, 3042, 3780, 3783}
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.
\(\displaystyle \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \int \frac {\sin \left (b x^2+a\right )}{x^3}dx\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle \frac {1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \int \frac {\sin \left (b x^2+a\right )}{x^4}dx^2\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \int \frac {\sin \left (b x^2+a\right )}{x^4}dx^2\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (b \int \frac {\cos \left (b x^2+a\right )}{x^2}dx^2-\frac {\sin \left (a+b x^2\right )}{x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (b \int \frac {\sin \left (b x^2+a+\frac {\pi }{2}\right )}{x^2}dx^2-\frac {\sin \left (a+b x^2\right )}{x^2}\right )\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (b \left (\cos (a) \int \frac {\cos \left (b x^2\right )}{x^2}dx^2-\sin (a) \int \frac {\sin \left (b x^2\right )}{x^2}dx^2\right )-\frac {\sin \left (a+b x^2\right )}{x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (b \left (\cos (a) \int \frac {\sin \left (b x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\sin (a) \int \frac {\sin \left (b x^2\right )}{x^2}dx^2\right )-\frac {\sin \left (a+b x^2\right )}{x^2}\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (b \left (\cos (a) \int \frac {\sin \left (b x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\sin (a) \text {Si}\left (b x^2\right )\right )-\frac {\sin \left (a+b x^2\right )}{x^2}\right )\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (b \left (\cos (a) \operatorname {CosIntegral}\left (b x^2\right )-\sin (a) \text {Si}\left (b x^2\right )\right )-\frac {\sin \left (a+b x^2\right )}{x^2}\right )\) |
Input:
Int[(c*Sin[a + b*x^2]^3)^(1/3)/x^3,x]
Output:
(Csc[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3)*(-(Sin[a + b*x^2]/x^2) + b*(Cos [a]*CosIntegral[b*x^2] - Sin[a]*SinIntegral[b*x^2])))/2
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 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 complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {\left (i c \,{\mathrm e}^{-3 i \left (b \,x^{2}+a \right )} \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3}\right )^{\frac {1}{3}} \left (i \operatorname {expIntegral}_{1}\left (-i b \,x^{2}\right ) b \,x^{2} {\mathrm e}^{i \left (b \,x^{2}+2 a \right )}+i {\mathrm e}^{i b \,x^{2}} b \,\operatorname {expIntegral}_{1}\left (i b \,x^{2}\right ) x^{2}+{\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right ) x^{2}}\) | \(119\) |
Input:
int((c*sin(b*x^2+a)^3)^(1/3)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/4*(I*c*exp(-3*I*(b*x^2+a))*(exp(2*I*(b*x^2+a))-1)^3)^(1/3)*(I*Ei(1,-I*b *x^2)*b*x^2*exp(I*(b*x^2+2*a))+I*exp(I*b*x^2)*b*Ei(1,I*b*x^2)*x^2+exp(2*I* (b*x^2+a))-1)/(exp(2*I*(b*x^2+a))-1)/x^2
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx=\frac {{\left (b x^{2} {\rm Ei}\left (i \, b x^{2}\right ) e^{\left (i \, a\right )} + b x^{2} {\rm Ei}\left (-i \, b x^{2}\right ) e^{\left (-i \, a\right )} - 2 \, \sin \left (b x^{2} + a\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {1}{3}}}{4 \, x^{2} \sin \left (b x^{2} + a\right )} \] Input:
integrate((c*sin(b*x^2+a)^3)^(1/3)/x^3,x, algorithm="fricas")
Output:
1/4*(b*x^2*Ei(I*b*x^2)*e^(I*a) + b*x^2*Ei(-I*b*x^2)*e^(-I*a) - 2*sin(b*x^2 + a))*(-(c*cos(b*x^2 + a)^2 - c)*sin(b*x^2 + a))^(1/3)/(x^2*sin(b*x^2 + a ))
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx=\int \frac {\sqrt [3]{c \sin ^{3}{\left (a + b x^{2} \right )}}}{x^{3}}\, dx \] Input:
integrate((c*sin(b*x**2+a)**3)**(1/3)/x**3,x)
Output:
Integral((c*sin(a + b*x**2)**3)**(1/3)/x**3, x)
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx=-\frac {1}{8} \, {\left ({\left (\Gamma \left (-1, i \, b x^{2}\right ) + \Gamma \left (-1, -i \, b x^{2}\right )\right )} \cos \left (a\right ) - {\left (i \, \Gamma \left (-1, i \, b x^{2}\right ) - i \, \Gamma \left (-1, -i \, b x^{2}\right )\right )} \sin \left (a\right )\right )} b c^{\frac {1}{3}} \] Input:
integrate((c*sin(b*x^2+a)^3)^(1/3)/x^3,x, algorithm="maxima")
Output:
-1/8*((gamma(-1, I*b*x^2) + gamma(-1, -I*b*x^2))*cos(a) - (I*gamma(-1, I*b *x^2) - I*gamma(-1, -I*b*x^2))*sin(a))*b*c^(1/3)
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx=\int { \frac {\left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac {1}{3}}}{x^{3}} \,d x } \] Input:
integrate((c*sin(b*x^2+a)^3)^(1/3)/x^3,x, algorithm="giac")
Output:
integrate((c*sin(b*x^2 + a)^3)^(1/3)/x^3, x)
Timed out. \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx=\int \frac {{\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{1/3}}{x^3} \,d x \] Input:
int((c*sin(a + b*x^2)^3)^(1/3)/x^3,x)
Output:
int((c*sin(a + b*x^2)^3)^(1/3)/x^3, x)
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^2\right )}}{x^3} \, dx=c^{\frac {1}{3}} \left (\int \frac {\sin \left (b \,x^{2}+a \right )}{x^{3}}d x \right ) \] Input:
int((c*sin(b*x^2+a)^3)^(1/3)/x^3,x)
Output:
c**(1/3)*int(sin(a + b*x**2)/x**3,x)