Integrand size = 15, antiderivative size = 117 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}}+\frac {a+b \tan ^2(x)}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}} \]
-1/2*arctanh((a-b*tan(x)^2)/(a+b)^(1/2)/(a+b*tan(x)^4)^(1/2))/(a+b)^(5/2)+ 1/6*(3*a^2+b*(5*a+2*b)*tan(x)^2)/a^2/(a+b)^2/(a+b*tan(x)^4)^(1/2)+1/6*(a+b *tan(x)^2)/a/(a+b)/(a+b*tan(x)^4)^(3/2)
Time = 0.88 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\frac {1}{6} \left (-\frac {3 \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{5/2}}+\frac {a^2 (4 a+b)+3 a b (2 a+b) \tan ^2(x)+3 a^2 b \tan ^4(x)+b^2 (5 a+2 b) \tan ^6(x)}{a^2 (a+b)^2 \left (a+b \tan ^4(x)\right )^{3/2}}\right ) \]
((-3*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/(a + b) ^(5/2) + (a^2*(4*a + b) + 3*a*b*(2*a + b)*Tan[x]^2 + 3*a^2*b*Tan[x]^4 + b^ 2*(5*a + 2*b)*Tan[x]^6)/(a^2*(a + b)^2*(a + b*Tan[x]^4)^(3/2)))/6
Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4153, 1577, 496, 25, 686, 27, 488, 219}
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.
| \(\displaystyle \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (x)}{\left (a+b \tan (x)^4\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \int \frac {\tan (x)}{\left (\tan ^2(x)+1\right ) \left (a+b \tan ^4(x)\right )^{5/2}}d\tan (x)\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\left (\tan ^2(x)+1\right ) \left (b \tan ^4(x)+a\right )^{5/2}}d\tan ^2(x)\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {1}{2} \left (\frac {a+b \tan ^2(x)}{3 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {\int -\frac {2 b \tan ^2(x)+3 a+2 b}{\left (\tan ^2(x)+1\right ) \left (b \tan ^4(x)+a\right )^{3/2}}d\tan ^2(x)}{3 a (a+b)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 b \tan ^2(x)+3 a+2 b}{\left (\tan ^2(x)+1\right ) \left (b \tan ^4(x)+a\right )^{3/2}}d\tan ^2(x)}{3 a (a+b)}+\frac {a+b \tan ^2(x)}{3 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}-\frac {\int -\frac {3 a^2 b}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)}{a b (a+b)}}{3 a (a+b)}+\frac {a+b \tan ^2(x)}{3 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {3 a \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)}{a+b}+\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}}{3 a (a+b)}+\frac {a+b \tan ^2(x)}{3 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}-\frac {3 a \int \frac {1}{-\tan ^4(x)+a+b}d\frac {a-b \tan ^2(x)}{\sqrt {b \tan ^4(x)+a}}}{a+b}}{3 a (a+b)}+\frac {a+b \tan ^2(x)}{3 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}-\frac {3 a \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{3/2}}}{3 a (a+b)}+\frac {a+b \tan ^2(x)}{3 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}\right )\) |
((a + b*Tan[x]^2)/(3*a*(a + b)*(a + b*Tan[x]^4)^(3/2)) + ((-3*a*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/(a + b)^(3/2) + (3*a^2 + b*(5*a + 2*b)*Tan[x]^2)/(a*(a + b)*Sqrt[a + b*Tan[x]^4]))/(3*a*(a + b)) )/2
3.5.4.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(585\) vs. \(2(101)=202\).
