\(\int \frac {x^m}{\cos ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [131]

Optimal result

Integrand size = 19, antiderivative size = 130 \[ \int \frac {x^m}{\cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{1+m} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {2 i+2 i m-5 b n}{4 b n},-\frac {2 i+2 i m-9 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m+5 i b n) \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

2*x^(1+m)*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*hypergeom([5/2, 1/4*(-2*I-2*I*m+5*b*n)/b/n],[1/4*(-2*I-2*I*m+9*
b*n)/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))/(2+2*m+5*I*b*n)/cos(a+b*ln(c*x^n))^(5/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4582, 4580, 371} \[ \int \frac {x^m}{\cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{m+1} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i (m+1)}{b n}\right ),-\frac {2 i m-9 b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(5 i b n+2 m+2) \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

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

[Out]

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

Rule 4582

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Cos[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*} \text {integral}& = \frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\cos ^{\frac {5}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {5 i b}{2}-\frac {1+m}{n}} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {5 i b}{2}+\frac {1+m}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^{5/2}} \, dx,x,c x^n\right )}{n \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ & = \frac {2 x^{1+m} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i (1+m)}{b n}\right ),-\frac {2 i+2 i m-9 b n}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m+5 i b n) \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.98 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.58 \[ \int \frac {x^m}{\cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{1+m} \left ((2+2 m-i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos \left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i+2 i m-3 b n}{4 b n},-\frac {2 i+2 i m-5 b n}{4 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+b n \sec \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \sin (b n \log (x))+\cos \left (a+b \log \left (c x^n\right )\right ) \left (-2 (1+m)+b n \tan \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )}{3 b^2 n^2 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

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

[Out]

(2*x^(1 + m)*((2 + 2*m - I*b*n)*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Cos[a + b*Log[c*x^n]]*Hypergeometric2F1[1,
-1/4*(2*I + (2*I)*m - 3*b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))] + b*n*
Sec[a - b*n*Log[x] + b*Log[c*x^n]]*Sin[b*n*Log[x]] + Cos[a + b*Log[c*x^n]]*(-2*(1 + m) + b*n*Tan[a - b*n*Log[x
] + b*Log[c*x^n]])))/(3*b^2*n^2*Cos[a + b*Log[c*x^n]]^(3/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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