Integrand size = 18, antiderivative size = 86 \[ \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx=-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x}+b \operatorname {CosIntegral}(2 b x) \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3}+b \cos (2 a) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \text {Si}(2 b x) \] Output:
-(c*sin(b*x+a)^3)^(2/3)/x+b*Ci(2*b*x)*csc(b*x+a)^2*sin(2*a)*(c*sin(b*x+a)^ 3)^(2/3)+b*cos(2*a)*csc(b*x+a)^2*(c*sin(b*x+a)^3)^(2/3)*Si(2*b*x)
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx=\frac {\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} (-1+\cos (2 (a+b x))+2 b x \operatorname {CosIntegral}(2 b x) \sin (2 a)+2 b x \cos (2 a) \text {Si}(2 b x))}{2 x} \] Input:
Integrate[(c*Sin[a + b*x]^3)^(2/3)/x^2,x]
Output:
(Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3)*(-1 + Cos[2*(a + b*x)] + 2*b*x*Co sIntegral[2*b*x]*Sin[2*a] + 2*b*x*Cos[2*a]*SinIntegral[2*b*x]))/(2*x)
Time = 0.58 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {7271, 3042, 3794, 27, 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.
\(\displaystyle \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \int \frac {\sin ^2(a+b x)}{x^2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \int \frac {\sin (a+b x)^2}{x^2}dx\) |
\(\Big \downarrow \) 3794 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (2 b \int \frac {\sin (2 a+2 b x)}{2 x}dx-\frac {\sin ^2(a+b x)}{x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (b \int \frac {\sin (2 a+2 b x)}{x}dx-\frac {\sin ^2(a+b x)}{x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (b \int \frac {\sin (2 a+2 b x)}{x}dx-\frac {\sin ^2(a+b x)}{x}\right )\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (b \left (\sin (2 a) \int \frac {\cos (2 b x)}{x}dx+\cos (2 a) \int \frac {\sin (2 b x)}{x}dx\right )-\frac {\sin ^2(a+b x)}{x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (b \left (\sin (2 a) \int \frac {\sin \left (2 b x+\frac {\pi }{2}\right )}{x}dx+\cos (2 a) \int \frac {\sin (2 b x)}{x}dx\right )-\frac {\sin ^2(a+b x)}{x}\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (b \left (\sin (2 a) \int \frac {\sin \left (2 b x+\frac {\pi }{2}\right )}{x}dx+\cos (2 a) \text {Si}(2 b x)\right )-\frac {\sin ^2(a+b x)}{x}\right )\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (b (\sin (2 a) \operatorname {CosIntegral}(2 b x)+\cos (2 a) \text {Si}(2 b x))-\frac {\sin ^2(a+b x)}{x}\right )\) |
Input:
Int[(c*Sin[a + b*x]^3)^(2/3)/x^2,x]
Output:
Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3)*(-(Sin[a + b*x]^2/x) + b*(CosInteg ral[2*b*x]*Sin[2*a] + Cos[2*a]*SinIntegral[2*b*x]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
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 = 1.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.30
method | result | size |
risch | \(\frac {\left (i c \,{\mathrm e}^{-3 i \left (b x +a \right )} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}\right )^{\frac {2}{3}} \left (-2 i {\mathrm e}^{2 i \left (b x +2 a \right )} \operatorname {expIntegral}_{1}\left (-2 i b x \right ) b x +2 i {\mathrm e}^{2 i b x} \operatorname {expIntegral}_{1}\left (2 i b x \right ) b x -{\mathrm e}^{4 i \left (b x +a \right )}-1+2 \,{\mathrm e}^{2 i \left (b x +a \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2} x}\) | \(112\) |
Input:
int((c*sin(b*x+a)^3)^(2/3)/x^2,x,method=_RETURNVERBOSE)
Output:
1/4*(I*c*exp(-3*I*(b*x+a))*(exp(2*I*(b*x+a))-1)^3)^(2/3)*(-2*I*exp(2*I*(b* x+2*a))*Ei(1,-2*I*b*x)*b*x+2*I*exp(2*I*b*x)*Ei(1,2*I*b*x)*b*x-exp(4*I*(b*x +a))-1+2*exp(2*I*(b*x+a)))/(exp(2*I*(b*x+a))-1)^2/x
