Integrand size = 17, antiderivative size = 158 \[ \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{\left (16+40 b^2 n^2+9 b^4 n^4\right ) x^2}-\frac {12 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{\left (16+40 b^2 n^2+9 b^4 n^4\right ) x^2}-\frac {3 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (4+9 b^2 n^2\right ) x^2}-\frac {2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (4+9 b^2 n^2\right ) x^2} \]
-6*b^3*n^3*cos(a+b*ln(c*x^n))/(9*b^4*n^4+40*b^2*n^2+16)/x^2-12*b^2*n^2*sin (a+b*ln(c*x^n))/(9*b^4*n^4+40*b^2*n^2+16)/x^2-3*b*n*cos(a+b*ln(c*x^n))*sin (a+b*ln(c*x^n))^2/(9*b^2*n^2+4)/x^2-2*sin(a+b*ln(c*x^n))^3/(9*b^2*n^2+4)/x ^2
Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {-3 b n \left (4+9 b^2 n^2\right ) \cos \left (a+b \log \left (c x^n\right )\right )+3 b n \left (4+b^2 n^2\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+4 \left (-4-13 b^2 n^2+\left (4+b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 \left (16+40 b^2 n^2+9 b^4 n^4\right ) x^2} \]
(-3*b*n*(4 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + 3*b*n*(4 + b^2*n^2)*Cos[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)*x^2)
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
\(\displaystyle \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 4990 |
\(\displaystyle \frac {6 b^2 n^2 \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^3}dx}{9 b^2 n^2+4}-\frac {2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^2 n^2+4\right )}-\frac {3 b n \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^2 n^2+4\right )}\) |
\(\Big \downarrow \) 4988 |
\(\displaystyle -\frac {2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^2 n^2+4\right )}-\frac {3 b n \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^2 n^2+4\right )}+\frac {6 b^2 n^2 \left (-\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (b^2 n^2+4\right )}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (b^2 n^2+4\right )}\right )}{9 b^2 n^2+4}\) |
(-3*b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/((4 + 9*b^2*n^2)*x^ 2) - (2*Sin[a + b*Log[c*x^n]]^3)/((4 + 9*b^2*n^2)*x^2) + (6*b^2*n^2*(-((b* n*Cos[a + b*Log[c*x^n]])/((4 + b^2*n^2)*x^2)) - (2*Sin[a + b*Log[c*x^n]])/ ((4 + b^2*n^2)*x^2)))/(4 + 9*b^2*n^2)
3.1.18.3.1 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]
Time = 6.38 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.40
method | result | size |
parallelrisch | \(\frac {6 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{6} b^{3} n^{3}-24 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{5} b^{2} n^{2}+\left (18 b^{3} n^{3}+48 b n \right ) {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4}+\left (-64 b^{2} n^{2}-64\right ) {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}+\left (-18 b^{3} n^{3}-48 b n \right ) {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}-24 \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right ) b^{2} n^{2}-6 b^{3} n^{3}}{9 x^{2} \left (b^{2} n^{2}+4\right ) \left (b^{2} n^{2}+\frac {4}{9}\right ) {\left (1+{\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}\right )}^{3}}\) | \(221\) |
1/9*(6*tan(1/2*a+b*ln((c*x^n)^(1/2)))^6*b^3*n^3-24*tan(1/2*a+b*ln((c*x^n)^ (1/2)))^5*b^2*n^2+(18*b^3*n^3+48*b*n)*tan(1/2*a+b*ln((c*x^n)^(1/2)))^4+(-6 4*b^2*n^2-64)*tan(1/2*a+b*ln((c*x^n)^(1/2)))^3+(-18*b^3*n^3-48*b*n)*tan(1/ 2*a+b*ln((c*x^n)^(1/2)))^2-24*tan(1/2*a+b*ln((c*x^n)^(1/2)))*b^2*n^2-6*b^3 *n^3)/x^2/(b^2*n^2+4)/(b^2*n^2+4/9)/(1+tan(1/2*a+b*ln((c*x^n)^(1/2)))^2)^3
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {3 \, {\left (b^{3} n^{3} + 4 \, b n\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \, {\left (3 \, b^{3} n^{3} + 4 \, b n\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 2 \, {\left (7 \, b^{2} n^{2} - {\left (b^{2} n^{2} + 4\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{{\left (9 \, b^{4} n^{4} + 40 \, b^{2} n^{2} + 16\right )} x^{2}} \]
(3*(b^3*n^3 + 4*b*n)*cos(b*n*log(x) + b*log(c) + a)^3 - 3*(3*b^3*n^3 + 4*b *n)*cos(b*n*log(x) + b*log(c) + a) - 2*(7*b^2*n^2 - (b^2*n^2 + 4)*cos(b*n* log(x) + b*log(c) + a)^2 + 4)*sin(b*n*log(x) + b*log(c) + a))/((9*b^4*n^4 + 40*b^2*n^2 + 16)*x^2)
Result contains complex when optimal does not.
