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

Optimal. Leaf size=86 \[ \frac{\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^{p+1}\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2},\frac{p+3}{2},\sin ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n (p+1) \sqrt{\cos ^2\left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

(Cos[a + b*Log[c*x^n]]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Sin[a + b*Log[c*x^n]]^2]*Sin[a + b*Log[c*x
^n]]^(1 + p))/(b*n*(1 + p)*Sqrt[Cos[a + b*Log[c*x^n]]^2])

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Rubi [A]  time = 0.0601127, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2643} \[ \frac{\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^{p+1}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\sin ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n (p+1) \sqrt{\cos ^2\left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^p/x,x]

[Out]

(Cos[a + b*Log[c*x^n]]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Sin[a + b*Log[c*x^n]]^2]*Sin[a + b*Log[c*x
^n]]^(1 + p))/(b*n*(1 + p)*Sqrt[Cos[a + b*Log[c*x^n]]^2])

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int \frac{\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sin ^p(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 ) \, _2F_1\left (\frac{1}{2},\frac{1+p}{2};\frac{3+p}{2};\sin ^2\left (a+b \log \left (c x^n\right )\right )\right ) \sin ^{1+p}\left (a+b \log \left (c x^n\right )\right )}{b n (1+p) \sqrt{\cos ^2\left (a+b \log \left (c x^n\right )\right )}}\\ \end{align*}

Mathematica [A]  time = 0.149624, size = 86, normalized size = 1. \[ \frac{\sec \left (a+b \log \left (c x^n\right )\right ) \sqrt{\cos ^2\left (a+b \log \left (c x^n\right )\right )} \sin ^{p+1}\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{p+1}{2},\frac{p+3}{2},\sin ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Cos[a + b*Log[c*x^n]]^2]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Sin[a + b*Log[c*x^n]]^2]*Sec[a + b
*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^(1 + p))/(b*n*(1 + p))

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Maple [F]  time = 0.13, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{p}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin(b*log(c*x^n) + a)^p/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*log(c*x**n))**p/x, x)

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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 )^{p}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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