Integrand size = 28, antiderivative size = 169 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx=\frac {2 c^3 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} f}+\frac {2 \sqrt {2} c^3 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} f}-\frac {4 c^3 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}+\frac {c^3 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^2(e+f x)}{f (a+a \sec (e+f x))^{3/2}} \]
2*c^3*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/a^(3/2)/f+2*c^3*ar ctan(1/2*a^(1/2)*tan(f*x+e)*2^(1/2)/(a+a*sec(f*x+e))^(1/2))*2^(1/2)/a^(3/2 )/f-4*c^3*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^(1/2)+c^3*sec(1/2*f*x+1/2*e)^2*s in(f*x+e)*tan(f*x+e)^2/f/(a+a*sec(f*x+e))^(3/2)
Time = 2.76 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.78 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx=\frac {2 c^3 \left (-3+\arctan \left (\sqrt {-1+\sec (e+f x)}\right ) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {-1+\sec (e+f x)}+\sqrt {2} \arctan \left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {-1+\sec (e+f x)}-\sec (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{a f \sqrt {a (1+\sec (e+f x))}} \]
(2*c^3*(-3 + ArcTan[Sqrt[-1 + Sec[e + f*x]]]*Cot[(e + f*x)/2]^2*Sqrt[-1 + Sec[e + f*x]] + Sqrt[2]*ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cot[(e + f *x)/2]^2*Sqrt[-1 + Sec[e + f*x]] - Sec[e + f*x])*Tan[(e + f*x)/2])/(a*f*Sq rt[a*(1 + Sec[e + f*x])])
Time = 0.45 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.05, 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, 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.
| \(\displaystyle \int \frac {(c-c \sec (e+f x))^3}{(a \sec (e+f x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4392 |
| \(\displaystyle -a^3 c^3 \int \frac {\tan ^6(e+f x)}{(\sec (e+f x) a+a)^{9/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^3 c^3 \int \frac {\cot \left (e+f x+\frac {\pi }{2}\right )^6}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4375 |
| \(\displaystyle \frac {2 a^2 c^3 \int \frac {\tan ^6(e+f x)}{(\sec (e+f x) a+a)^3 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {2 a^2 c^3 \left (\frac {\int \frac {2 \tan ^2(e+f x) \left (\frac {2 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+3\right )}{(\sec (e+f x) a+a) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{2 a^2}+\frac {\tan ^3(e+f x)}{a^2 (a \sec (e+f x)+a)^{3/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^2 c^3 \left (\frac {\int \frac {\tan ^2(e+f x) \left (\frac {2 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+3\right )}{(\sec (e+f x) a+a) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a^2}+\frac {\tan ^3(e+f x)}{a^2 (a \sec (e+f x)+a)^{3/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {2 a^2 c^3 \left (\frac {-\frac {\int \frac {a \left (\frac {3 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+4\right )}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a^2}-\frac {2 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}}{a^2}+\frac {\tan ^3(e+f x)}{a^2 (a \sec (e+f x)+a)^{3/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^2 c^3 \left (\frac {-\frac {\int \frac {\frac {3 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+4}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a}-\frac {2 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}}{a^2}+\frac {\tan ^3(e+f x)}{a^2 (a \sec (e+f x)+a)^{3/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {2 a^2 c^3 \left (\frac {-\frac {\int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )+2 \int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a}-\frac {2 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}}{a^2}+\frac {\tan ^3(e+f x)}{a^2 (a \sec (e+f x)+a)^{3/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 a^2 c^3 \left (\frac {-\frac {-\frac {\arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a}}}{a}-\frac {2 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}}{a^2}+\frac {\tan ^3(e+f x)}{a^2 (a \sec (e+f x)+a)^{3/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}\right )}{f}\) |
(2*a^2*c^3*((-((-(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]]/ Sqrt[a]) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[ e + f*x]])])/Sqrt[a])/a) - (2*Tan[e + f*x])/(a*Sqrt[a + a*Sec[e + f*x]]))/ a^2 + Tan[e + f*x]^3/(a^2*(a + a*Sec[e + f*x])^(3/2)*(2 + (a*Tan[e + f*x]^ 2)/(a + a*Sec[e + f*x])))))/f
