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

Optimal. Leaf size=57 \[ -\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )} \]

[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]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4485} \[ -\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[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 4485

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*} \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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 40, normalized size = 0.70 \[ -\frac {\sin \left (a+b \log \left (c x^n\right )\right )+b n \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2 x+x} \]

Antiderivative was successfully verified.

[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))

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 44, normalized size = 0.77 \[ -\frac {b n \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{{\left (b^{2} n^{2} + 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 209, normalized size = 3.67 \[ -\frac {{\left ({\left (b \cos \left (2 \, b \log \relax (c)\right ) \cos \left (b \log \relax (c)\right ) + b \sin \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + b \cos \left (b \log \relax (c)\right )\right )} n + \cos \left (b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) - \cos \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + \sin \left (b \log \relax (c)\right )\right )} \cos \left (b \log \left (x^{n}\right ) + a\right ) - {\left ({\left (b \cos \left (b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) - b \cos \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + b \sin \left (b \log \relax (c)\right )\right )} n - \cos \left (2 \, b \log \relax (c)\right ) \cos \left (b \log \relax (c)\right ) - \sin \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) - \cos \left (b \log \relax (c)\right )\right )} \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \relax (c)\right )^{2} + b^{2} \sin \left (b \log \relax (c)\right )^{2}\right )} n^{2} + \cos \left (b \log \relax (c)\right )^{2} + \sin \left (b \log \relax (c)\right )^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*log(c*x^n))/x^2,x)

[Out]

int(sin(a + b*log(c*x^n))/x^2, x)

________________________________________________________________________________________

sympy [A]  time = 7.54, size = 287, normalized size = 5.04 \[ \begin {cases} - \frac {\log {\relax (x )} \sin {\left (- a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 x} - \frac {i \log {\relax (x )} \cos {\left (- a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 x} + \frac {\sin {\left (- a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 x} - \frac {\log {\relax (c )} \sin {\left (- a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 n x} - \frac {i \log {\relax (c )} \cos {\left (- a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 n x} & \text {for}\: b = - \frac {i}{n} \\\frac {\log {\relax (x )} \sin {\left (a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 x} + \frac {i \log {\relax (x )} \cos {\left (a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 x} + \frac {i \cos {\left (a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 x} + \frac {\log {\relax (c )} \sin {\left (a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 n x} + \frac {i \log {\relax (c )} \cos {\left (a + i \log {\relax (x )} + \frac {i \log {\relax (c )}}{n} \right )}}{2 n x} & \text {for}\: b = \frac {i}{n} \\- \frac {b n \cos {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b^{2} n^{2} x + x} - \frac {\sin {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b^{2} n^{2} x + x} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________