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
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)
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)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]]))
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]
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]
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]
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]
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]
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]
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) ))
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)
\[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{x \sqrt {\tan {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \] 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)
\[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\tan \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \] 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)
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
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)
\[ \int \frac {1}{x \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {\sqrt {\tan \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{\tan \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) x}d x \] 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)