3.59 \(\int \sin ^{\frac{3}{2}}(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=109 \[ \frac{2 x \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{1}{4} \left (-3-\frac{2 i}{b n}\right ),\frac{1}{4} \left (1-\frac{2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0727374, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4483, 4491, 364} \[ \frac{2 x \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-3-\frac{2 i}{b n}\right );\frac{1}{4} \left (1-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^(3/2),x]

[Out]

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

Rule 4483

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, 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 \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \sin ^{\frac{3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{\frac{3 i b}{2}-\frac{1}{n}} \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname{Subst}\left (\int x^{-1-\frac{3 i b}{2}+\frac{1}{n}} \left (1-e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ &=\frac{2 x \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-3-\frac{2 i}{b n}\right );\frac{1}{4} \left (1-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.79785, size = 161, normalized size = 1.48 \[ \frac{x \left (6 i b^2 n^2 \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{4}-\frac{i}{2 b n},\frac{5}{4}-\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(b n-2 i) \left (4 \sin ^2\left (a+b \log \left (c x^n\right )\right )-3 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )\right )}{(b n-2 i) \left (9 b^2 n^2+4\right ) \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[a + b*Log[c*x^n]]^(3/2),x]

[Out]

(x*((6*I)*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Hypergeometric2F1[1, 3/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*
n), E^((2*I)*(a + b*Log[c*x^n]))] + (-2*I + b*n)*(4*Sin[a + b*Log[c*x^n]]^2 - 3*b*n*Sin[2*(a + b*Log[c*x^n])])
))/((-2*I + b*n)*(4 + 9*b^2*n^2)*Sqrt[Sin[a + b*Log[c*x^n]]])

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Maple [F]  time = 0.166, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b*ln(c*x^n))^(3/2),x)

[Out]

int(sin(a+b*ln(c*x^n))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

integrate(sin(b*log(c*x^n) + a)^(3/2), 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))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*ln(c*x**n))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*log(c*x^n))^(3/2),x, algorithm="giac")

[Out]

integrate(sin(b*log(c*x^n) + a)^(3/2), x)