Integrand size = 28, antiderivative size = 185 \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {2 c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}-\frac {16 \sqrt {2} c^4 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} f}+\frac {14 c^4 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {2 a c^4 \tan ^3(e+f x)}{f (a+a \sec (e+f x))^{3/2}}+\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}} \]
2*c^4*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/f/a^(1/2)-16*c^4*a rctan(1/2*a^(1/2)*tan(f*x+e)*2^(1/2)/(a+a*sec(f*x+e))^(1/2))*2^(1/2)/f/a^( 1/2)+14*c^4*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)-2*a*c^4*tan(f*x+e)^3/f/(a+ a*sec(f*x+e))^(3/2)+2/5*a^2*c^4*tan(f*x+e)^5/f/(a+a*sec(f*x+e))^(5/2)
Time = 5.43 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83 \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\frac {c^4 \cot \left (\frac {1}{2} (e+f x)\right ) \left (100-155 \cos (e+f x)+96 \cos (2 (e+f x))-41 \cos (3 (e+f x))+20 \arctan \left (\sqrt {-1+\sec (e+f x)}\right ) \cos ^3(e+f x) \sqrt {-1+\sec (e+f x)}-160 \sqrt {2} \arctan \left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \cos ^3(e+f x) \sqrt {-1+\sec (e+f x)}\right ) \sec ^3(e+f x)}{10 f \sqrt {a (1+\sec (e+f x))}} \]
(c^4*Cot[(e + f*x)/2]*(100 - 155*Cos[e + f*x] + 96*Cos[2*(e + f*x)] - 41*C os[3*(e + f*x)] + 20*ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cos[e + f*x]^3*Sqrt[- 1 + Sec[e + f*x]] - 160*Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Co s[e + f*x]^3*Sqrt[-1 + Sec[e + f*x]])*Sec[e + f*x]^3)/(10*f*Sqrt[a*(1 + Se c[e + f*x])])
Time = 0.51 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4392, 3042, 4375, 381, 27, 444, 27, 444, 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.
| \(\displaystyle \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a \sec (e+f x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}{\sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a}}dx\) |
| \(\Big \downarrow \) 4392 |
| \(\displaystyle a^4 c^4 \int \frac {\tan ^8(e+f x)}{(\sec (e+f x) a+a)^{9/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \int \frac {\cot \left (e+f x+\frac {\pi }{2}\right )^8}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4375 |
| \(\displaystyle -\frac {2 a^4 c^4 \int \frac {\tan ^8(e+f x)}{(\sec (e+f x) a+a)^4 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{f}\) |
| \(\Big \downarrow \) 381 |
| \(\displaystyle -\frac {2 a^4 c^4 \left (-\frac {\int \frac {5 \tan ^4(e+f x) \left (\frac {3 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}{(\sec (e+f x) a+a)^2 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{5 a^2}-\frac {\tan ^5(e+f x)}{5 a^2 (a \sec (e+f x)+a)^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 a^4 c^4 \left (-\frac {\int \frac {\tan ^4(e+f x) \left (\frac {3 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}{(\sec (e+f x) a+a)^2 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a^2}-\frac {\tan ^5(e+f x)}{5 a^2 (a \sec (e+f x)+a)^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle -\frac {2 a^4 c^4 \left (-\frac {-\frac {\int \frac {3 a \tan ^2(e+f x) \left (\frac {7 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+6\right )}{(\sec (e+f x) a+a) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{3 a^2}-\frac {\tan ^3(e+f x)}{a (a \sec (e+f x)+a)^{3/2}}}{a^2}-\frac {\tan ^5(e+f x)}{5 a^2 (a \sec (e+f x)+a)^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 a^4 c^4 \left (-\frac {-\frac {\int \frac {\tan ^2(e+f x) \left (\frac {7 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+6\right )}{(\sec (e+f x) a+a) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a}-\frac {\tan ^3(e+f x)}{a (a \sec (e+f x)+a)^{3/2}}}{a^2}-\frac {\tan ^5(e+f x)}{5 a^2 (a \sec (e+f x)+a)^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle -\frac {2 a^4 c^4 \left (-\frac {-\frac {-\frac {\int \frac {a \left (\frac {15 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+14\right )}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a^2}-\frac {7 