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

Optimal. Leaf size=39 \[ \frac{\log (x)}{2}-\frac{\sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

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

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Rubi [A]  time = 0.0304173, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2635, 8} \[ \frac{\log (x)}{2}-\frac{\sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^2/x,x]

[Out]

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sin ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac{\operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac{\log (x)}{2}-\frac{\cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end{align*}

Mathematica [A]  time = 0.0714815, size = 36, normalized size = 0.92 \[ -\frac{\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-2*(a + b*Log[c*x^n]) + Sin[2*(a + b*Log[c*x^n])])/(4*b*n)

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Maple [A]  time = 0.022, size = 52, normalized size = 1.3 \begin{align*} -{\frac{\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{2\,bn}}+{\frac{\ln \left ( c{x}^{n} \right ) }{2\,n}}+{\frac{a}{2\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14662, size = 74, normalized size = 1.9 \begin{align*} \frac{2 \, b n \log \left (x\right ) - \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{4 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*b*n*log(x) - cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) - cos(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/(b*n)

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Fricas [A]  time = 0.49112, size = 119, normalized size = 3.05 \begin{align*} \frac{b n \log \left (x\right ) - \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 )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 23.0007, size = 56, normalized size = 1.44 \begin{align*} - \frac{\begin{cases} \log{\left (x \right )} \cos{\left (2 a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log{\left (x \right )} \cos{\left (2 a + 2 b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\sin{\left (2 a + 2 b n \log{\left (x \right )} + 2 b \log{\left (c \right )} \right )}}{2 b n} & \text{otherwise} \end{cases}}{2} + \frac{\log{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((log(x)*cos(2*a), Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*cos(2*a + 2*b*log(c)), Eq(n, 0)), (sin
(2*a + 2*b*n*log(x) + 2*b*log(c))/(2*b*n), True))/2 + log(x)/2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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