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

Optimal. Leaf size=111 \[ \frac{2 x^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{4} \left (-1-\frac{4 i}{b n}\right ),\frac{1}{4} \left (3-\frac{4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}}{(4-i b n) \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

[Out]

(2*x^2*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)])

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Rubi [A]  time = 0.0860572, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4493, 4491, 364} \[ \frac{2 x^2 \, _2F_1\left (-\frac{1}{2},\frac{1}{4} \left (-1-\frac{4 i}{b n}\right );\frac{1}{4} \left (3-\frac{4 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}}{(4-i b n) \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

Antiderivative was successfully verified.

[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 4493

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])

Rule 4491

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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{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 ) \operatorname{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 \, _2F_1\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]  time = 1.39445, size = 94, normalized size = 0.85 \[ \frac{2 x^2 \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,\frac{5}{4}-\frac{i}{b n},\frac{3}{4}-\frac{i}{b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-4+i b n} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*x^2*Hypergeometric2F1[1, 5/4 - I/(b*n), 3/4 - I/(b*n), E^((2*I)*(a + b*
Log[c*x^n]))]*Sqrt[Sin[a + b*Log[c*x^n]]])/(-4 + I*b*n)

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Maple [F]  time = 0.349, size = 0, normalized size = 0. \begin{align*} \int x\sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)