Integrand size = 24, antiderivative size = 88 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{12} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x) \] Output:
1/12*(-1/n^2)^(1/2)*n*x^3*(c*x^n)^(3/n)/exp(a*(-1/n^2)^(1/2)*n)-1/2*exp(a* (-1/n^2)^(1/2)*n)*(-1/n^2)^(1/2)*n*x^3*ln(x)/((c*x^n)^(3/n))
\[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \] Input:
Integrate[x^2*Sin[a + 3*Sqrt[-n^(-2)]*Log[c*x^n]],x]
Output:
Integrate[x^2*Sin[a + 3*Sqrt[-n^(-2)]*Log[c*x^n]], x]
Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
\(\displaystyle \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \int \left (c x^n\right )^{\frac {3}{n}-1} \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4992 |
\(\displaystyle -\frac {1}{2} \sqrt {-\frac {1}{n^2}} x^3 \left (c x^n\right )^{-3/n} \int \left (\frac {e^{a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}-e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {6}{n}-1}\right )d\left (c x^n\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \sqrt {-\frac {1}{n^2}} x^3 \left (c x^n\right )^{-3/n} \left (e^{a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )-\frac {1}{6} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{6/n}\right )\) |
Input:
Int[x^2*Sin[a + 3*Sqrt[-n^(-2)]*Log[c*x^n]],x]
Output:
-1/2*(Sqrt[-n^(-2)]*x^3*(-1/6*(n*(c*x^n)^(6/n))/E^(a*Sqrt[-n^(-2)]*n) + E^ (a*Sqrt[-n^(-2)]*n)*Log[c*x^n]))/(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])
Leaf count of result is larger than twice the leaf count of optimal. \(618\) vs. \(2(77)=154\).
Time = 1.56 (sec) , antiderivative size = 619, normalized size of antiderivative = 7.03
method | result | size |
parts | \(\frac {3 n \,x^{2} \sqrt {-\frac {1}{n^{2}}}\, {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )\right )}{8}-\frac {x^{2} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )\right )}{8}-\frac {-\frac {n \left (-\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right )}{2}+c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) \tan \left (\frac {a}{2}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{2}\right )+\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3}}{6}-\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} c^{-\frac {1}{n}} x^{3} {\tan \left (\frac {a}{2}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}{6}+\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}{2}\right )}{1+{\tan \left (\frac {a}{2}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, n^{2} \left (\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right )}{2}+\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \tan \left (\frac {a}{2}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{2}\right )}{3 \sqrt {-\frac {1}{n^{2}}}\, n}-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}{2}-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) \tan \left (\frac {a}{2}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{2}\right )}{\sqrt {-\frac {1}{n^{2}}}\, n}\right )}{1+{\tan \left (\frac {a}{2}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}{4 n}\) | \(619\) |
Input:
int(x^2*sin(a+3*(-1/n^2)^(1/2)*ln(c*x^n)),x,method=_RETURNVERBOSE)
Output:
