\(\int \frac {1}{x \cos ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [121]

Optimal result

Integrand size = 19, antiderivative size = 63 \[ \int \frac {1}{x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{3 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2716, 2720} \[ \int \frac {1}{x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{3 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[1/(x*Cos[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

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

Rule 2716

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

Rule 2720

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\cos ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\cos (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n} \\ & = \frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{3 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \left (\operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )+\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{\cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\right )}{3 b n} \]

[In]

Integrate[1/(x*Cos[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(290\) vs. \(2(93)=186\).

Time = 2.61 (sec) , antiderivative size = 291, normalized size of antiderivative = 4.62

method	result	size
derivativedivides	\(-\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2} \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}{3 n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right )}^{\frac {3}{2}} \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b}\)	\(291\)
default	\(-\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2} \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (a +2 b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{2}}\, \sqrt {-1+2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}}{3 n \sqrt {-2 {\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, {\left (2 {\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right )}^{\frac {3}{2}} \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) b}\)	\(291\)

[In]

int(1/x/cos(a+b*ln(c*x^n))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.37 \[ \int \frac {1}{x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {-i \, \sqrt {2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + i \, \sqrt {2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + 2 \, \sqrt {\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}} \]

[In]

integrate(1/x/cos(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

1/3*(-I*sqrt(2)*cos(b*n*log(x) + b*log(c) + a)^2*weierstrassPInverse(-4, 0, cos(b*n*log(x) + b*log(c) + a) + I
*sin(b*n*log(x) + b*log(c) + a)) + I*sqrt(2)*cos(b*n*log(x) + b*log(c) + a)^2*weierstrassPInverse(-4, 0, cos(b
*n*log(x) + b*log(c) + a) - I*sin(b*n*log(x) + b*log(c) + a)) + 2*sqrt(cos(b*n*log(x) + b*log(c) + a))*sin(b*n
*log(x) + b*log(c) + a))/(b*n*cos(b*n*log(x) + b*log(c) + a)^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x/cos(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x/cos(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x*cos(b*log(c*x^n) + a)^(5/2)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x/cos(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 27.86 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{3\,b\,n\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}\,\sqrt {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2}} \]

[In]

int(1/(x*cos(a + b*log(c*x^n))^(5/2)),x)

[Out]

(2*sin(a + b*log(c*x^n))*hypergeom([-3/4, 1/2], 1/4, cos(a + b*log(c*x^n))^2))/(3*b*n*cos(a + b*log(c*x^n))^(3
/2)*(sin(a + b*log(c*x^n))^2)^(1/2))