Integrand size = 15, antiderivative size = 158 \[ \int x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {6 b^3 n^3 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{16+40 b^2 n^2+9 b^4 n^4}+\frac {12 b^2 n^2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{16+40 b^2 n^2+9 b^4 n^4}-\frac {3 b n x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}+\frac {2 x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2} \]
-6*b^3*n^3*x^2*cos(a+b*ln(c*x^n))/(9*b^4*n^4+40*b^2*n^2+16)+12*b^2*n^2*x^2 *sin(a+b*ln(c*x^n))/(9*b^4*n^4+40*b^2*n^2+16)-3*b*n*x^2*cos(a+b*ln(c*x^n)) *sin(a+b*ln(c*x^n))^2/(9*b^2*n^2+4)+2*x^2*sin(a+b*ln(c*x^n))^3/(9*b^2*n^2+ 4)
Time = 0.39 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.79 \[ \int x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (-3 b n \left (4+9 b^2 n^2\right ) \cos \left (a+b \log \left (c x^n\right )\right )+3 b n \left (4+b^2 n^2\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-4 \left (-4-13 b^2 n^2+\left (4+b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )\right )}{4 \left (16+40 b^2 n^2+9 b^4 n^4\right )} \]
(x^2*(-3*b*n*(4 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + 3*b*n*(4 + b^2*n^2)*C os[3*(a + b*Log[c*x^n])] - 4*(-4 - 13*b^2*n^2 + (4 + b^2*n^2)*Cos[2*(a + b *Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(4*(16 + 40*b^2*n^2 + 9*b^4*n^4))
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4990, 4988}
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.
| \(\displaystyle \int x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 4990 |
| \(\displaystyle \frac {6 b^2 n^2 \int x \sin \left (a+b \log \left (c x^n\right )\right )dx}{9 b^2 n^2+4}+\frac {2 x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4}-\frac {3 b n x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4}\) |
| \(\Big \downarrow \) 4988 |
| \(\displaystyle \frac {2 x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4}-\frac {3 b n x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4}+\frac {6 b^2 n^2 \left (\frac {2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4}-\frac {b n x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4}\right )}{9 b^2 n^2+4}\) |
(-3*b*n*x^2*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/(4 + 9*b^2*n^2) + (2*x^2*Sin[a + b*Log[c*x^n]]^3)/(4 + 9*b^2*n^2) + (6*b^2*n^2*(-((b*n*x^ 2*Cos[a + b*Log[c*x^n]])/(4 + b^2*n^2)) + (2*x^2*Sin[a + b*Log[c*x^n]])/(4 + b^2*n^2)))/(4 + 9*b^2*n^2)
3.1.14.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_ Symbol] :> Simp[(m + 1)*(e*x)^(m + 1)*(Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*e *n^2 + e*(m + 1)^2)), x] - Simp[b*d*n*(e*x)^(m + 1)*(Cos[d*(a + b*Log[c*x^n ])]/(b^2*d^2*e*n^2 + e*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] & & NeQ[b^2*d^2*n^2 + (m + 1)^2, 0]
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_ ), x_Symbol] :> Simp[(m + 1)*(e*x)^(m + 1)*(Sin[d*(a + b*Log[c*x^n])]^p/(b^ 2*d^2*e*n^2*p^2 + e*(m + 1)^2)), x] + (-Simp[b*d*n*p*(e*x)^(m + 1)*Cos[d*(a + b*Log[c*x^n])]*(Sin[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 + e *(m + 1)^2)), x] + Simp[b^2*d^2*n^2*p*((p - 1)/(b^2*d^2*n^2*p^2 + (m + 1)^2 )) Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]
