Integrand size = 28, antiderivative size = 189 \[ \int \frac {(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {2 c^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} f}-\frac {11 c^2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{4 \sqrt {2} a^{5/2} f}-\frac {3 c^2 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{8 a^2 f \sqrt {a+a \sec (e+f x)}}-\frac {c^2 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{4 a^2 f \sqrt {a+a \sec (e+f x)}} \]
2*c^2*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/a^(5/2)/f-11/8*c^2 *arctan(1/2*a^(1/2)*tan(f*x+e)*2^(1/2)/(a+a*sec(f*x+e))^(1/2))/a^(5/2)/f*2 ^(1/2)-3/8*c^2*sec(1/2*f*x+1/2*e)^2*sin(f*x+e)/a^2/f/(a+a*sec(f*x+e))^(1/2 )-1/4*c^2*cos(f*x+e)*sec(1/2*f*x+1/2*e)^4*sin(f*x+e)/a^2/f/(a+a*sec(f*x+e) )^(1/2)
Time = 1.52 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.77 \[ \int \frac {(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=-\frac {c^2 \left (22 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x)-8 \text {arctanh}\left (\sqrt {1-\sec (e+f x)}\right ) (1+\sec (e+f x))^2+\sqrt {1-\sec (e+f x)} (7+3 \sec (e+f x))\right ) \tan (e+f x)}{4 f \sqrt {1-\sec (e+f x)} (a (1+\sec (e+f x)))^{5/2}} \]
-1/4*(c^2*(22*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]]*Cos[(e + f*x )/2]^4*Sec[e + f*x]^2 - 8*ArcTanh[Sqrt[1 - Sec[e + f*x]]]*(1 + Sec[e + f*x ])^2 + Sqrt[1 - Sec[e + f*x]]*(7 + 3*Sec[e + f*x]))*Tan[e + f*x])/(f*Sqrt[ 1 - Sec[e + f*x]]*(a*(1 + Sec[e + f*x]))^(5/2))
Time = 0.44 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3042, 4392, 3042, 4375, 372, 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.
| \(\displaystyle \int \frac {(c-c \sec (e+f x))^2}{(a \sec (e+f x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^2}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4392 |
| \(\displaystyle a^2 c^2 \int \frac {\tan ^4(e+f x)}{(\sec (e+f x) a+a)^{9/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \frac {\cot \left (e+f x+\frac {\pi }{2}\right )^4}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4375 |
| \(\displaystyle -\frac {2 c^2 \int \frac {\tan ^4(e+f x)}{(\sec (e+f x) a+a)^2 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^3}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle -\frac {2 c^2 \left (\frac {\int \frac {2 \left (1-\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}\right )}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{4 a^2}+\frac {\tan (e+f x)}{2 a^2 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 c^2 \left (\frac {\int \frac {1-\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{2 a^2}+\frac {\tan (e+f x)}{2 a^2 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {2 c^2 \left (\frac {\frac {\int \frac {a \left (5-\frac {3 a \tan ^2(e+f x)}{\sec (e+f x) a+a}\right )}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{4 a}+\frac {3 \tan (e+f x)}{4 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan (e+f x)}{2 a^2 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 c^2 \left (\frac {\frac {1}{4} \int \frac {5-\frac {3 a \tan ^2(e+f x)}{\sec (e+f x) a+a}}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )+\frac {3 \tan (e+f x)}{4 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan (e+f x)}{2 a^2 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {2 c^2 \left (\frac {\frac {1}{4} \left (8 \int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )-11 \int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )\right )+\frac {3 \tan (e+f x)}{4 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan (e+f x)}{2 a^2 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2 c^2 \left (\frac {\frac {1}{4} \left (\frac {11 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {2} \sqrt {a}}-\frac {8 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a}}\right )+\frac {3 \tan (e+f x)}{4 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan (e+f x)}{2 a^2 \sqrt {a \sec (e+f x)+a} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
(-2*c^2*(Tan[e + f*x]/(2*a^2*Sqrt[a + a*Sec[e + f*x]]*(2 + (a*Tan[e + f*x] ^2)/(a + a*Sec[e + f*x]))^2) + (((-8*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/Sqrt[a] + (11*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*S qrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*Sqrt[a]))/4 + (3*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]))))/(2*a^ 2)))/f
3.1.82.3.1 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[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && 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[((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]
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 = 3.55 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.30
| method | result | size |
