\(\int \frac {1}{\cos ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [118]

Optimal result

Integrand size = 15, antiderivative size = 109 \[ \int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (7-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

2*x*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*hypergeom([3/2, 3/4-1/2*I/b/n],[7/4-1/2*I/b/n],-exp(2*I*a)*(c*x^n)^(2
*I*b))/(2+3*I*b*n)/cos(a+b*ln(c*x^n))^(3/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 109, 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} \[ \int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (7-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[Cos[a + b*Log[c*x^n]]^(-3/2),x]

[Out]

(2*x*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))^(3/2)*Hypergeometric2F1[3/2, (3 - (2*I)/(b*n))/4, (7 - (2*I)/(b*n))/4
, -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/((2 + (3*I)*b*n)*Cos[a + b*Log[c*x^n]]^(3/2))

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 \frac {x^{-1+\frac {1}{n}}}{\cos ^{\frac {3}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-\frac {3 i b}{2}-\frac {1}{n}} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {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 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ & = \frac {2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (7-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(431\) vs. \(2(109)=218\).

Time = 6.57 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.95 \[ \int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {x \left (-\left (\left (4+b^2 n^2\right ) x^{i b n} \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n},\frac {7}{4}-\frac {i}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )+(-2 i+3 b n) x^{-i b n} \left (-\left ((-2 i+b n) \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {3}{4}-\frac {i}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )+2 x^{i b n} \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} (b n \cos (b n \log (x))-2 \sin (b n \log (x)))\right )\right )}{b n (-2 i+3 b n) \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \left (-2 \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+b n \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )} \]

[In]

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

[Out]

(x*(-((4 + b^2*n^2)*x^(I*b*n)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeome
tric2F1[1/2, 3/4 - (I/2)/(b*n), 7/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]) + ((-2*I + 3*b*n)*(-((-2
*I + b*n)*Sqrt[2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[1/2, -1/4*(2
*I + b*n)/(b*n), 3/4 - (I/2)/(b*n), -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]) + 2*x^(I*b*n)*Sqrt[1/(E^(I*a)*(c*x^n)^(
I*b)) + E^(I*a)*(c*x^n)^(I*b)]*(b*n*Cos[b*n*Log[x]] - 2*Sin[b*n*Log[x]])))/x^(I*b*n)))/(b*n*(-2*I + 3*b*n)*Sqr
t[1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b)]*Sqrt[Cos[a + b*Log[c*x^n]]]*(-2*Cos[a - b*n*Log[x] + b*Lo
g[c*x^n]] + b*n*Sin[a - b*n*Log[x] + b*Log[c*x^n]]))

Maple [F]

\[\int \frac {1}{{\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}d x\]

[In]

int(1/cos(a+b*ln(c*x^n))^(3/2),x)

[Out]

int(1/cos(a+b*ln(c*x^n))^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/cos(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{\cos ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

[In]

integrate(1/cos(a+b*ln(c*x**n))**(3/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/cos(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/cos(a+b*log(c*x^n))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \,d x \]

[In]

int(1/cos(a + b*log(c*x^n))^(3/2),x)

[Out]

int(1/cos(a + b*log(c*x^n))^(3/2), x)