Integrand size = 23, antiderivative size = 177 \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{9/2} f}+\frac {91 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{32 \sqrt {2} a^{9/2} f}+\frac {\tan (e+f x)}{3 a f (a+a \sec (e+f x))^{7/2}}+\frac {11 \tan (e+f x)}{24 a^2 f (a+a \sec (e+f x))^{5/2}}+\frac {27 \tan (e+f x)}{32 a^3 f (a+a \sec (e+f x))^{3/2}} \]
-2*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/a^(9/2)/f+91/64*arcta n(1/2*a^(1/2)*tan(f*x+e)*2^(1/2)/(a+a*sec(f*x+e))^(1/2))/a^(9/2)/f*2^(1/2) +1/3*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^(7/2)+11/24*tan(f*x+e)/a^2/f/(a+a*sec (f*x+e))^(5/2)+27/32*tan(f*x+e)/a^3/f/(a+a*sec(f*x+e))^(3/2)
Time = 6.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.36 \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\frac {\cos ^3\left (\frac {1}{2} (e+f x)\right ) \sec ^{\frac {9}{2}}(e+f x) \sqrt {a (1+\sec (e+f x))} \left (-384 \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {1}{1+\sec (e+f x)}}}\right ) \cos ^7\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{(1+\cos (e+f x))^2}} \sqrt {1+\sec (e+f x)}+273 \arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ) \cos ^7\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {1+\sec (e+f x)}+\frac {(319+412 \cos (e+f x)+157 \cos (2 (e+f x))) \sin \left (\frac {1}{2} (e+f x)\right )}{8 \sqrt {\sec (e+f x)}}\right )}{6 a^5 f (1+\sec (e+f x))^5} \]
(Cos[(e + f*x)/2]^3*Sec[e + f*x]^(9/2)*Sqrt[a*(1 + Sec[e + f*x])]*(-384*Ar cTan[Tan[(e + f*x)/2]/Sqrt[(1 + Sec[e + f*x])^(-1)]]*Cos[(e + f*x)/2]^7*Sq rt[Cos[e + f*x]/(1 + Cos[e + f*x])^2]*Sqrt[1 + Sec[e + f*x]] + 273*ArcSin[ Tan[(e + f*x)/2]]*Cos[(e + f*x)/2]^7*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[(1 + Se c[e + f*x])^(-1)]*Sqrt[1 + Sec[e + f*x]] + ((319 + 412*Cos[e + f*x] + 157* Cos[2*(e + f*x)])*Sin[(e + f*x)/2])/(8*Sqrt[Sec[e + f*x]])))/(6*a^5*f*(1 + Sec[e + f*x])^5)
Time = 0.35 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.46, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 4375, 373, 402, 27, 402, 27, 397, 216}
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.
\(\displaystyle \int \frac {\tan ^2(e+f x)}{(a \sec (e+f x)+a)^{9/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cot \left (e+f x+\frac {\pi }{2}\right )^2}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{9/2}}dx\) |
\(\Big \downarrow \) 4375 |
\(\displaystyle -\frac {2 \int \frac {\tan ^2(e+f x)}{(\sec (e+f x) a+a) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^4}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a^3 f}\) |
\(\Big \downarrow \) 373 |
\(\displaystyle -\frac {2 \left (-\frac {\int \frac {1-\frac {5 a \tan ^2(e+f x)}{\sec (e+f x) a+a}}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^3}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{6 a}-\frac {\tan (e+f x)}{6 a \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^3}\right )}{a^3 f}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle -\frac {2 \left (-\frac {\frac {\int \frac {3 a \left (5-\frac {11 a \tan ^2(e+f x)}{\sec (e+f x) a+a}\right )}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{8 a}+\frac {11 \tan (e+f x)}{8 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}}{6 a}-\frac {\tan (e+f x)}{6 a \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^3}\right )}{a^3 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 \left (-\frac {\frac {3}{8} \int \frac {5-\frac {11 a \tan ^2(e+f x)}{\sec (e+f x) a+a}}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )+\frac {11 \tan (e+f x)}{8 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}}{6 a}-\frac {\tan (e+f x)}{6 a \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^3}\right )}{a^3 f}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle -\frac {2 \left (-\frac {\frac {3}{8} \left (\frac {\int \frac {a \left (37-\frac {27 a \tan ^2(e+f x)}{\sec (e+f x) a+a}\right )}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{4 a}+\frac {27 \tan (e+f x)}{4 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )+\frac {11 \tan (e+f x)}{8 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}}{6 a}-\frac {\tan (e+f x)}{6 a \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^3}\right )}{a^3 f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 \left (-\frac {\frac {3}{8} \left (\frac {1}{4} \int \frac {37-\frac {27 a \tan ^2(e+f x)}{\sec (e+f x) a+a}}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )+\frac {27 \tan (e+f x)}{4 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )+\frac {11 \tan (e+f x)}{8 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}}{6 a}-\frac {\tan (e+f x)}{6 a \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^3}\right )}{a^3 f}\) |
\(\Big \downarrow \) 397 |
\(\displaystyle -\frac {2 \left (-\frac {\frac {3}{8} \left (\frac {1}{4} \left (64 \int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )-91 \int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )\right )+\frac {27 \tan (e+f x)}{4 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )+\frac {11 \tan (e+f x)}{8 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}}{6 a}-\frac {\tan (e+f x)}{6 a \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^3}\right )}{a^3 f}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {2 \left (-\frac {\frac {3}{8} \left (\frac {1}{4} \left (\frac {91 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {64 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a}}\right )+\frac {27 \tan (e+f x)}{4 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )+\frac {11 \tan (e+f x)}{8 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}}{6 a}-\frac {\tan (e+f x)}{6 a \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^3}\right )}{a^3 f}\) |
(-2*(-1/6*Tan[e + f*x]/(a*Sqrt[a + a*Sec[e + f*x]]*(2 + (a*Tan[e + f*x]^2) /(a + a*Sec[e + f*x]))^3) - ((11*Tan[e + f*x])/(8*Sqrt[a + a*Sec[e + f*x]] *(2 + (a*Tan[e + f*x]^2)/(a + a*Sec[e + f*x]))^2) + (3*(((-64*ArcTan[(Sqrt [a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/Sqrt[a] + (91*ArcTan[(Sqrt[a] *Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*Sqrt[a]))/4 + (27*Tan[e + f*x])/(4*Sqrt[a + a*Sec[e + f*x]]*(2 + (a*Tan[e + f*x]^2)/(a + a*Sec[e + f*x])))))/8)/(6*a)))/(a^3*f)
3.3.7.3.1 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]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[-2*(a^(m/2 + n + 1/2)/d) Subst[Int[x^m*((2 + a*x^2 )^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x] ]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && I ntegerQ[n - 1/2]
Leaf count of result is larger than twice the leaf count of optimal. \(330\) vs. \(2(148)=296\).
