Integrand size = 18, antiderivative size = 165 \[ \int x^3 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=-\frac {3 \left (c \sin ^3(a+b x)\right )^{2/3}}{8 b^4}+\frac {3 x^2 \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^2}+\frac {3 x \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b^3}-\frac {x^3 \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}-\frac {3 x^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{8 b^2}+\frac {1}{8} x^4 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \]
-3/8*(c*sin(b*x+a)^3)^(2/3)/b^4+3/4*x^2*(c*sin(b*x+a)^3)^(2/3)/b^2+3/4*x*c ot(b*x+a)*(c*sin(b*x+a)^3)^(2/3)/b^3-1/2*x^3*cot(b*x+a)*(c*sin(b*x+a)^3)^( 2/3)/b-3/8*x^2*csc(b*x+a)^2*(c*sin(b*x+a)^3)^(2/3)/b^2+1/8*x^4*csc(b*x+a)^ 2*(c*sin(b*x+a)^3)^(2/3)
Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.48 \[ \int x^3 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=\frac {\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (2 b^4 x^4+\left (3-6 b^2 x^2\right ) \cos (2 (a+b x))+\left (6 b x-4 b^3 x^3\right ) \sin (2 (a+b x))\right )}{16 b^4} \]
(Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3)*(2*b^4*x^4 + (3 - 6*b^2*x^2)*Cos[ 2*(a + b*x)] + (6*b*x - 4*b^3*x^3)*Sin[2*(a + b*x)]))/(16*b^4)
Time = 0.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {7271, 3042, 3792, 15, 3042, 3791, 15}
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.
\(\displaystyle \int x^3 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \int x^3 \sin ^2(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \int x^3 \sin (a+b x)^2dx\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (-\frac {3 \int x \sin ^2(a+b x)dx}{2 b^2}+\frac {\int x^3dx}{2}+\frac {3 x^2 \sin ^2(a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (-\frac {3 \int x \sin ^2(a+b x)dx}{2 b^2}+\frac {3 x^2 \sin ^2(a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^4}{8}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (-\frac {3 \int x \sin (a+b x)^2dx}{2 b^2}+\frac {3 x^2 \sin ^2(a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^4}{8}\right )\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (-\frac {3 \left (\frac {\int xdx}{2}+\frac {\sin ^2(a+b x)}{4 b^2}-\frac {x \sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {3 x^2 \sin ^2(a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^4}{8}\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (\frac {3 x^2 \sin ^2(a+b x)}{4 b^2}-\frac {3 \left (\frac {\sin ^2(a+b x)}{4 b^2}-\frac {x \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^2}{4}\right )}{2 b^2}-\frac {x^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {x^4}{8}\right )\) |
Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3)*(x^4/8 - (x^3*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (3*x^2*Sin[a + b*x]^2)/(4*b^2) - (3*(x^2/4 - (x*Cos[a + b*x ]*Sin[a + b*x])/(2*b) + Sin[a + b*x]^2/(4*b^2)))/(2*b^2))
3.4.35.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[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 = 0.39 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.26
method | result | size |
risch | \(-\frac {x^{4} \left (i c \,{\mathrm e}^{-3 i \left (b x +a \right )} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{8 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {i \left (4 b^{3} x^{3}+6 i x^{2} b^{2}-6 b x -3 i\right ) \left (i c \,{\mathrm e}^{-3 i \left (b x +a \right )} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}\right )^{\frac {2}{3}} {\mathrm e}^{4 i \left (b x +a \right )}}{32 b^{4} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {i \left (i c \,{\mathrm e}^{-3 i \left (b x +a \right )} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}\right )^{\frac {2}{3}} \left (4 b^{3} x^{3}-6 i x^{2} b^{2}-6 b x +3 i\right )}{32 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2} b^{4}}\) | \(208\) |
-1/8*x^4/(exp(2*I*(b*x+a))-1)^2*(I*c*exp(-3*I*(b*x+a))*(exp(2*I*(b*x+a))-1 )^3)^(2/3)*exp(2*I*(b*x+a))-1/32*I/b^4*(4*b^3*x^3+6*I*x^2*b^2-6*b*x-3*I)/( exp(2*I*(b*x+a))-1)^2*(I*c*exp(-3*I*(b*x+a))*(exp(2*I*(b*x+a))-1)^3)^(2/3) *exp(4*I*(b*x+a))+1/32*I*(I*c*exp(-3*I*(b*x+a))*(exp(2*I*(b*x+a))-1)^3)^(2 /3)/(exp(2*I*(b*x+a))-1)^2*(4*b^3*x^3-6*I*b^2*x^2-6*b*x+3*I)/b^4
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.67 \[ \int x^3 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=-\frac {{\left (2 \, b^{4} x^{4} + 6 \, b^{2} x^{2} - 6 \, {\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (b x + a\right )^{2} - 4 \, {\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 3\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{16 \, {\left (b^{4} \cos \left (b x + a\right )^{2} - b^{4}\right )}} \]
-1/16*(2*b^4*x^4 + 6*b^2*x^2 - 6*(2*b^2*x^2 - 1)*cos(b*x + a)^2 - 4*(2*b^3 *x^3 - 3*b*x)*cos(b*x + a)*sin(b*x + a) - 3)*(-(c*cos(b*x + a)^2 - c)*sin( b*x + a))^(2/3)/(b^4*cos(b*x + a)^2 - b^4)
\[ \int x^3 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=\int x^{3} \left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac {2}{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (141) = 282\).
Time = 0.33 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.73 \[ \int x^3 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=-\frac {32 \, {\left (c^{\frac {2}{3}} \arctan \left (\frac {\sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1}\right ) - \frac {\frac {c^{\frac {2}{3}} \sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1} - \frac {c^{\frac {2}{3}} \sin \left (b x + a\right )^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}}}{\frac {2 \, \sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {\sin \left (b x + a\right )^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 1}\right )} a^{3} + 6 \, {\left (2 \, {\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right )\right )} a^{2} c^{\frac {2}{3}} - 2 \, {\left (4 \, {\left (b x + a\right )}^{3} - 6 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a c^{\frac {2}{3}} + {\left (2 \, {\left (b x + a\right )}^{4} - 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c^{\frac {2}{3}}}{32 \, b^{4}} \]
-1/32*(32*(c^(2/3)*arctan(sin(b*x + a)/(cos(b*x + a) + 1)) - (c^(2/3)*sin( b*x + a)/(cos(b*x + a) + 1) - c^(2/3)*sin(b*x + a)^3/(cos(b*x + a) + 1)^3) /(2*sin(b*x + a)^2/(cos(b*x + a) + 1)^2 + sin(b*x + a)^4/(cos(b*x + a) + 1 )^4 + 1))*a^3 + 6*(2*(b*x + a)^2 - 2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b* x + 2*a))*a^2*c^(2/3) - 2*(4*(b*x + a)^3 - 6*(b*x + a)*cos(2*b*x + 2*a) - 3*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*a*c^(2/3) + (2*(b*x + a)^4 - 3*(2* (b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 2*(2*(b*x + a)^3 - 3*b*x - 3*a)*sin(2* b*x + 2*a))*c^(2/3))/b^4
\[ \int x^3 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=\int { \left (c \sin \left (b x + a\right )^{3}\right )^{\frac {2}{3}} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=\int x^3\,{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{2/3} \,d x \]