3/8*n*x^2*(-1/n^2)^(1/2)*exp(1/n*ln(c*x^n)-1/n*ln(c))*cos(a+3*(-1/n^2)^(1/ 2)*ln(c*x^n))-1/8*x^2*exp(1/n*ln(c*x^n)-1/n*ln(c))*sin(a+3*(-1/n^2)^(1/2)* ln(c*x^n))-1/4/n*(-n*(-1/2*(-1/n^2)^(1/2)*n/(c^(1/n))*exp(1/n*(ln(c*x^n)-n *ln(x)))*x^3*ln(x)+1/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*ln(x)*tan( 1/2*a+3/2*(-1/n^2)^(1/2)*ln(c*x^n))+1/6*(-1/n^2)^(1/2)*n/(c^(1/n))*exp(1/n *(ln(c*x^n)-n*ln(x)))*x^3-1/6*(-1/n^2)^(1/2)*n*exp(1/n*(ln(c*x^n)-n*ln(x)) )/(c^(1/n))*x^3*tan(1/2*a+3/2*(-1/n^2)^(1/2)*ln(c*x^n))^2+1/2*(-1/n^2)^(1/ 2)*n/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*ln(x)*tan(1/2*a+3/2*(-1/n^ 2)^(1/2)*ln(c*x^n))^2)/(1+tan(1/2*a+3/2*(-1/n^2)^(1/2)*ln(c*x^n))^2)+3*(-1 /n^2)^(1/2)*n^2*(1/2/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*ln(x)+1/3/ (-1/n^2)^(1/2)/(c^(1/n))/n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*tan(1/2*a+3/2* (-1/n^2)^(1/2)*ln(c*x^n))-1/2/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*l n(x)*tan(1/2*a+3/2*(-1/n^2)^(1/2)*ln(c*x^n))^2-1/(-1/n^2)^(1/2)/(c^(1/n))/ n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*ln(x)*tan(1/2*a+3/2*(-1/n^2)^(1/2)*ln(c *x^n)))/(1+tan(1/2*a+3/2*(-1/n^2)^(1/2)*ln(c*x^n))^2))
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.48 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{12} \, {\left (i \, x^{6} - 6 i \, e^{\left (\frac {2 \, {\left (i \, a n - 3 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {i \, a n - 3 \, \log \left (c\right )}{n}\right )} \] Input:
integrate(x^2*sin(a+3*(-1/n^2)^(1/2)*log(c*x^n)),x, algorithm="fricas")
Output:
1/12*(I*x^6 - 6*I*e^(2*(I*a*n - 3*log(c))/n)*log(x))*e^(-(I*a*n - 3*log(c) )/n)
\[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^{2} \sin {\left (a + 3 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \] Input:
integrate(x**2*sin(a+3*(-1/n**2)**(1/2)*ln(c*x**n)),x)
Output:
Integral(x**2*sin(a + 3*sqrt(-1/n**2)*log(c*x**n)), x)
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {c^{\frac {6}{n}} x^{6} \sin \left (a\right ) + 6 \, \log \left (x\right ) \sin \left (a\right )}{12 \, c^{\frac {3}{n}}} \] Input:
integrate(x^2*sin(a+3*(-1/n^2)^(1/2)*log(c*x^n)),x, algorithm="maxima")
Output:
1/12*(c^(6/n)*x^6*sin(a) + 6*log(x)*sin(a))/c^(3/n)
Time = 0.28 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \] Input:
integrate(x^2*sin(a+3*(-1/n^2)^(1/2)*log(c*x^n)),x, algorithm="giac")
Output:
+Infinity
Time = 20.53 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {x^3\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}}{6\,n\,\sqrt {-\frac {1}{n^2}}+6{}\mathrm {i}}-\frac {x^3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}{6\,n\,\sqrt {-\frac {1}{n^2}}-6{}\mathrm {i}} \] Input:
int(x^2*sin(a + 3*log(c*x^n)*(-1/n^2)^(1/2)),x)
Output:
- (x^3*exp(-a*1i)/(c*x^n)^((-1/n^2)^(1/2)*3i))/(6*n*(-1/n^2)^(1/2) + 6i) - (x^3*exp(a*1i)*(c*x^n)^((-1/n^2)^(1/2)*3i))/(6*n*(-1/n^2)^(1/2) - 6i)
Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^{3} \left (-3 \cos \left (\frac {3 \,\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \mathrm {log}\left (x^{n} c \right ) i +\cos \left (\frac {3 \,\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) i n +3 \,\mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {3 \,\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )\right )}{6 n} \] Input:
int(x^2*sin(a+3*(-1/n^2)^(1/2)*log(c*x^n)),x)
Output:
(x**3*( - 3*cos((3*log(x**n*c)*i + a*n)/n)*log(x**n*c)*i + cos((3*log(x**n *c)*i + a*n)/n)*i*n + 3*log(x**n*c)*sin((3*log(x**n*c)*i + a*n)/n)))/(6*n)