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx=\frac {{\left (i \, b x {\rm Ei}\left (2 i \, b x\right ) e^{\left (2 i \, a\right )} - i \, b x {\rm Ei}\left (-2 i \, b x\right ) e^{\left (-2 i \, a\right )} - 2 \, \cos \left (b x + a\right )^{2} + 2\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{2 \, {\left (x \cos \left (b x + a\right )^{2} - x\right )}} \] Input:
integrate((c*sin(b*x+a)^3)^(2/3)/x^2,x, algorithm="fricas")
Output:
1/2*(I*b*x*Ei(2*I*b*x)*e^(2*I*a) - I*b*x*Ei(-2*I*b*x)*e^(-2*I*a) - 2*cos(b *x + a)^2 + 2)*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))^(2/3)/(x*cos(b*x + a )^2 - x)
\[ \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx=\int \frac {\left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac {2}{3}}}{x^{2}}\, dx \] Input:
integrate((c*sin(b*x+a)**3)**(2/3)/x**2,x)
Output:
Integral((c*sin(a + b*x)**3)**(2/3)/x**2, x)
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.08 \[ \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx=\frac {{\left ({\left ({\left (-i \, \sqrt {3} + 1\right )} E_{2}\left (2 i \, b x\right ) + {\left (i \, \sqrt {3} + 1\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} - {\left ({\left (\sqrt {3} + i\right )} E_{2}\left (2 i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + {\left ({\left ({\left (-i \, \sqrt {3} + 1\right )} E_{2}\left (2 i \, b x\right ) + {\left (i \, \sqrt {3} + 1\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 4\right )} \sin \left (2 \, a\right )^{2} + {\left ({\left (i \, \sqrt {3} + 1\right )} E_{2}\left (2 i \, b x\right ) + {\left (-i \, \sqrt {3} + 1\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 4 \, \cos \left (2 \, a\right )^{2} - {\left ({\left ({\left (\sqrt {3} + i\right )} E_{2}\left (2 i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - {\left (\sqrt {3} - i\right )} E_{2}\left (2 i \, b x\right ) - {\left (\sqrt {3} + i\right )} E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} b c^{\frac {2}{3}}}{16 \, {\left (a \cos \left (2 \, a\right )^{2} + a \sin \left (2 \, a\right )^{2} - {\left (b x + a\right )} {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )}\right )}} \] Input:
integrate((c*sin(b*x+a)^3)^(2/3)/x^2,x, algorithm="maxima")
Output:
1/16*(((-I*sqrt(3) + 1)*exp_integral_e(2, 2*I*b*x) + (I*sqrt(3) + 1)*exp_i ntegral_e(2, -2*I*b*x))*cos(2*a)^3 - ((sqrt(3) + I)*exp_integral_e(2, 2*I* b*x) + (sqrt(3) - I)*exp_integral_e(2, -2*I*b*x))*sin(2*a)^3 + (((-I*sqrt( 3) + 1)*exp_integral_e(2, 2*I*b*x) + (I*sqrt(3) + 1)*exp_integral_e(2, -2* I*b*x))*cos(2*a) - 4)*sin(2*a)^2 + ((I*sqrt(3) + 1)*exp_integral_e(2, 2*I* b*x) + (-I*sqrt(3) + 1)*exp_integral_e(2, -2*I*b*x))*cos(2*a) - 4*cos(2*a) ^2 - (((sqrt(3) + I)*exp_integral_e(2, 2*I*b*x) + (sqrt(3) - I)*exp_integr al_e(2, -2*I*b*x))*cos(2*a)^2 - (sqrt(3) - I)*exp_integral_e(2, 2*I*b*x) - (sqrt(3) + I)*exp_integral_e(2, -2*I*b*x))*sin(2*a))*b*c^(2/3)/(a*cos(2*a )^2 + a*sin(2*a)^2 - (b*x + a)*(cos(2*a)^2 + sin(2*a)^2))
\[ \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )^{3}\right )^{\frac {2}{3}}}{x^{2}} \,d x } \] Input:
integrate((c*sin(b*x+a)^3)^(2/3)/x^2,x, algorithm="giac")
Output:
integrate((c*sin(b*x + a)^3)^(2/3)/x^2, x)
Timed out. \[ \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx=\int \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{2/3}}{x^2} \,d x \] Input:
int((c*sin(a + b*x)^3)^(2/3)/x^2,x)
Output:
int((c*sin(a + b*x)^3)^(2/3)/x^2, x)
\[ \int \frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{x^2} \, dx=\frac {c^{\frac {2}{3}} \left (\left (\int \frac {\sin \left (b x +a \right )^{2}}{x^{2}}d x \right ) x -2 \left (\int \frac {1}{x^{2}}d x \right ) x -2\right )}{x} \] Input:
int((c*sin(b*x+a)^3)^(2/3)/x^2,x)
Output:
(c**(2/3)*(int(sin(a + b*x)**2/x**2,x)*x - 2*int(1/x**2,x)*x - 2))/x