Integrand size = 14, antiderivative size = 302 \[ \int \frac {1}{a+b \tan ^4(c+d x)} \, dx=\frac {x}{a+b}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b) d}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b) d}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b) d}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b) d} \]
x/(a+b)+1/4*b^(1/4)*arctan(1-b^(1/4)*2^(1/2)*tan(d*x+c)/a^(1/4))*(a^(1/2)- b^(1/2))/a^(3/4)/(a+b)/d*2^(1/2)-1/4*b^(1/4)*arctan(1+b^(1/4)*2^(1/2)*tan( d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))/a^(3/4)/(a+b)/d*2^(1/2)-1/8*b^(1/4)*ln(a ^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*tan(d*x+c)+b^(1/2)*tan(d*x+c)^2)*(a^(1/2)+b ^(1/2))/a^(3/4)/(a+b)/d*2^(1/2)+1/8*b^(1/4)*ln(a^(1/2)+a^(1/4)*b^(1/4)*2^( 1/2)*tan(d*x+c)+b^(1/2)*tan(d*x+c)^2)*(a^(1/2)+b^(1/2))/a^(3/4)/(a+b)/d*2^ (1/2)
Time = 0.62 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75 \[ \int \frac {1}{a+b \tan ^4(c+d x)} \, dx=\frac {8 a^{3/4} \arctan (\tan (c+d x))+\sqrt {2} \sqrt [4]{b} \left (2 \left (\sqrt {a}-\sqrt {b}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )-2 \left (\sqrt {a}-\sqrt {b}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )-\left (\sqrt {a}+\sqrt {b}\right ) \left (\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )-\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )\right )\right )}{8 a^{3/4} (a+b) d} \]
(8*a^(3/4)*ArcTan[Tan[c + d*x]] + Sqrt[2]*b^(1/4)*(2*(Sqrt[a] - Sqrt[b])*A rcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)] - 2*(Sqrt[a] - Sqrt[b])* ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)] - (Sqrt[a] + Sqrt[b])*( Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^ 2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])))/(8*a^(3/4)*(a + b)*d)
Time = 0.45 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4144, 1485, 2009}
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.
| \(\displaystyle \int \frac {1}{a+b \tan ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a+b \tan (c+d x)^4}dx\) |
| \(\Big \downarrow \) 4144 |
| \(\displaystyle \frac {\int \frac {1}{\left (\tan ^2(c+d x)+1\right ) \left (b \tan ^4(c+d x)+a\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 1485 |
| \(\displaystyle \frac {\int \left (\frac {b-b \tan ^2(c+d x)}{(a+b) \left (b \tan ^4(c+d x)+a\right )}+\frac {1}{(a+b) \left (\tan ^2(c+d x)+1\right )}\right )d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b)}+\frac {\arctan (\tan (c+d x))}{a+b}}{d}\) |
(ArcTan[Tan[c + d*x]]/(a + b) + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (S qrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)) - ((Sqr t[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)] )/(2*Sqrt[2]*a^(3/4)*(a + b)) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2 ]*a^(3/4)*(a + b)) + ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^ (1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*( a + b)))/d
3.4.85.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[Expa ndIntegrand[(d + e*x^2)^q/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(a + b* (ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {x}{a +b}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (256 a^{5} d^{4}+512 a^{4} b \,d^{4}+256 a^{3} b^{2} d^{4}\right ) \textit {\_Z}^{4}-64 \textit {\_Z}^{2} a^{2} b \,d^{2}+b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (-\frac {32 a^{3} d^{2}}{a -b}-\frac {32 a^{2} b \,d^{2}}{a -b}\right ) \textit {\_R}^{2}+\left (\frac {8 i a^{2} d}{a -b}-\frac {8 i a b d}{a -b}\right ) \textit {\_R} +\frac {a}{a -b}+\frac {b}{a -b}\right )\right )\) | \(157\) |
| derivativedivides | \(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a +b}-\frac {b \left (-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}{\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )\right )}{8 a}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}{\tan \left (d x +c \right )^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a +b}}{d}\) | \(292\) |
| default | \(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a +b}-\frac {b \left (-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}{\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )\right )}{8 a}+\frac {\sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}{\tan \left (d x +c \right )^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a +b}}{d}\) | \(292\) |
x/(a+b)+sum(_R*ln(exp(2*I*(d*x+c))+(-32/(a-b)*a^3*d^2-32/(a-b)*a^2*b*d^2)* _R^2+(8*I/(a-b)*a^2*d-8*I/(a-b)*a*b*d)*_R+a/(a-b)+b/(a-b)),_R=RootOf((256* a^5*d^4+512*a^4*b*d^4+256*a^3*b^2*d^4)*_Z^4-64*_Z^2*a^2*b*d^2+b))
Leaf count of result is larger than twice the leaf count of optimal. 1541 vs. \(2 (222) = 444\).
