3.114 \(\int \cos ^{\frac{5}{2}}(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=110 \[ \frac{2 x \text{Hypergeometric2F1}\left (-\frac{5}{2},\frac{1}{4} \left (-5-\frac{2 i}{b n}\right ),-\frac{b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0718108, antiderivative size = 110, 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 = {4484, 4492, 364} \[ \frac{2 x \, _2F_1\left (-\frac{5}{2},\frac{1}{4} \left (-5-\frac{2 i}{b n}\right );-\frac{b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*Log[c*x^n]]^(5/2),x]

[Out]

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

Rule 4484

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Cos[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 4492

Int[Cos[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(Cos[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 \cos ^{\frac{5}{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}} \cos ^{\frac{5}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{\frac{5 i b}{2}-\frac{1}{n}} \cos ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname{Subst}\left (\int x^{-1-\frac{5 i b}{2}+\frac{1}{n}} \left (1+e^{2 i a} x^{2 i b}\right )^{5/2} \, dx,x,c x^n\right )}{n \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}}\\ &=\frac{2 x \cos ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac{5}{2},\frac{1}{4} \left (-5-\frac{2 i}{b n}\right );-\frac{2 i+b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}}\\ \end{align*}

Mathematica [B]  time = 7.1712, size = 696, normalized size = 6.33 \[ \frac{30 b^3 n^3 x^{1-i b n} e^{i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} \sqrt{2+2 x^{2 i b n} e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}} \left ((b n+2 i) x^{2 i b n} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n},\frac{7}{4}-\frac{i}{2 b n},x^{2 i b n} \left (-e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}\right )\right )+(3 b n-2 i) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{b n+2 i}{4 b n},\frac{3}{4}-\frac{i}{2 b n},x^{2 i b n} \left (-e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}\right )\right )\right )}{(2-5 i b n) (b n+2 i) (3 b n-2 i) (5 b n-2 i) \left ((b n-2 i) e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}-b n-2 i\right ) \sqrt{x^{-i b n} e^{-i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} \left (1+x^{2 i b n} e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}\right )}}+\sqrt{\cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x)\right )} \left (-\frac{2 x \left (15 b^2 n^2 \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )-b n \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+2 \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}{(5 b n-2 i) (5 b n+2 i) \left (b n \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )-2 \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}+\frac{x \sin (2 b n \log (x)) \left (5 b n \cos \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )-2 \sin \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )}{(5 b n-2 i) (5 b n+2 i)}+\frac{x \cos (2 b n \log (x)) \left (5 b n \sin \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )+2 \cos \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )}{(5 b n-2 i) (5 b n+2 i)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[a + b*Log[c*x^n]]^(5/2),x]

[Out]

(30*b^3*E^(I*(a + b*(-(n*Log[x]) + Log[c*x^n])))*n^3*x^(1 - I*b*n)*Sqrt[2 + 2*E^((2*I)*(a + b*(-(n*Log[x]) + L
og[c*x^n])))*x^((2*I)*b*n)]*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(
b*n), -(E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))] + (-2*I + 3*b*n)*Hypergeometric2F1[1/2, -
(2*I + b*n)/(4*b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))]))/((2 -
(5*I)*b*n)*(2*I + b*n)*(-2*I + 3*b*n)*(-2*I + 5*b*n)*(-2*I - b*n + E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])
))*(-2*I + b*n))*Sqrt[(1 + E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))/(E^(I*(a + b*(-(n*Log[x
]) + Log[c*x^n])))*x^(I*b*n))]) + Sqrt[Cos[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]]*((-2*x*(2*Cos[a + b
*(-(n*Log[x]) + Log[c*x^n])] + 15*b^2*n^2*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] - b*n*Sin[a + b*(-(n*Log[x]) +
Log[c*x^n])]))/((-2*I + 5*b*n)*(2*I + 5*b*n)*(-2*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + b*n*Sin[a + b*(-(n*L
og[x]) + Log[c*x^n])])) + (x*Sin[2*b*n*Log[x]]*(5*b*n*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] - 2*Sin[2*(a +
b*(-(n*Log[x]) + Log[c*x^n]))]))/((-2*I + 5*b*n)*(2*I + 5*b*n)) + (x*Cos[2*b*n*Log[x]]*(2*Cos[2*(a + b*(-(n*L
og[x]) + Log[c*x^n]))] + 5*b*n*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/((-2*I + 5*b*n)*(2*I + 5*b*n)))

________________________________________________________________________________________

Maple [F]  time = 0.151, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+b*ln(c*x^n))^(5/2),x)

[Out]

int(cos(a+b*ln(c*x^n))^(5/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(b*log(c*x^n) + a)^(5/2), x)

________________________________________________________________________________________

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(cos(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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(cos(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

integrate(cos(b*log(c*x^n) + a)^(5/2), x)