Integrand size = 25, antiderivative size = 172 \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {3}{16} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-1/n}+\frac {3}{32} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{48} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x) \] Output:
-3/16*exp(a*(-1/n^2)^(1/2)*n)*(-1/n^2)^(1/2)*n*x^3/((c*x^n)^(1/n))+3/32*(- 1/n^2)^(1/2)*n*x^3*(c*x^n)^(1/n)/exp(a*(-1/n^2)^(1/2)*n)-1/48*(-1/n^2)^(1/ 2)*n*x^3*(c*x^n)^(3/n)/exp(3*a*(-1/n^2)^(1/2)*n)+1/8*exp(3*a*(-1/n^2)^(1/2 )*n)*(-1/n^2)^(1/2)*n*x^3*ln(x)/((c*x^n)^(3/n))
\[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \] Input:
Integrate[x^2*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^3,x]
Output:
Integrate[x^2*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^3, x]
Time = 0.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4996, 4992, 2009}
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.
\(\displaystyle \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \int \left (c x^n\right )^{\frac {3}{n}-1} \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4992 |
\(\displaystyle \frac {1}{8} \sqrt {-\frac {1}{n^2}} x^3 \left (c x^n\right )^{-3/n} \int \left (-3 e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {2}{n}-1}+3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {4}{n}-1}-e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {6}{n}-1}+\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}\right )d\left (c x^n\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} \sqrt {-\frac {1}{n^2}} x^3 \left (c x^n\right )^{-3/n} \left (-\frac {3}{2} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}+\frac {3}{4} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{4/n}-\frac {1}{6} n e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{6/n}+e^{3 a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )\right )\) |
Input:
Int[x^2*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^3,x]
Output:
(Sqrt[-n^(-2)]*x^3*((-3*E^(a*Sqrt[-n^(-2)]*n)*n*(c*x^n)^(2/n))/2 + (3*n*(c *x^n)^(4/n))/(4*E^(a*Sqrt[-n^(-2)]*n)) - (n*(c*x^n)^(6/n))/(6*E^(3*a*Sqrt[ -n^(-2)]*n)) + E^(3*a*Sqrt[-n^(-2)]*n)*Log[c*x^n]))/(8*(c*x^n)^(3/n))
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Simp[(m + 1)^p/(2^p*b^p*d^p*p^p) Int[ExpandIntegrand[(e*x)^m*(E^(a*b*d ^2*(p/(m + 1)))/x^((m + 1)/p) - x^((m + 1)/p)/E^(a*b*d^2*(p/(m + 1))))^p, x ], x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + ( m + 1)^2, 0]
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_ .), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)) Subst[Int[x ^((m + 1)/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
\[\int x^{2} {\sin \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )\right )}^{3}d x\]
Input:
int(x^2*sin(a+(-1/n^2)^(1/2)*ln(c*x^n))^3,x)
Output:
int(x^2*sin(a+(-1/n^2)^(1/2)*ln(c*x^n))^3,x)
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.48 \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{96} \, {\left (-2 i \, x^{6} + 9 i \, x^{4} e^{\left (\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} - 18 i \, x^{2} e^{\left (\frac {4 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} + 12 i \, e^{\left (\frac {6 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {3 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \] Input:
integrate(x^2*sin(a+(-1/n^2)^(1/2)*log(c*x^n))^3,x, algorithm="fricas")
Output:
1/96*(-2*I*x^6 + 9*I*x^4*e^(2*(I*a*n - log(c))/n) - 18*I*x^2*e^(4*(I*a*n - log(c))/n) + 12*I*e^(6*(I*a*n - log(c))/n)*log(x))*e^(-3*(I*a*n - log(c)) /n)
Timed out. \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x**2*sin(a+(-1/n**2)**(1/2)*ln(c*x**n))**3,x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.52 \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {18 \, c^{\frac {2}{n}} x^{3} \sin \left (a\right ) - 12 \, {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} \log \left (x\right ) \sin \left (3 \, a\right ) - {\left (2 \, c^{\frac {6}{n}} x^{6} \sin \left (3 \, a\right ) - 9 \, c^{\frac {4}{n}} x^{4} \sin \left (a\right )\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}}{96 \, c^{\frac {3}{n}} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}} \] Input:
integrate(x^2*sin(a+(-1/n^2)^(1/2)*log(c*x^n))^3,x, algorithm="maxima")
Output:
1/96*(18*c^(2/n)*x^3*sin(a) - 12*(x^n)^(1/n)*log(x)*sin(3*a) - (2*c^(6/n)* x^6*sin(3*a) - 9*c^(4/n)*x^4*sin(a))*(x^n)^(1/n))/(c^(3/n)*(x^n)^(1/n))
Exception generated. \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(x^2*sin(a+(-1/n^2)^(1/2)*log(c*x^n))^3,x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: ((-9 *i)*sageVARn^4*sageVARx^3*exp((-3*i)*sageVARa)*exp((3*sageVARn*abs(sageVAR n)*ln(sageVARx)+3*abs(sageVARn)*ln(sageVARc))/sageVARn^2)+27*i*sageVARn^4* sageVARx^3*exp((-i)
Timed out. \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^2\,{\sin \left (a+\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}\right )}^3 \,d x \] Input:
int(x^2*sin(a + log(c*x^n)*(-1/n^2)^(1/2))^3,x)
Output:
int(x^2*sin(a + log(c*x^n)*(-1/n^2)^(1/2))^3, x)
Time = 0.17 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.25 \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^{3} \left (-12 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}^{2} i +3 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \mathrm {log}\left (x^{n} c \right ) i -5 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}^{2} i n -\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) i n +12 \,\mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}^{3}-9 \,\mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )+9 {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}^{3} n \right )}{24 n} \] Input:
int(x^2*sin(a+(-1/n^2)^(1/2)*log(c*x^n))^3,x)
Output:
(x**3*( - 12*cos((log(x**n*c)*i + a*n)/n)*log(x**n*c)*sin((log(x**n*c)*i + a*n)/n)**2*i + 3*cos((log(x**n*c)*i + a*n)/n)*log(x**n*c)*i - 5*cos((log( x**n*c)*i + a*n)/n)*sin((log(x**n*c)*i + a*n)/n)**2*i*n - cos((log(x**n*c) *i + a*n)/n)*i*n + 12*log(x**n*c)*sin((log(x**n*c)*i + a*n)/n)**3 - 9*log( x**n*c)*sin((log(x**n*c)*i + a*n)/n) + 9*sin((log(x**n*c)*i + a*n)/n)**3*n ))/(24*n)