\[\int x {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}d x\]
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.89 \[ \int x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, {\left (b^{3} n^{3} + 4 \, b n\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \, {\left (3 \, b^{3} n^{3} + 4 \, b n\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 2 \, {\left ({\left (b^{2} n^{2} + 4\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - {\left (7 \, b^{2} n^{2} + 4\right )} x^{2}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{9 \, b^{4} n^{4} + 40 \, b^{2} n^{2} + 16} \]
(3*(b^3*n^3 + 4*b*n)*x^2*cos(b*n*log(x) + b*log(c) + a)^3 - 3*(3*b^3*n^3 + 4*b*n)*x^2*cos(b*n*log(x) + b*log(c) + a) - 2*((b^2*n^2 + 4)*x^2*cos(b*n* log(x) + b*log(c) + a)^2 - (7*b^2*n^2 + 4)*x^2)*sin(b*n*log(x) + b*log(c) + a))/(9*b^4*n^4 + 40*b^2*n^2 + 16)
\[ \int x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int x \sin ^{3}{\left (a - \frac {2 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {2 i}{n} \\\int x \sin ^{3}{\left (a - \frac {2 i \log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = - \frac {2 i}{3 n} \\\int x \sin ^{3}{\left (a + \frac {2 i \log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = \frac {2 i}{3 n} \\\int x \sin ^{3}{\left (a + \frac {2 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {2 i}{n} \\- \frac {9 b^{3} n^{3} x^{2} \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 40 b^{2} n^{2} + 16} - \frac {6 b^{3} n^{3} x^{2} \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 40 b^{2} n^{2} + 16} + \frac {14 b^{2} n^{2} x^{2} \sin ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 40 b^{2} n^{2} + 16} + \frac {12 b^{2} n^{2} x^{2} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 40 b^{2} n^{2} + 16} - \frac {12 b n x^{2} \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 40 b^{2} n^{2} + 16} + \frac {8 x^{2} \sin ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 40 b^{2} n^{2} + 16} & \text {otherwise} \end {cases} \]
Piecewise((Integral(x*sin(a - 2*I*log(c*x**n)/n)**3, x), Eq(b, -2*I/n)), ( Integral(x*sin(a - 2*I*log(c*x**n)/(3*n))**3, x), Eq(b, -2*I/(3*n))), (Int egral(x*sin(a + 2*I*log(c*x**n)/(3*n))**3, x), Eq(b, 2*I/(3*n))), (Integra l(x*sin(a + 2*I*log(c*x**n)/n)**3, x), Eq(b, 2*I/n)), (-9*b**3*n**3*x**2*s in(a + b*log(c*x**n))**2*cos(a + b*log(c*x**n))/(9*b**4*n**4 + 40*b**2*n** 2 + 16) - 6*b**3*n**3*x**2*cos(a + b*log(c*x**n))**3/(9*b**4*n**4 + 40*b** 2*n**2 + 16) + 14*b**2*n**2*x**2*sin(a + b*log(c*x**n))**3/(9*b**4*n**4 + 40*b**2*n**2 + 16) + 12*b**2*n**2*x**2*sin(a + b*log(c*x**n))*cos(a + b*lo g(c*x**n))**2/(9*b**4*n**4 + 40*b**2*n**2 + 16) - 12*b*n*x**2*sin(a + b*lo g(c*x**n))**2*cos(a + b*log(c*x**n))/(9*b**4*n**4 + 40*b**2*n**2 + 16) + 8 *x**2*sin(a + b*log(c*x**n))**3/(9*b**4*n**4 + 40*b**2*n**2 + 16), True))
Leaf count of result is larger than twice the leaf count of optimal. 1016 vs. \(2 (158) = 316\).
