\(\int \cos ^{\frac {5}{2}}(a+b \log (c x^n)) \, dx\) [114]

Optimal result

Integrand size = 15, antiderivative size = 110 \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right ),-\frac {2 i+b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}} \]

[Out]

2*x*cos(a+b*ln(c*x^n))^(5/2)*hypergeom([-5/2, -5/4-1/2*I/b/n],[1/4*(-2*I-b*n)/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b)
)/(2-5*I*b*n)/(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4572, 4580, 371} \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right ),-\frac {b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}} \]

[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, -1/4*(2*I + b*n)/(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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4572

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 4580

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {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 ) \text {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 ) \operatorname {Hypergeometric2F1}\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] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(696\) vs. \(2(110)=220\).

Time = 8.49 (sec) , antiderivative size = 696, normalized size of antiderivative = 6.33 \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {30 b^3 e^{i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} n^3 x^{1-i b n} \sqrt {2+2 e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}} \left ((2 i+b n) x^{2 i b n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n},\frac {7}{4}-\frac {i}{2 b n},-e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}\right )+(-2 i+3 b n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {3}{4}-\frac {i}{2 b n},-e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}\right )\right )}{(2-5 i b n) (2 i+b n) (-2 i+3 b n) (-2 i+5 b n) \left (-2 i-b n+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} (-2 i+b n)\right ) \sqrt {e^{-i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{-i b n} \left (1+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}\right )}}+\sqrt {\cos \left (a+b n \log (x)+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} \left (-\frac {2 x \left (2 \cos \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+15 b^2 n^2 \cos \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )-b n \sin \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}{(-2 i+5 b n) (2 i+5 b n) \left (-2 \cos \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+b n \sin \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}+\frac {x \sin (2 b n \log (x)) \left (5 b n \cos \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )-2 \sin \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{(-2 i+5 b n) (2 i+5 b n)}+\frac {x \cos (2 b n \log (x)) \left (2 \cos \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+5 b n \sin \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{(-2 i+5 b n) (2 i+5 b n)}\right ) \]

[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, -
1/4*(2*I + b*n)/(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
*Log[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
*Log[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]

\[\int {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {5}{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}} \,d x } \]

[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)

Giac [F]

\[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2} \,d x \]

[In]

int(cos(a + b*log(c*x^n))^(5/2),x)

[Out]

int(cos(a + b*log(c*x^n))^(5/2), x)