Integrand size = 19, antiderivative size = 199 \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b n} \]
-1/2*arctan(-1+2^(1/2)*tan(a+b*ln(c*x^n))^(1/2))/b/n*2^(1/2)-1/2*arctan(1+ 2^(1/2)*tan(a+b*ln(c*x^n))^(1/2))/b/n*2^(1/2)+1/4*ln(1-2^(1/2)*tan(a+b*ln( c*x^n))^(1/2)+tan(a+b*ln(c*x^n)))/b/n*2^(1/2)-1/4*ln(1+2^(1/2)*tan(a+b*ln( c*x^n))^(1/2)+tan(a+b*ln(c*x^n)))/b/n*2^(1/2)+2*tan(a+b*ln(c*x^n))^(1/2)/b /n
Time = 0.23 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2}}+2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b n} \]
(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] + Log[1 - Sqrt[2]*Sqrt[Tan[a + b* Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[T an[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]) + 2*Sqrt[Tan[a + b*Log[c*x^n]]])/(b*n)
Time = 0.43 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {3039, 3042, 3954, 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.
| \(\displaystyle \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \tan \left (a+b \log \left (c x^n\right )\right )^{3/2}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\int \frac {1}{\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\int \frac {1}{\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {\int \frac {1}{\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )} \left (\tan ^2\left (a+b \log \left (c x^n\right )\right )+1\right )}d\tan \left (a+b \log \left (c x^n\right )\right )}{b}}{n}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \int \frac {1}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}}{n}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \left (\frac {1}{2} \int \frac {1-\tan \left (a+b \log \left (c x^n\right )\right )}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \int \frac {\tan \left (a+b \log \left (c x^n\right )\right )+1}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{b}}{n}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \left (\frac {1}{2} \int \frac {1-\tan \left (a+b \log \left (c x^n\right )\right )}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \int \frac {1}{\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b}}{n}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \left (\frac {1}{2} \int \frac {1-\tan \left (a+b \log \left (c x^n\right )\right )}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan \left (a+b \log \left (c x^n\right )\right )-1}d\left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan \left (a+b \log \left (c x^n\right )\right )-1}d\left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}\right )\right )}{b}}{n}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \left (\frac {1}{2} \int \frac {1-\tan \left (a+b \log \left (c x^n\right )\right )}{\tan ^2\left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b}}{n}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b}}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b}}{n}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}{\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1}d\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )\right )}{b}}{n}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2}}\right )\right )}{b}}{n}\) |
((-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 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]))/ 2))/b + (2*Sqrt[Tan[a + b*Log[c*x^n]]])/b)/n
3.2.81.3.1 Defintions of rubi rules used
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*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
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 = 0.85 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\) | \(139\) |
| default | \(\frac {2 \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\) | \(139\) |
1/n/b*(2*tan(a+b*ln(c*x^n))^(1/2)-1/4*2^(1/2)*(ln((1+2^(1/2)*tan(a+b*ln(c* x^n))^(1/2)+tan(a+b*ln(c*x^n)))/(1-2^(1/2)*tan(a+b*ln(c*x^n))^(1/2)+tan(a+ b*ln(c*x^n))))+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))))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.63 \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}\right ) + i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}\right ) - i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (-i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}\right ) - b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (-b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}\right ) - 4 \, \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}}{2 \, b n} \]
-1/2*(b*n*(-1/(b^4*n^4))^(1/4)*log(b*n*(-1/(b^4*n^4))^(1/4) + 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))) + I*b*n*(-1/(b^4*n^4))^(1/4)*log(I*b*n*(-1/(b^4*n^4))^(1/4) + 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)) ) - I*b*n*(-1/(b^4*n^4))^(1/4)*log(-I*b*n*(-1/(b^4*n^4))^(1/4) + 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 ))) - b*n*(-1/(b^4*n^4))^(1/4)*log(-b*n*(-1/(b^4*n^4))^(1/4) + 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)) ) - 4*sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(cos(2*b*n*log(x) + 2*b*lo g(c) + 2*a) + 1)))/(b*n)
\[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\tan ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
\[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\tan \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x} \,d x } \]
Timed out. \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
Time = 29.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.39 \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{b\,n}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n} \]