Time = 0.33 (sec) , antiderivative size = 1541, normalized size of antiderivative = 5.10 \[ \int \frac {1}{a+b \tan ^4(c+d x)} \, dx=\text {Too large to display} \]
1/8*((a + b)*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^ 3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log((2*(a^3 - a*b^2)*d*sqrt(((a^3 + 2*a^2*b + a*b^ 2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b ^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) + ( a*b - b^2)*tan(d*x + c)^2 + a^2 - a*b + ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan (d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3) /((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) - (a + b)*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a ^3 + 2*a^2*b + a*b^2)*d^2))*log(-(2*(a^3 - a*b^2)*d*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4* a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c ) - (a*b - b^2)*tan(d*x + c)^2 - a^2 + a*b - ((a^4 + 2*a^3*b + a^2*b^2)*d^ 2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) + (a + b)*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2* a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2* b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log(-(2*(a^3 - a*b^2)*d*sqrt(-((a^3 + 2* a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^...
\[ \int \frac {1}{a+b \tan ^4(c+d x)} \, dx=\int \frac {1}{a + b \tan ^{4}{\left (c + d x \right )}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+b \tan ^4(c+d x)} \, dx=-\frac {\frac {b {\left (\frac {2 \, \sqrt {2} {\left (\sqrt {a} - \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} \tan \left (d x + c\right ) + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} {\left (\sqrt {a} - \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} \tan \left (d x + c\right ) - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} {\left (\sqrt {a} + \sqrt {b}\right )} \log \left (\sqrt {b} \tan \left (d x + c\right )^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (\sqrt {a} + \sqrt {b}\right )} \log \left (\sqrt {b} \tan \left (d x + c\right )^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}}\right )}}{a + b} - \frac {8 \, {\left (d x + c\right )}}{a + b}}{8 \, d} \]
-1/8*(b*(2*sqrt(2)*(sqrt(a) - sqrt(b))*arctan(1/2*sqrt(2)*(2*sqrt(b)*tan(d *x + c) + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sq rt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*(sqrt(a) - sqrt(b))*arctan(1/2*sqrt(2) *(2*sqrt(b)*tan(d*x + c) - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b))) /(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*(sqrt(a) + sqrt(b))*log (sqrt(b)*tan(d*x + c)^2 + sqrt(2)*a^(1/4)*b^(1/4)*tan(d*x + c) + sqrt(a))/ (a^(3/4)*b^(3/4)) + sqrt(2)*(sqrt(a) + sqrt(b))*log(sqrt(b)*tan(d*x + c)^2 - sqrt(2)*a^(1/4)*b^(1/4)*tan(d*x + c) + sqrt(a))/(a^(3/4)*b^(3/4)))/(a + b) - 8*(d*x + c)/(a + b))/d
Time = 0.82 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.17 \[ \int \frac {1}{a+b \tan ^4(c+d x)} \, dx=\frac {\frac {2 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (a b^{3}\right )^{\frac {3}{4}}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \tan \left (d x + c\right )\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}}{\sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} + \frac {2 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (a b^{3}\right )^{\frac {3}{4}}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \tan \left (d x + c\right )\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}}{\sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} + \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\tan \left (d x + c\right )^{2} + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {\frac {a}{b}}\right )}{\sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} - \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\tan \left (d x + c\right )^{2} - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {\frac {a}{b}}\right )}{\sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} + \frac {4 \, {\left (d x + c\right )}}{a + b}}{4 \, d} \]
1/4*(2*((a*b^3)^(1/4)*b^2 - (a*b^3)^(3/4))*(pi*floor((d*x + c)/pi + 1/2) + arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*tan(d*x + c))/(a/b)^(1/4)))/( sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + 2*((a*b^3)^(1/4)*b^2 - (a*b^3)^(3/4))*( pi*floor((d*x + c)/pi + 1/2) + arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*tan(d*x + c))/(a/b)^(1/4)))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + ((a*b^3) ^(1/4)*b^2 + (a*b^3)^(3/4))*log(tan(d*x + c)^2 + sqrt(2)*(a/b)^(1/4)*tan(d *x + c) + sqrt(a/b))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) - ((a*b^3)^(1/4)*b^ 2 + (a*b^3)^(3/4))*log(tan(d*x + c)^2 - sqrt(2)*(a/b)^(1/4)*tan(d*x + c) + sqrt(a/b))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + 4*(d*x + c)/(a + b))/d
Time = 15.50 (sec) , antiderivative size = 4038, normalized size of antiderivative = 13.37 \[ \int \frac {1}{a+b \tan ^4(c+d x)} \, dx=\text {Too large to display} \]
(2*atan(((((20*a*b^5 + 4*b^6 - ((((128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 - (tan(c + d*x)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512 *a^5*b^4)*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) + tan(c + d*x)*(32*a*b^6 + 16*b ^7 - 240*a^2*b^5))*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) - 6*b^5*tan(c + d*x))/ (2*a + 2*b) - (((20*a*b^5 + 4*b^6 - ((((128*a^2*b^6 - 64*a*b^7 + 448*a^3*b ^5 + 256*a^4*b^4 + (tan(c + d*x)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4)*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) - tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) + 6*b^5*tan(c + d *x))/(2*a + 2*b))/(((((20*a*b^5 + 4*b^6 - ((((128*a^2*b^6 - 64*a*b^7 + 448 *a^3*b^5 + 256*a^4*b^4 - (tan(c + d*x)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^ 4*b^5 - 512*a^5*b^4)*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) + tan(c + d*x)*(32*a *b^6 + 16*b^7 - 240*a^2*b^5))*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) - 6*b^5*tan (c + d*x))*1i)/(2*a + 2*b) + ((((20*a*b^5 + 4*b^6 - ((((128*a^2*b^6 - 64*a *b^7 + 448*a^3*b^5 + 256*a^4*b^4 + (tan(c + d*x)*(512*a^2*b^7 + 512*a^3*b^ 6 - 512*a^4*b^5 - 512*a^5*b^4)*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) - tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) + 6*b^5*tan(c + d*x))*1i)/(2*a + 2*b))))/(d*(2*a + 2*b)) - (atan((((20*a*b^ 5 - (((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 + ta n(c + d*x)*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*...