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}}{a}-\frac {\tan ^3(e+f x)}{a (a \sec (e+f x)+a)^{3/2}}}{a^2}-\frac {\tan ^5(e+f x)}{5 a^2 (a \sec (e+f x)+a)^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 a^4 c^4 \left (-\frac {-\frac {-\frac {\int \frac {\frac {15 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+14}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a}-\frac {7 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}}{a}-\frac {\tan ^3(e+f x)}{a (a \sec (e+f x)+a)^{3/2}}}{a^2}-\frac {\tan ^5(e+f x)}{5 a^2 (a \sec (e+f x)+a)^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {2 a^4 c^4 \left (-\frac {-\frac {-\frac {16 \int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )-\int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a}-\frac {7 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}}{a}-\frac {\tan ^3(e+f x)}{a (a \sec (e+f x)+a)^{3/2}}}{a^2}-\frac {\tan ^5(e+f x)}{5 a^2 (a \sec (e+f x)+a)^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 a^4 c^4 \left (-\frac {-\frac {-\frac {\frac {\arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a}}-\frac {8 \sqrt {2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a}}}{a}-\frac {7 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}}{a}-\frac {\tan ^3(e+f x)}{a (a \sec (e+f x)+a)^{3/2}}}{a^2}-\frac {\tan ^5(e+f x)}{5 a^2 (a \sec (e+f x)+a)^{5/2}}\right )}{f}\) |
(-2*a^4*c^4*(-1/5*Tan[e + f*x]^5/(a^2*(a + a*Sec[e + f*x])^(5/2)) - (-(Tan [e + f*x]^3/(a*(a + a*Sec[e + f*x])^(3/2))) - (-((ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]]/Sqrt[a] - (8*Sqrt[2]*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/Sqrt[a])/a) - (7*Tan[e + f*x ])/(a*Sqrt[a + a*Sec[e + f*x]]))/a)/a^2))/f
3.1.65.3.1 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^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q) + 1))), x] - Simp[e^4/(b*d*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 4)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + b*c*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q }, x] && NeQ[b*c - a*d, 0] && GtQ[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[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 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]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[Cot[e + f*x]^(2*m)*( c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && !( IntegerQ[n] && GtQ[m - n, 0])
Time = 6.49 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.52
| method | result | size |
| default | \(\frac {c^{4} \left (5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {5}{2}}-80 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {5}{2}}+98 \left (1-\cos \left (f x +e \right )\right )^{5} \csc \left (f x +e \right )^{5}-160 \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}+70 \csc \left (f x +e \right )-70 \cot \left (f x +e \right )\right ) \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}}{5 f a \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )^{2} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )^{2}}\) | \(282\) |
| parts | \(-\frac {c^{4} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\sqrt {2}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right )-2 \,\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right )\right )}{f a}-\frac {c^{4} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (15 \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 ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {5}{2}}-34 \left (1-\cos \left (f x +e \right )\right )^{5} \csc \left (f x +e \right )^{5}+40 \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-30 \csc \left (f x +e \right )+30 \cot \left (f x +e \right )\right )}{15 f a \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{2}}-\frac {4 c^{4} \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {2}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}}{f a}-\frac {6 c^{4} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right )+2 \cot \left (f x +e \right )-2 \csc \left (f x +e \right )\right )}{f a}-\frac {4 c^{4} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (3 \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 ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}}-4 \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}\right )}{3 f a \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )}\) | \(702\) |
