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

Optimal. Leaf size=111 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (\frac {2 i}{b n}-3\right );\frac {1}{4} \left (1+\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x (2+3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (\frac {2 i}{b n}-3\right );\frac {1}{4} \left (1+\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x (2+3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*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)*x*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^(3/2))

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])

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 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])

Rubi steps

\begin {align*} \int \frac {\sin ^{\frac {3}{2}}\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}} \sin ^{\frac {3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n x}\\ &=\frac {\left (\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 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ &=-\frac {2 \, _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) x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.18, size = 172, normalized size = 1.55 \[ \frac {6 i b^2 n^2 \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \, _2F_1\left (1,\frac {3}{4}+\frac {i}{2 b n};\frac {5}{4}+\frac {i}{2 b n};e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-(b n+2 i) \left (4 \sin ^2\left (a+b \log \left (c x^n\right )\right )+3 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{x (b n+2 i) (3 b n-2 i) (3 b n+2 i) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((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)*(-2*I + 3*b*n)*(2*I + 3*b*n)*x*Sqrt[Sin[a + b*Log[c*x^n]]])

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{\frac {3}{2}}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________