3.1.73.3.1 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[((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 = 5.67 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {c^{3} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}+2 \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}+2 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right ) \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}-4 \csc \left (f x +e \right )+4 \cot \left (f x +e \right )\right )}{a^{2} f}\) | \(222\) |
| parts | \(-\frac {c^{3} \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 (-4 \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 )-\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 )}{4 f \,a^{2}}-\frac {3 c^{3} \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 (-\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 )+\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 )}{4 f \,a^{2}}+\frac {3 c^{3} \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 (\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 )}{4 f \,a^{2}}+\frac {c^{3} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (\left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}+7 \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 ) \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}-9 \csc \left (f x +e \right )+9 \cot \left (f x +e \right )\right )}{4 f \,a^{2}}\) | \(652\) |
1/a^2*c^3/f*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)*(2^(1/2)*arctan h(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)^(1/2)+2*(1-cos(f*x+e))^3*csc(f*x+e)^3+ 2*ln(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))*((1-co s(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)-4*csc(f*x+e)+4*cot(f*x+e))
Time = 0.56 (sec) , antiderivative size = 550, normalized size of antiderivative = 3.25 \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx=\left [\frac {\sqrt {2} {\left (a c^{3} \cos \left (f x + e\right )^{2} + 2 \, a c^{3} \cos \left (f x + e\right ) + a c^{3}\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 ) - {\left (c^{3} \cos \left (f x + e\right )^{2} + 2 \, c^{3} \cos \left (f x + e\right ) + c^{3}\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 (3 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f}, -\frac {2 \, {\left ({\left (c^{3} \cos \left (f x + e\right )^{2} + 2 \, c^{3} \cos \left (f x + e\right ) + c^{3}\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 (3 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + \frac {\sqrt {2} {\left (a c^{3} \cos \left (f x + e\right )^{2} + 2 \, a c^{3} \cos \left (f x + e\right ) + a c^{3}\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 )}}{a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f}\right ] \]
[(sqrt(2)*(a*c^3*cos(f*x + e)^2 + 2*a*c^3*cos(f*x + e) + a*c^3)*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)) - (c^3*cos(f*x + e)^2 + 2*c^3*cos(f*x + e) + c^3 )*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*(3*c^3*cos(f*x + e) + c^3)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 + 2*a^2*f*cos(f*x + e) + a^2*f), -2*((c^3*cos(f*x + e)^2 + 2*c^3*cos(f*x + e) + c^3)*sqrt(a)*arctan(sqrt((a *cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) + (3 *c^3*cos(f*x + e) + c^3)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) + sqrt(2)*(a*c^3*cos(f*x + e)^2 + 2*a*c^3*cos(f*x + e) + a*c^3)*arctan (sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin (f*x + e)))/sqrt(a))/(a^2*f*cos(f*x + e)^2 + 2*a^2*f*cos(f*x + e) + a^2*f) ]
\[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx=- c^{3} \left (\int \frac {3 \sec {\left (e + f x \right )}}{a \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {3 \sec ^{2}{\left (e + f x \right )}}{a \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{a \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {1}{a \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx\right ) \]
-c**3*(Integral(3*sec(e + f*x)/(a*sqrt(a*sec(e + f*x) + a)*sec(e + f*x) + a*sqrt(a*sec(e + f*x) + a)), x) + Integral(-3*sec(e + f*x)**2/(a*sqrt(a*se c(e + f*x) + a)*sec(e + f*x) + a*sqrt(a*sec(e + f*x) + a)), x) + Integral( sec(e + f*x)**3/(a*sqrt(a*sec(e + f*x) + a)*sec(e + f*x) + a*sqrt(a*sec(e + f*x) + a)), x) + Integral(-1/(a*sqrt(a*sec(e + f*x) + a)*sec(e + f*x) + a*sqrt(a*sec(e + f*x) + a)), x))
\[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx=\int { -\frac {{\left (c \sec \left (f x + e\right ) - c\right )}^{3}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^3}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]