\(\int \frac {\sin ^p(a+b \log (c x^n))}{x} \, dx\) [83]

Optimal result

Integrand size = 17, antiderivative size = 86 \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\sin ^2\left (a+b \log \left (c x^n\right )\right )\right ) \sin ^{1+p}\left (a+b \log \left (c x^n\right )\right )}{b n (1+p) \sqrt {\cos ^2\left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

cos(a+b*ln(c*x^n))*hypergeom([1/2, 1/2+1/2*p],[3/2+1/2*p],sin(a+b*ln(c*x^n))^2)*sin(a+b*ln(c*x^n))^(p+1)/b/n/(
p+1)/(cos(a+b*ln(c*x^n))^2)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2722} \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^{p+1}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {p+1}{2},\frac {p+3}{2},\sin ^2\left (a+b \log \left (c x^n\right )\right )\right )}{b n (p+1) \sqrt {\cos ^2\left (a+b \log \left (c x^n\right )\right )}} \]

[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 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 ) \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sqrt {\cos ^2\left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+p}{2},\frac {3+p}{2},\sin ^2\left (a+b \log \left (c x^n\right )\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right ) \sin ^{1+p}\left (a+b \log \left (c x^n\right )\right )}{b n (1+p)} \]

[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))

Maple [F]

\[\int \frac {{\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}}{x}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 27.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )\,{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{p+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {1}{2}-\frac {p}{2};\ \frac {3}{2};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{b\,n\,{\left ({\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}^{\frac {p}{2}+\frac {1}{2}}} \]

[In]

int(sin(a + b*log(c*x^n))^p/x,x)

[Out]

-(cos(a + b*log(c*x^n))*sin(a + b*log(c*x^n))^(p + 1)*hypergeom([1/2, 1/2 - p/2], 3/2, cos(a + b*log(c*x^n))^2
))/(b*n*(sin(a + b*log(c*x^n))^2)^(p/2 + 1/2))