\(\int \frac {\sin (a+b \log (c x^n))}{x^2} \, dx\) [5]

Optimal result

Integrand size = 15, antiderivative size = 57 \[ \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right )}{\left (1+b^2 n^2\right ) x}-\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+b^2 n^2\right ) x} \]

[Out]

-b*n*cos(a+b*ln(c*x^n))/(b^2*n^2+1)/x-sin(a+b*ln(c*x^n))/(b^2*n^2+1)/x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4573} \[ \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )} \]

[In]

Int[Sin[a + b*Log[c*x^n]]/x^2,x]

[Out]

-((b*n*Cos[a + b*Log[c*x^n]])/((1 + b^2*n^2)*x)) - Sin[a + b*Log[c*x^n]]/((1 + b^2*n^2)*x)

Rule 4573

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {b n \cos \left (a+b \log \left (c x^n\right )\right )}{\left (1+b^2 n^2\right ) x}-\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+b^2 n^2\right ) x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.70 \[ \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right )+\sin \left (a+b \log \left (c x^n\right )\right )}{x+b^2 n^2 x} \]

[In]

Integrate[Sin[a + b*Log[c*x^n]]/x^2,x]

[Out]

-((b*n*Cos[a + b*Log[c*x^n]] + Sin[a + b*Log[c*x^n]])/(x + b^2*n^2*x))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {-\cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b n -\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{x \left (b^{2} n^{2}+1\right )}\)	\(45\)

[In]

int(sin(a+b*ln(c*x^n))/x^2,x,method=_RETURNVERBOSE)

[Out]

1/x/(b^2*n^2+1)*(-cos(a+b*ln(c*x^n))*b*n-sin(a+b*ln(c*x^n)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.77 \[ \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{{\left (b^{2} n^{2} + 1\right )} x} \]

[In]

integrate(sin(a+b*log(c*x^n))/x^2,x, algorithm="fricas")

[Out]

-(b*n*cos(b*n*log(x) + b*log(c) + a) + sin(b*n*log(x) + b*log(c) + a))/((b^2*n^2 + 1)*x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.02 (sec) , antiderivative size = 192, normalized size of antiderivative = 3.37 \[ \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\begin {cases} - \frac {i \cos {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 x} + \frac {\log {\left (c x^{n} \right )} \sin {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 n x} - \frac {i \log {\left (c x^{n} \right )} \cos {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 n x} & \text {for}\: b = - \frac {i}{n} \\- \frac {\sin {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 x} + \frac {\log {\left (c x^{n} \right )} \sin {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 n x} + \frac {i \log {\left (c x^{n} \right )} \cos {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{2 n x} & \text {for}\: b = \frac {i}{n} \\- \frac {b n \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} x + x} - \frac {\sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} x + x} & \text {otherwise} \end {cases} \]

[In]

integrate(sin(a+b*ln(c*x**n))/x**2,x)

[Out]

Piecewise((-I*cos(a - I*log(c*x**n)/n)/(2*x) + log(c*x**n)*sin(a - I*log(c*x**n)/n)/(2*n*x) - I*log(c*x**n)*co
s(a - I*log(c*x**n)/n)/(2*n*x), Eq(b, -I/n)), (-sin(a + I*log(c*x**n)/n)/(2*x) + log(c*x**n)*sin(a + I*log(c*x
**n)/n)/(2*n*x) + I*log(c*x**n)*cos(a + I*log(c*x**n)/n)/(2*n*x), Eq(b, I/n)), (-b*n*cos(a + b*log(c*x**n))/(b
**2*n**2*x + x) - sin(a + b*log(c*x**n))/(b**2*n**2*x + x), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (57) = 114\).

Time = 0.22 (sec) , antiderivative size = 209, normalized size of antiderivative = 3.67 \[ \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {{\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \cos \left (b \log \left (c\right )\right )\right )} n + \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + \sin \left (b \log \left (c\right )\right )\right )} \cos \left (b \log \left (x^{n}\right ) + a\right ) - {\left ({\left (b \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \sin \left (b \log \left (c\right )\right )\right )} n - \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) - \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) - \cos \left (b \log \left (c\right )\right )\right )} \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \left (c\right )\right )^{2} + b^{2} \sin \left (b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (b \log \left (c\right )\right )^{2} + \sin \left (b \log \left (c\right )\right )^{2}\right )} x} \]

[In]

integrate(sin(a+b*log(c*x^n))/x^2,x, algorithm="maxima")

[Out]

-1/2*(((b*cos(2*b*log(c))*cos(b*log(c)) + b*sin(2*b*log(c))*sin(b*log(c)) + b*cos(b*log(c)))*n + cos(b*log(c))
*sin(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(c)) + sin(b*log(c)))*cos(b*log(x^n) + a) - ((b*cos(b*log(c))*sin(
2*b*log(c)) - b*cos(2*b*log(c))*sin(b*log(c)) + b*sin(b*log(c)))*n - cos(2*b*log(c))*cos(b*log(c)) - sin(2*b*l
og(c))*sin(b*log(c)) - cos(b*log(c)))*sin(b*log(x^n) + a))/(((b^2*cos(b*log(c))^2 + b^2*sin(b*log(c))^2)*n^2 +
cos(b*log(c))^2 + sin(b*log(c))^2)*x)

Giac [F]

\[ \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )}{x^{2}} \,d x } \]

[In]

integrate(sin(a+b*log(c*x^n))/x^2,x, algorithm="giac")

[Out]

integrate(sin(b*log(c*x^n) + a)/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]

[In]

int(sin(a + b*log(c*x^n))/x^2,x)

[Out]

int(sin(a + b*log(c*x^n))/x^2, x)