\(\int \frac {1}{x \sqrt {\tan (a+b \log (c x^n))}} \, dx\) [183]

Optimal result

Integrand size = 19, antiderivative size = 124 \[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{1+\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n} \] Output:

1/2*arctan(-1+2^(1/2)*tan(a+b*ln(c*x^n))^(1/2))*2^(1/2)/b/n+1/2*arctan(1+2 
^(1/2)*tan(a+b*ln(c*x^n))^(1/2))*2^(1/2)/b/n+1/2*arctanh(2^(1/2)*tan(a+b*l 
n(c*x^n))^(1/2)/(1+tan(a+b*ln(c*x^n))))*2^(1/2)/b/n
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {-2 \arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )+2 \arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )-\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )+\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \] Input:

Integrate[1/(x*Sqrt[Tan[a + b*Log[c*x^n]]]),x]
 

Output:

(-2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]] + 2*ArcTan[1 + Sqrt[2] 
*Sqrt[Tan[a + b*Log[c*x^n]]]] - Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]] 
] + Tan[a + b*Log[c*x^n]]] + Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + 
 Tan[a + b*Log[c*x^n]]])/(2*Sqrt[2]*b*n)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.37, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {3039, 3042, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \frac {1}{\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {1}{\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {\int \frac {1}{\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )} \left (\tan ^2\left (a+b \log \left (c x^n\right )\right )+1\right )}d\tan \left (a+b \log \left (c x^n\right )\right )}{b n}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 \int \frac {1}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b n}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1-\tan \left (a+b \log \left (c x^n\right )\right )}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \int \frac {\tan \left (a+b \log \left (c x^n\right )\right )+1}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1-\tan \left (a+b \log \left (c x^n\right )\right )}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \int \frac {1}{\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b n}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1-\tan \left (a+b \log \left (c x^n\right )\right )}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan \left (a+b \log \left (c x^n\right )\right )-1}d\left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan \left (a+b \log \left (c x^n\right )\right )-1}d\left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}\right )\right )}{b n}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1-\tan \left (a+b \log \left (c x^n\right )\right )}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b n}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}{\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b n}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2}}\right )\right )}{b n}\)

Input:

Int[1/(x*Sqrt[Tan[a + b*Log[c*x^n]]]),x]
 

Output:

(2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/Sqrt[2]) + ArcTan[1 
 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2] 
*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/Sqrt[2] + Log[1 + Sq 
rt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]))/2) 
)/(b*n)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3039

Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst 
[[3]]   Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /;  !FalseQ[lst]] /; 
NonsumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+1}{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4 n b}\)	\(122\)
default	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+1}{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4 n b}\)	\(122\)

Input:

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

Output:

1/4/n/b*2^(1/2)*(ln((tan(a+b*ln(c*x^n))+2^(1/2)*tan(a+b*ln(c*x^n))^(1/2)+1 
)/(tan(a+b*ln(c*x^n))-2^(1/2)*tan(a+b*ln(c*x^n))^(1/2)+1))+2*arctan(1+2^(1 
/2)*tan(a+b*ln(c*x^n))^(1/2))+2*arctan(-1+2^(1/2)*tan(a+b*ln(c*x^n))^(1/2) 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (107) = 214\).

Time = 0.08 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.94 \[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}} + 1\right ) + 2 \, \sqrt {2} \arctan \left (\sqrt {2} \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}} - 1\right ) + \sqrt {2} \log \left (\frac {{\left (\sqrt {2} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \sqrt {2}\right )} \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}} + \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}\right ) - \sqrt {2} \log \left (-\frac {{\left (\sqrt {2} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \sqrt {2}\right )} \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}} - \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - 1}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}\right )}{4 \, b n} \] Input:

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

Output:

1/4*(2*sqrt(2)*arctan(sqrt(2)*sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(c 
os(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)) + 1) + 2*sqrt(2)*arctan(sqrt(2)* 
sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(cos(2*b*n*log(x) + 2*b*log(c) + 
 2*a) + 1)) - 1) + sqrt(2)*log(((sqrt(2)*cos(2*b*n*log(x) + 2*b*log(c) + 2 
*a) + sqrt(2))*sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(cos(2*b*n*log(x) 
 + 2*b*log(c) + 2*a) + 1)) + cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + sin(2* 
b*n*log(x) + 2*b*log(c) + 2*a) + 1)/(cos(2*b*n*log(x) + 2*b*log(c) + 2*a) 
+ 1)) - sqrt(2)*log(-((sqrt(2)*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + sqrt 
(2))*sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(cos(2*b*n*log(x) + 2*b*log 
(c) + 2*a) + 1)) - cos(2*b*n*log(x) + 2*b*log(c) + 2*a) - sin(2*b*n*log(x) 
 + 2*b*log(c) + 2*a) - 1)/(cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)))/(b* 
n)
 

Sympy [F]

\[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{x \sqrt {\tan {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \] Input:

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

Output:

Integral(1/(x*sqrt(tan(a + b*log(c*x**n)))), x)
 

Maxima [F]

\[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\tan \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \] Input:

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

Output:

integrate(1/(x*sqrt(tan(b*log(c*x^n) + a))), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 20.71 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n} \] Input:

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

Output:

- ((-1)^(1/4)*atan((-1)^(1/4)*tan(a + b*log(c*x^n))^(1/2))*1i)/(b*n) - ((- 
1)^(1/4)*atanh((-1)^(1/4)*tan(a + b*log(c*x^n))^(1/2))*1i)/(b*n)
 

Reduce [F]

\[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {\sqrt {\tan \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{\tan \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) x}d x \] Input:

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

Output:

int(sqrt(tan(log(x**n*c)*b + a))/(tan(log(x**n*c)*b + a)*x),x)