| default | \(-\frac {c^{2} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (2 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-3 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-8 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )+11 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{8 a^{3} f}\) | \(245\) |
| parts | \(-\frac {c^{2} \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 (2 \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}-32 \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 )-13 \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 )+43 \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 )}{32 f \,a^{3}}+\frac {c^{2} \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 (-2 \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 )-5 \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 )+5 \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 )}{32 f \,a^{3}}-\frac {c^{2} \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 (2 \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 )-3 \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 )+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 )\right )}{16 f \,a^{3}}\) | \(639\) |
-1/8/a^3*c^2/f*(-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)*(2*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*(-c ot(f*x+e)+csc(f*x+e))-3*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-cot(f*x+ e)+csc(f*x+e))-8*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)))+11*ln(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e ))^2*csc(f*x+e)^2-1)^(1/2)))
Time = 0.84 (sec) , antiderivative size = 645, normalized size of antiderivative = 3.41 \[ \int \frac {(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\left [-\frac {11 \, \sqrt {2} {\left (c^{2} \cos \left (f x + e\right )^{3} + 3 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, c^{2} \cos \left (f x + e\right ) + c^{2}\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 ) + 16 \, {\left (c^{2} \cos \left (f x + e\right )^{3} + 3 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, c^{2} \cos \left (f x + e\right ) + c^{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 ) + 4 \, {\left (7 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, c^{2} \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 )}{16 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}, \frac {11 \, \sqrt {2} {\left (c^{2} \cos \left (f x + e\right )^{3} + 3 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, c^{2} \cos \left (f x + e\right ) + c^{2}\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 ) - 16 \, {\left (c^{2} \cos \left (f x + e\right )^{3} + 3 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, c^{2} \cos \left (f x + e\right ) + c^{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 ) - 2 \, {\left (7 \, c^{2} \cos \left (f x + e\right )^{2} + 3 \, c^{2} \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 )}{8 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} f \cos \left (f x + e\right ) + a^{3} f\right )}}\right ] \]
[-1/16*(11*sqrt(2)*(c^2*cos(f*x + e)^3 + 3*c^2*cos(f*x + e)^2 + 3*c^2*cos( f*x + e) + c^2)*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)) + 16*(c^2*cos(f*x + e) ^3 + 3*c^2*cos(f*x + e)^2 + 3*c^2*cos(f*x + e) + c^2)*sqrt(-a)*log((2*a*co s(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*(7*c^2*cos (f*x + e)^2 + 3*c^2*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))* sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 + 3*a^3*f*cos (f*x + e) + a^3*f), 1/8*(11*sqrt(2)*(c^2*cos(f*x + e)^3 + 3*c^2*cos(f*x + e)^2 + 3*c^2*cos(f*x + e) + c^2)*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))) - 16*(c^2*cos(f *x + e)^3 + 3*c^2*cos(f*x + e)^2 + 3*c^2*cos(f*x + e) + c^2)*sqrt(a)*arcta n(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - 2*(7*c^2*cos(f*x + e)^2 + 3*c^2*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 + 3*a^3*f*cos(f*x + e) + a^3*f)]
\[ \int \frac {(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=c^{2} \left (\int \left (- \frac {2 \sec {\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \frac {1}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx\right ) \]
c**2*(Integral(-2*sec(e + f*x)/(a**2*sqrt(a*sec(e + f*x) + a)*sec(e + f*x) **2 + 2*a**2*sqrt(a*sec(e + f*x) + a)*sec(e + f*x) + a**2*sqrt(a*sec(e + f *x) + a)), x) + Integral(sec(e + f*x)**2/(a**2*sqrt(a*sec(e + f*x) + a)*se c(e + f*x)**2 + 2*a**2*sqrt(a*sec(e + f*x) + a)*sec(e + f*x) + a**2*sqrt(a *sec(e + f*x) + a)), x) + Integral(1/(a**2*sqrt(a*sec(e + f*x) + a)*sec(e + f*x)**2 + 2*a**2*sqrt(a*sec(e + f*x) + a)*sec(e + f*x) + a**2*sqrt(a*sec (e + f*x) + a)), x))
\[ \int \frac {(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\int { \frac {{\left (c \sec \left (f x + e\right ) - c\right )}^{2}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {(c-c \sec (e+f x))^2}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^2}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]