1/5*c^4/f/a*(5*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)))*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(5/2)-80* ln(csc(f*x+e)-cot(f*x+e)+((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)^(5/2)+98*(1-cos(f*x+e))^5*csc(f*x+e)^5-160*(1-co s(f*x+e))^3*csc(f*x+e)^3+70*csc(f*x+e)-70*cot(f*x+e))*(-2*a/((1-cos(f*x+e) )^2*csc(f*x+e)^2-1))^(1/2)/(-cot(f*x+e)+csc(f*x+e)+1)^2/(-cot(f*x+e)+csc(f *x+e)-1)^2
Time = 0.70 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.98 \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\left [\frac {40 \, \sqrt {2} {\left (a c^{4} \cos \left (f x + e\right )^{3} + a c^{4} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, \cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 5 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + c^{4} \cos \left (f x + e\right )^{2}\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 ) + 2 \, {\left (41 \, c^{4} \cos \left (f x + e\right )^{2} - 7 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{5 \, {\left (a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )^{2}\right )}}, -\frac {2 \, {\left (5 \, {\left (c^{4} \cos \left (f x + e\right )^{3} + c^{4} \cos \left (f x + e\right )^{2}\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 ) - {\left (41 \, c^{4} \cos \left (f x + e\right )^{2} - 7 \, c^{4} \cos \left (f x + e\right ) + c^{4}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - \frac {40 \, \sqrt {2} {\left (a c^{4} \cos \left (f x + e\right )^{3} + a c^{4} \cos \left (f x + e\right )^{2}\right )} \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 )}{\sqrt {a}}\right )}}{5 \, {\left (a f \cos \left (f x + e\right )^{3} + a f \cos \left (f x + e\right )^{2}\right )}}\right ] \]
[1/5*(40*sqrt(2)*(a*c^4*cos(f*x + e)^3 + a*c^4*cos(f*x + e)^2)*sqrt(-1/a)* log((2*sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(-1/a)*cos(f*x + e)*sin(f*x + e) + 3*cos(f*x + e)^2 + 2*cos(f*x + e) - 1)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) - 5*(c^4*cos(f*x + e)^3 + c^4*cos(f*x + e)^2)*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) ) + 2*(41*c^4*cos(f*x + e)^2 - 7*c^4*cos(f*x + e) + c^4)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a*f*cos(f*x + e)^3 + a*f*cos(f*x + e )^2), -2/5*(5*(c^4*cos(f*x + e)^3 + c^4*cos(f*x + e)^2)*sqrt(a)*arctan(sqr t((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - (41*c^4*cos(f*x + e)^2 - 7*c^4*cos(f*x + e) + c^4)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) - 40*sqrt(2)*(a*c^4*cos(f*x + e)^3 + a*c^4 *cos(f*x + e)^2)*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*co s(f*x + e)/(sqrt(a)*sin(f*x + e)))/sqrt(a))/(a*f*cos(f*x + e)^3 + a*f*cos( f*x + e)^2)]
\[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=c^{4} \left (\int \left (- \frac {4 \sec {\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {6 \sec ^{2}{\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {4 \sec ^{3}{\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \frac {1}{\sqrt {a \sec {\left (e + f x \right )} + a}}\, dx\right ) \]
c**4*(Integral(-4*sec(e + f*x)/sqrt(a*sec(e + f*x) + a), x) + Integral(6*s ec(e + f*x)**2/sqrt(a*sec(e + f*x) + a), x) + Integral(-4*sec(e + f*x)**3/ sqrt(a*sec(e + f*x) + a), x) + Integral(sec(e + f*x)**4/sqrt(a*sec(e + f*x ) + a), x) + Integral(1/sqrt(a*sec(e + f*x) + a), x))
\[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\int { \frac {{\left (c \sec \left (f x + e\right ) - c\right )}^{4}}{\sqrt {a \sec \left (f x + e\right ) + a}} \,d x } \]
Exception generated. \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {(c-c \sec (e+f x))^4}{\sqrt {a+a \sec (e+f x)}} \, dx=\int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^4}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}} \,d x \]