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

Optimal. Leaf size=63 \[ \frac {6 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n} \]

[Out]

6/5*(cos(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/cos(1/2*a+1/2*b*ln(c*x^n))*EllipticE(sin(1/2*a+1/2*b*ln(c*x^n)),2^(1/
2))/b/n+2/5*cos(a+b*ln(c*x^n))^(3/2)*sin(a+b*ln(c*x^n))/b/n

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Rubi [A]  time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2635, 2639} \[ \frac {6 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*Log[c*x^n]]^(5/2)/x,x]

[Out]

(6*EllipticE[(a + b*Log[c*x^n])/2, 2])/(5*b*n) + (2*Cos[a + b*Log[c*x^n]]^(3/2)*Sin[a + b*Log[c*x^n]])/(5*b*n)

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin {align*} \int \frac {\cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cos ^{\frac {5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac {3 \operatorname {Subst}\left (\int \sqrt {\cos (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac {6 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac {2 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 58, normalized size = 0.92 \[ \frac {6 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right ) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{5 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*EllipticE[(a + b*Log[c*x^n])/2, 2] + Sqrt[Cos[a + b*Log[c*x^n]]]*Sin[2*(a + b*Log[c*x^n])])/(5*b*n)

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 280, normalized size = 4.44 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \left (-8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{6}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right ) \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{5 n \sqrt {-2 \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5/n*((2*cos(1/2*a+1/2*b*ln(c*x^n))^2-1)*sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-8*cos(1/2*a+1/2*b*ln(c*x^n))*
sin(1/2*a+1/2*b*ln(c*x^n))^6+8*cos(1/2*a+1/2*b*ln(c*x^n))*sin(1/2*a+1/2*b*ln(c*x^n))^4-3*(sin(1/2*a+1/2*b*ln(c
*x^n))^2)^(1/2)*(2*sin(1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)*EllipticE(cos(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))-2*sin(1
/2*a+1/2*b*ln(c*x^n))^2*cos(1/2*a+1/2*b*ln(c*x^n)))/(-2*sin(1/2*a+1/2*b*ln(c*x^n))^4+sin(1/2*a+1/2*b*ln(c*x^n)
)^2)^(1/2)/sin(1/2*a+1/2*b*ln(c*x^n))/(2*cos(1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.38, size = 65, normalized size = 1.03 \[ -\frac {2\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{7/2}\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{7\,b\,n\,\sqrt {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*log(c*x^n))^(5/2)/x,x)

[Out]

-(2*cos(a + b*log(c*x^n))^(7/2)*sin(a + b*log(c*x^n))*hypergeom([1/2, 7/4], 11/4, cos(a + b*log(c*x^n))^2))/(7
*b*n*(sin(a + b*log(c*x^n))^2)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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