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

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

[Out]

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

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Rubi [A]  time = 0.0880578, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4493, 4491, 364} \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},\frac{1}{4} \left (\frac{2 i}{b n}-1\right );\frac{1}{4} \left (3+\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}}{x (2+i b n) \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Hypergeometric2F1[-1/2, (-1 + (2*I)/(b*n))/4, (3 + (2*I)/(b*n))/4, E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Sin
[a + b*Log[c*x^n]]])/((2 + I*b*n)*x*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 \frac{\sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}}{x^2} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int x^{-1-\frac{1}{n}} \sqrt{\sin (a+b \log (x))} \, dx,x,c x^n\right )}{n x}\\ &=\frac{\left (\left (c x^n\right )^{\frac{i b}{2}+\frac{1}{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{1}{n}} \sqrt{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n x \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}}}\\ &=-\frac{2 \, _2F_1\left (-\frac{1}{2},\frac{1}{4} \left (-1+\frac{2 i}{b n}\right );\frac{1}{4} \left (3+\frac{2 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 )}}{(2+i b n) x \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}}}\\ \end{align*}

Mathematica [A]  time = 1.40171, size = 99, normalized size = 0.89 \[ -\frac{2 i \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,\frac{5}{4}+\frac{i}{2 b n},\frac{3}{4}+\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{x (b n-2 i)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.168, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\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(sin(a+b*ln(c*x^n))^(1/2)/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sin(a + b*log(c*x**n)))/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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