Time = 0.23 (sec) , antiderivative size = 1016, normalized size of antiderivative = 6.43 \[ \int x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/8*((3*(b^3*cos(6*b*log(c))*cos(3*b*log(c)) + b^3*sin(6*b*log(c))*sin(3*b *log(c)) + b^3*cos(3*b*log(c)))*n^3 - 2*(b^2*cos(3*b*log(c))*sin(6*b*log(c )) - b^2*cos(6*b*log(c))*sin(3*b*log(c)) + b^2*sin(3*b*log(c)))*n^2 + 12*( b*cos(6*b*log(c))*cos(3*b*log(c)) + b*sin(6*b*log(c))*sin(3*b*log(c)) + b* cos(3*b*log(c)))*n - 8*cos(3*b*log(c))*sin(6*b*log(c)) + 8*cos(6*b*log(c)) *sin(3*b*log(c)) - 8*sin(3*b*log(c)))*x^2*cos(3*b*log(x^n) + 3*a) - 3*(9*( b^3*cos(4*b*log(c))*cos(3*b*log(c)) + b^3*cos(3*b*log(c))*cos(2*b*log(c)) + b^3*sin(4*b*log(c))*sin(3*b*log(c)) + b^3*sin(3*b*log(c))*sin(2*b*log(c) ))*n^3 - 18*(b^2*cos(3*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin (3*b*log(c)) + b^2*cos(2*b*log(c))*sin(3*b*log(c)) - b^2*cos(3*b*log(c))*s in(2*b*log(c)))*n^2 + 4*(b*cos(4*b*log(c))*cos(3*b*log(c)) + b*cos(3*b*log (c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(3*b*log(c)) + b*sin(3*b*log(c ))*sin(2*b*log(c)))*n - 8*cos(3*b*log(c))*sin(4*b*log(c)) + 8*cos(4*b*log( c))*sin(3*b*log(c)) - 8*cos(2*b*log(c))*sin(3*b*log(c)) + 8*cos(3*b*log(c) )*sin(2*b*log(c)))*x^2*cos(b*log(x^n) + a) - (3*(b^3*cos(3*b*log(c))*sin(6 *b*log(c)) - b^3*cos(6*b*log(c))*sin(3*b*log(c)) + b^3*sin(3*b*log(c)))*n^ 3 + 2*(b^2*cos(6*b*log(c))*cos(3*b*log(c)) + b^2*sin(6*b*log(c))*sin(3*b*l og(c)) + b^2*cos(3*b*log(c)))*n^2 + 12*(b*cos(3*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(3*b*log(c)) + b*sin(3*b*log(c)))*n + 8*cos(6*b*log (c))*cos(3*b*log(c)) + 8*sin(6*b*log(c))*sin(3*b*log(c)) + 8*cos(3*b*lo...
Leaf count of result is larger than twice the leaf count of optimal. 18117 vs. \(2 (158) = 316\).
Time = 1.16 (sec) , antiderivative size = 18117, normalized size of antiderivative = 114.66 \[ \int x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/8*(3*b^3*n^3*x^2*e^(3/2*pi*b*n*sgn(x) - 3/2*pi*b*n + 3/2*pi*b*sgn(c) - 3 /2*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c)))^2*tan(1/2*b*n*log(ab s(x)) + 1/2*b*log(abs(c)))^2*tan(3/2*a)^2*tan(1/2*a)^2 - 27*b^3*n^3*x^2*e^ (1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(3/2*b*n* log(abs(x)) + 3/2*b*log(abs(c)))^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs (c)))^2*tan(3/2*a)^2*tan(1/2*a)^2 - 27*b^3*n^3*x^2*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b* log(abs(c)))^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(3/2*a)^2 *tan(1/2*a)^2 + 3*b^3*n^3*x^2*e^(-3/2*pi*b*n*sgn(x) + 3/2*pi*b*n - 3/2*pi* b*sgn(c) + 3/2*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c)))^2*tan(1/ 2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(3/2*a)^2*tan(1/2*a)^2 + 3*b^3 *n^3*x^2*e^(3/2*pi*b*n*sgn(x) - 3/2*pi*b*n + 3/2*pi*b*sgn(c) - 3/2*pi*b)*t an(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c)))^2*tan(1/2*b*n*log(abs(x)) + 1/ 2*b*log(abs(c)))^2*tan(3/2*a)^2 + 27*b^3*n^3*x^2*e^(1/2*pi*b*n*sgn(x) - 1/ 2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log (abs(c)))^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(3/2*a)^2 + 27*b^3*n^3*x^2*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2* pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c)))^2*tan(1/2*b*n*log(abs(x )) + 1/2*b*log(abs(c)))^2*tan(3/2*a)^2 + 3*b^3*n^3*x^2*e^(-3/2*pi*b*n*sgn( x) + 3/2*pi*b*n - 3/2*pi*b*sgn(c) + 3/2*pi*b)*tan(3/2*b*n*log(abs(x)) +...
Time = 27.39 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {x^2\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}\,3{}\mathrm {i}}{-16+b\,n\,8{}\mathrm {i}}-\frac {3\,x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{8\,b\,n-16{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}\,1{}\mathrm {i}}{-16+b\,n\,24{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}{24\,b\,n-16{}\mathrm {i}} \]