Time = 0.07 (sec) , antiderivative size = 586, normalized size of antiderivative = 5.01
| method | result | size |
| derivativedivides | \(-\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}-\frac {-\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right ) a}+\frac {\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right ) a}+\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}\) | \(586\) |
| default | \(-\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}-\frac {-\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right ) a}+\frac {\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right ) a}+\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}\) | \(586\) |
-1/2*b^2/((-a*b)^(1/2)+b)^2/((-a*b)^(1/2)-b)^2/(a+b)^(1/2)*ln((2*a+2*b-2*b *(1+tan(x)^2)+2*(a+b)^(1/2)*(b*(1+tan(x)^2)^2-2*b*(1+tan(x)^2)+a+b)^(1/2)) /(1+tan(x)^2))-1/8/((-a*b)^(1/2)+b)/a*(-1/3/(-a*b)^(1/2)/(tan(x)^2-(-a*b)^ (1/2)/b)^2*(b*(tan(x)^2-(-a*b)^(1/2)/b)^2+2*(-a*b)^(1/2)*(tan(x)^2-(-a*b)^ (1/2)/b))^(1/2)-1/3/a/(tan(x)^2-(-a*b)^(1/2)/b)*(b*(tan(x)^2-(-a*b)^(1/2)/ b)^2+2*(-a*b)^(1/2)*(tan(x)^2-(-a*b)^(1/2)/b))^(1/2))+1/8/((-a*b)^(1/2)-b) /a*(1/3/(-a*b)^(1/2)/(tan(x)^2+(-a*b)^(1/2)/b)^2*(b*(tan(x)^2+(-a*b)^(1/2) /b)^2-2*(-a*b)^(1/2)*(tan(x)^2+(-a*b)^(1/2)/b))^(1/2)-1/3/a/(tan(x)^2+(-a* b)^(1/2)/b)*(b*(tan(x)^2+(-a*b)^(1/2)/b)^2-2*(-a*b)^(1/2)*(tan(x)^2+(-a*b) ^(1/2)/b))^(1/2))+1/8*(2*(-a*b)^(1/2)+b)/((-a*b)^(1/2)+b)^2/a^2/(tan(x)^2- (-a*b)^(1/2)/b)*(b*(tan(x)^2-(-a*b)^(1/2)/b)^2+2*(-a*b)^(1/2)*(tan(x)^2-(- a*b)^(1/2)/b))^(1/2)-1/8*(2*(-a*b)^(1/2)-b)/((-a*b)^(1/2)-b)^2/a^2/(tan(x) ^2+(-a*b)^(1/2)/b)*(b*(tan(x)^2+(-a*b)^(1/2)/b)^2-2*(-a*b)^(1/2)*(tan(x)^2 +(-a*b)^(1/2)/b))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (103) = 206\).
Time = 0.46 (sec) , antiderivative size = 599, normalized size of antiderivative = 5.12 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a^{2} b^{2} \tan \left (x\right )^{8} + 2 \, a^{3} b \tan \left (x\right )^{4} + a^{4}\right )} \sqrt {a + b} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 2 \, {\left ({\left (5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right )} \tan \left (x\right )^{6} + 3 \, {\left (a^{3} b + a^{2} b^{2}\right )} \tan \left (x\right )^{4} + 4 \, a^{4} + 5 \, a^{3} b + a^{2} b^{2} + 3 \, {\left (2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{2}\right )} \sqrt {b \tan \left (x\right )^{4} + a}}{12 \, {\left ({\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} \tan \left (x\right )^{8} + a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} \tan \left (x\right )^{4}\right )}}, -\frac {3 \, {\left (a^{2} b^{2} \tan \left (x\right )^{8} + 2 \, a^{3} b \tan \left (x\right )^{4} + a^{4}\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) - {\left ({\left (5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right )} \tan \left (x\right )^{6} + 3 \, {\left (a^{3} b + a^{2} b^{2}\right )} \tan \left (x\right )^{4} + 4 \, a^{4} + 5 \, a^{3} b + a^{2} b^{2} + 3 \, {\left (2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{2}\right )} \sqrt {b \tan \left (x\right )^{4} + a}}{6 \, {\left ({\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} \tan \left (x\right )^{8} + a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} \tan \left (x\right )^{4}\right )}}\right ] \]
[1/12*(3*(a^2*b^2*tan(x)^8 + 2*a^3*b*tan(x)^4 + a^4)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*((5*a^2*b^ 2 + 7*a*b^3 + 2*b^4)*tan(x)^6 + 3*(a^3*b + a^2*b^2)*tan(x)^4 + 4*a^4 + 5*a ^3*b + a^2*b^2 + 3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*tan(x)^8 + a^7 + 3*a^6 *b + 3*a^5*b^2 + a^4*b^3 + 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*tan (x)^4), -1/6*(3*(a^2*b^2*tan(x)^8 + 2*a^3*b*tan(x)^4 + a^4)*sqrt(-a - b)*a rctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan( x)^4 + a^2 + a*b)) - ((5*a^2*b^2 + 7*a*b^3 + 2*b^4)*tan(x)^6 + 3*(a^3*b + a^2*b^2)*tan(x)^4 + 4*a^4 + 5*a^3*b + a^2*b^2 + 3*(2*a^3*b + 3*a^2*b^2 + a *b^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*tan(x)^8 + a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + 2*(a^6*b + 3*a^5 *b^2 + 3*a^4*b^3 + a^3*b^4)*tan(x)^4)]
\[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\int \frac {\tan {\left (x \right )}}{\left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (x\right )}{{\left (b \tan \left (x\right )^{4} + a\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (103) = 206\).
Time = 0.30 (sec) , antiderivative size = 618, normalized size of antiderivative = 5.28 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\frac {{\left ({\left (\frac {{\left (5 \, a^{7} b^{3} + 32 \, a^{6} b^{4} + 87 \, a^{5} b^{5} + 130 \, a^{4} b^{6} + 115 \, a^{3} b^{7} + 60 \, a^{2} b^{8} + 17 \, a b^{9} + 2 \, b^{10}\right )} \tan \left (x\right )^{2}}{a^{10} b + 8 \, a^{9} b^{2} + 28 \, a^{8} b^{3} + 56 \, a^{7} b^{4} + 70 \, a^{6} b^{5} + 56 \, a^{5} b^{6} + 28 \, a^{4} b^{7} + 8 \, a^{3} b^{8} + a^{2} b^{9}} + \frac {3 \, {\left (a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}\right )}}{a^{10} b + 8 \, a^{9} b^{2} + 28 \, a^{8} b^{3} + 56 \, a^{7} b^{4} + 70 \, a^{6} b^{5} + 56 \, a^{5} b^{6} + 28 \, a^{4} b^{7} + 8 \, a^{3} b^{8} + a^{2} b^{9}}\right )} \tan \left (x\right )^{2} + \frac {3 \, {\left (2 \, a^{8} b^{2} + 13 \, a^{7} b^{3} + 36 \, a^{6} b^{4} + 55 \, a^{5} b^{5} + 50 \, a^{4} b^{6} + 27 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}\right )}}{a^{10} b + 8 \, a^{9} b^{2} + 28 \, a^{8} b^{3} + 56 \, a^{7} b^{4} + 70 \, a^{6} b^{5} + 56 \, a^{5} b^{6} + 28 \, a^{4} b^{7} + 8 \, a^{3} b^{8} + a^{2} b^{9}}\right )} \tan \left (x\right )^{2} + \frac {4 \, a^{9} b + 25 \, a^{8} b^{2} + 66 \, a^{7} b^{3} + 95 \, a^{6} b^{4} + 80 \, a^{5} b^{5} + 39 \, a^{4} b^{6} + 10 \, a^{3} b^{7} + a^{2} b^{8}}{a^{10} b + 8 \, a^{9} b^{2} + 28 \, a^{8} b^{3} + 56 \, a^{7} b^{4} + 70 \, a^{6} b^{5} + 56 \, a^{5} b^{6} + 28 \, a^{4} b^{7} + 8 \, a^{3} b^{8} + a^{2} b^{9}}}{6 \, {\left (b \tan \left (x\right )^{4} + a\right )}^{\frac {3}{2}}} - \frac {\arctan \left (\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a - b}} \]
1/6*((((5*a^7*b^3 + 32*a^6*b^4 + 87*a^5*b^5 + 130*a^4*b^6 + 115*a^3*b^7 + 60*a^2*b^8 + 17*a*b^9 + 2*b^10)*tan(x)^2/(a^10*b + 8*a^9*b^2 + 28*a^8*b^3 + 56*a^7*b^4 + 70*a^6*b^5 + 56*a^5*b^6 + 28*a^4*b^7 + 8*a^3*b^8 + a^2*b^9) + 3*(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3*b ^7 + a^2*b^8)/(a^10*b + 8*a^9*b^2 + 28*a^8*b^3 + 56*a^7*b^4 + 70*a^6*b^5 + 56*a^5*b^6 + 28*a^4*b^7 + 8*a^3*b^8 + a^2*b^9))*tan(x)^2 + 3*(2*a^8*b^2 + 13*a^7*b^3 + 36*a^6*b^4 + 55*a^5*b^5 + 50*a^4*b^6 + 27*a^3*b^7 + 8*a^2*b^ 8 + a*b^9)/(a^10*b + 8*a^9*b^2 + 28*a^8*b^3 + 56*a^7*b^4 + 70*a^6*b^5 + 56 *a^5*b^6 + 28*a^4*b^7 + 8*a^3*b^8 + a^2*b^9))*tan(x)^2 + (4*a^9*b + 25*a^8 *b^2 + 66*a^7*b^3 + 95*a^6*b^4 + 80*a^5*b^5 + 39*a^4*b^6 + 10*a^3*b^7 + a^ 2*b^8)/(a^10*b + 8*a^9*b^2 + 28*a^8*b^3 + 56*a^7*b^4 + 70*a^6*b^5 + 56*a^5 *b^6 + 28*a^4*b^7 + 8*a^3*b^8 + a^2*b^9))/(b*tan(x)^4 + a)^(3/2) - arctan( (sqrt(b)*tan(x)^2 - sqrt(b*tan(x)^4 + a) + sqrt(b))/sqrt(-a - b))/((a^2 + 2*a*b + b^2)*sqrt(-a - b))
Timed out. \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\int \frac {\mathrm {tan}\left (x\right )}{{\left (b\,{\mathrm {tan}\left (x\right )}^4+a\right )}^{5/2}} \,d x \]