Time = 3.07 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.87
method | result | size |
default | \(\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-8 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {5}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+10 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+12 \left (1-\cos \left (f x +e \right )\right )^{3} \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \csc \left (f x +e \right )^{3}-192 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )-93 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+273 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{192 f \,a^{5}}\) | \(331\) |
1/192/f/a^5*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)*((1-cos(f*x+e)) ^2*csc(f*x+e)^2-1)^(1/2)*(-8*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(5/2)*(-cot (f*x+e)+csc(f*x+e))+10*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*(-cot(f*x+e )+csc(f*x+e))+12*(1-cos(f*x+e))^3*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)* csc(f*x+e)^3-192*2^(1/2)*arctanh(2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1) ^(1/2)*(-cot(f*x+e)+csc(f*x+e)))-93*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2 )*(-cot(f*x+e)+csc(f*x+e))+273*ln(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e))^2* csc(f*x+e)^2-1)^(1/2)))
Time = 0.42 (sec) , antiderivative size = 674, normalized size of antiderivative = 3.81 \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\left [-\frac {273 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, a \cos \left (f x + e\right )^{2} + 2 \, a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) + 384 \, {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) - 4 \, {\left (157 \, \cos \left (f x + e\right )^{3} + 206 \, \cos \left (f x + e\right )^{2} + 81 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{384 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{5} f \cos \left (f x + e\right )^{3} + 6 \, a^{5} f \cos \left (f x + e\right )^{2} + 4 \, a^{5} f \cos \left (f x + e\right ) + a^{5} f\right )}}, -\frac {273 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 384 \, {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 2 \, {\left (157 \, \cos \left (f x + e\right )^{3} + 206 \, \cos \left (f x + e\right )^{2} + 81 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{192 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{5} f \cos \left (f x + e\right )^{3} + 6 \, a^{5} f \cos \left (f x + e\right )^{2} + 4 \, a^{5} f \cos \left (f x + e\right ) + a^{5} f\right )}}\right ] \]
[-1/384*(273*sqrt(2)*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + 3*a*cos(f*x + e)^2 + 2*a *cos(f*x + e) - a)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + 384*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(-a) *log((2*a*cos(f*x + e)^2 - 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e) + 1)) - 4*(157*cos(f*x + e)^3 + 206*cos(f*x + e)^2 + 81*cos(f*x + e))*sqrt((a*cos( f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^5*f*cos(f*x + e)^4 + 4*a^5*f* cos(f*x + e)^3 + 6*a^5*f*cos(f*x + e)^2 + 4*a^5*f*cos(f*x + e) + a^5*f), - 1/192*(273*sqrt(2)*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos( f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - 384*(cos(f*x + e)^4 + 4*c os(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(a)*arctan(sqrt ((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - 2*(157*cos(f*x + e)^3 + 206*cos(f*x + e)^2 + 81*cos(f*x + e))*sqrt((a*cos (f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^5*f*cos(f*x + e)^4 + 4*a^5*f *cos(f*x + e)^3 + 6*a^5*f*cos(f*x + e)^2 + 4*a^5*f*cos(f*x + e) + a^5*f)]
Timed out. \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {9}{2}}} \,d x } \]
Time = 1.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.62 \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\frac {\sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \sqrt {2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} - \frac {19 \, \sqrt {2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \frac {111 \, \sqrt {2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{192 \, f} \]
1/192*sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a)*(2*(4*sqrt(2)*tan(1/2*f*x + 1/2* e)^2/(a^5*sgn(cos(f*x + e))) - 19*sqrt(2)/(a^5*sgn(cos(f*x + e))))*tan(1/2 *f*x + 1/2*e)^2 + 111*sqrt(2)/(a^5*sgn(cos(f*x + e))))*tan(1/2*f*x + 1/2*e )/f
Timed out. \[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{9/2}} \,d x \]