\(\int x \sqrt {\sin (a+b \log (c x^n))} \, dx\) [53]

Optimal result

Integrand size = 17, antiderivative size = 111 \[ \int x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {4 i}{b n}\right ),\frac {1}{4} \left (3-\frac {4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(4-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

[Out]

2*x^2*hypergeom([-1/2, -1/4-I/b/n],[3/4-I/b/n],exp(2*I*a)*(c*x^n)^(2*I*b))*sin(a+b*ln(c*x^n))^(1/2)/(4-I*b*n)/
(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4581, 4579, 371} \[ \int x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {4 i}{b n}\right ),\frac {1}{4} \left (3-\frac {4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(4-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

[In]

Int[x*Sqrt[Sin[a + b*Log[c*x^n]]],x]

[Out]

(2*x^2*Hypergeometric2F1[-1/2, (-1 - (4*I)/(b*n))/4, (3 - (4*I)/(b*n))/4, E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[
Sin[a + b*Log[c*x^n]]])/((4 - I*b*n)*Sqrt[1 - E^((2*I)*a)*(c*x^n)^((2*I)*b)])

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4579

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] :> Dist[Sin[d*(a + b*Log[x])]^p*(x^(
I*b*d*p)/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p), Int[(e*x)^m*((1 - E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)), x], x] /
; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rule 4581

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a
, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sqrt {\sin (a+b \log (x))} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^2 \left (c x^n\right )^{\frac {i b}{2}-\frac {2}{n}} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int x^{-1-\frac {i b}{2}+\frac {2}{n}} \sqrt {1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \\ & = \frac {2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {4 i}{b n}\right ),\frac {1}{4} \left (3-\frac {4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(4-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.00 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.31 \[ \int x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {i \sqrt {2} x^2 \sqrt {-i e^{-i a} \left (c x^n\right )^{-i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4}-\frac {i}{b n},\frac {3}{4}-\frac {i}{b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(4 i+b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

[In]

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

[Out]

(I*Sqrt[2]*x^2*Sqrt[((-I)*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)))/(E^(I*a)*(c*x^n)^(I*b))]*Hypergeometric2F1[-1/
2, -1/4 - I/(b*n), 3/4 - I/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)])/((4*I + b*n)*Sqrt[1 - E^((2*I)*a)*(c*x^n)^((
2*I)*b)])

Maple [F]

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

[In]

int(x*sin(a+b*ln(c*x^n))^(1/2),x)

[Out]

int(x*sin(a+b*ln(c*x^n))^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*sin(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

integrate(x*sin(a+b*ln(c*x**n))**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate(x*sin(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(x*sin(a+b*log(c*x^n))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(x*sin(a + b*log(c*x^n))^(1/2),x)

[Out]

int(x*sin(a + b*log(c*x^n))^(1/2), x)