Time = 35.84 (sec) , antiderivative size = 886, normalized size of antiderivative = 5.61 \[ \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Too large to display} \]
Piecewise((-sin(3*a - 6*I*log(c*x**n)/n)/(64*x**2) - 3*I*cos(a - 2*I*log(c *x**n)/n)/(16*x**2) + 3*I*cos(3*a - 6*I*log(c*x**n)/n)/(64*x**2) + 3*log(c *x**n)*sin(a - 2*I*log(c*x**n)/n)/(8*n*x**2) - 3*I*log(c*x**n)*cos(a - 2*I *log(c*x**n)/n)/(8*n*x**2), Eq(b, -2*I/n)), (-27*sin(a - 2*I*log(c*x**n)/( 3*n))/(64*x**2) + sin(3*a - 2*I*log(c*x**n)/n)/(16*x**2) + 9*I*cos(a - 2*I *log(c*x**n)/(3*n))/(64*x**2) - log(c*x**n)*sin(3*a - 2*I*log(c*x**n)/n)/( 8*n*x**2) + I*log(c*x**n)*cos(3*a - 2*I*log(c*x**n)/n)/(8*n*x**2), Eq(b, - 2*I/(3*n))), (-27*sin(a + 2*I*log(c*x**n)/(3*n))/(64*x**2) - 9*I*cos(a + 2 *I*log(c*x**n)/(3*n))/(64*x**2) - I*cos(3*a + 2*I*log(c*x**n)/n)/(16*x**2) - log(c*x**n)*sin(3*a + 2*I*log(c*x**n)/n)/(8*n*x**2) - I*log(c*x**n)*cos (3*a + 2*I*log(c*x**n)/n)/(8*n*x**2), Eq(b, 2*I/(3*n))), (-3*sin(a + 2*I*l og(c*x**n)/n)/(16*x**2) - sin(3*a + 6*I*log(c*x**n)/n)/(64*x**2) - 3*I*cos (3*a + 6*I*log(c*x**n)/n)/(64*x**2) + 3*log(c*x**n)*sin(a + 2*I*log(c*x**n )/n)/(8*n*x**2) + 3*I*log(c*x**n)*cos(a + 2*I*log(c*x**n)/n)/(8*n*x**2), E q(b, 2*I/n)), (-9*b**3*n**3*sin(a + b*log(c*x**n))**2*cos(a + b*log(c*x**n ))/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2) - 6*b**3*n**3*cos(a + b*log(c*x**n))**3/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2) - 14*b* *2*n**2*sin(a + b*log(c*x**n))**3/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2) - 12*b**2*n**2*sin(a + b*log(c*x**n))*cos(a + b*log(c*x**n))**2/( 9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2) - 12*b*n*sin(a + b*log(...
Leaf count of result is larger than twice the leaf count of optimal. 1007 vs. \(2 (158) = 316\).
Time = 0.25 (sec) , antiderivative size = 1007, normalized size of antiderivative = 6.37 \[ \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Too large to display} \]
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)))*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))*sin(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))*s in(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))*si n(2*b*log(c)))*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*log(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*log(c)))*s...
\[ \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \]