Integrand size = 28, antiderivative size = 260 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {2 c^5 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} f}-\frac {23 \sqrt {2} c^5 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} f}+\frac {21 c^5 \tan (e+f x)}{a^2 f \sqrt {a+a \sec (e+f x)}}-\frac {19 c^5 \tan ^3(e+f x)}{6 a f (a+a \sec (e+f x))^{3/2}}+\frac {3 c^5 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x) \tan ^4(e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}+\frac {a c^5 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin ^2(e+f x) \tan ^5(e+f x)}{4 f (a+a \sec (e+f x))^{7/2}} \]
2*c^5*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/a^(5/2)/f-23*c^5*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)/a^(5/ 2)/f+21*c^5*tan(f*x+e)/a^2/f/(a+a*sec(f*x+e))^(1/2)-19/6*c^5*tan(f*x+e)^3/ a/f/(a+a*sec(f*x+e))^(3/2)+3/4*c^5*sec(1/2*f*x+1/2*e)^2*sin(f*x+e)*tan(f*x +e)^4/f/(a+a*sec(f*x+e))^(5/2)+1/4*a*c^5*sec(1/2*f*x+1/2*e)^4*sin(f*x+e)^2 *tan(f*x+e)^5/f/(a+a*sec(f*x+e))^(7/2)
Time = 8.45 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.69 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {c^5 \cot \left (\frac {1}{2} (e+f x)\right ) \left ((81-30 \cos (e+f x)+52 \cos (2 (e+f x))-66 \cos (3 (e+f x))-37 \cos (4 (e+f x))) \sec ^4\left (\frac {1}{2} (e+f x)\right )+96 \arctan \left (\sqrt {-1+\sec (e+f x)}\right ) \cos ^2(e+f x) \sqrt {-1+\sec (e+f x)}-1104 \sqrt {2} \arctan \left (\frac {\sqrt {-1+\sec (e+f x)}}{\sqrt {2}}\right ) \cos ^2(e+f x) \sqrt {-1+\sec (e+f x)}\right ) \sec ^2(e+f x)}{48 a^2 f \sqrt {a (1+\sec (e+f x))}} \]
(c^5*Cot[(e + f*x)/2]*((81 - 30*Cos[e + f*x] + 52*Cos[2*(e + f*x)] - 66*Co s[3*(e + f*x)] - 37*Cos[4*(e + f*x)])*Sec[(e + f*x)/2]^4 + 96*ArcTan[Sqrt[ -1 + Sec[e + f*x]]]*Cos[e + f*x]^2*Sqrt[-1 + Sec[e + f*x]] - 1104*Sqrt[2]* ArcTan[Sqrt[-1 + Sec[e + f*x]]/Sqrt[2]]*Cos[e + f*x]^2*Sqrt[-1 + Sec[e + f *x]])*Sec[e + f*x]^2)/(48*a^2*f*Sqrt[a*(1 + Sec[e + f*x])])
Time = 0.57 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {3042, 4392, 3042, 4375, 372, 27, 440, 25, 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.
| \(\displaystyle \int \frac {(c-c \sec (e+f x))^5}{(a \sec (e+f x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^5}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4392 |
| \(\displaystyle -a^5 c^5 \int \frac {\tan ^{10}(e+f x)}{(\sec (e+f x) a+a)^{15/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^5 c^5 \int \frac {\cot \left (e+f x+\frac {\pi }{2}\right )^{10}}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{15/2}}dx\) |
| \(\Big \downarrow \) 4375 |
| \(\displaystyle \frac {2 a^3 c^5 \int \frac {\tan ^{10}(e+f x)}{(\sec (e+f x) a+a)^5 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^3}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\int \frac {2 \tan ^6(e+f x) \left (\frac {5 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+7\right )}{(\sec (e+f x) a+a)^3 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{4 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\int \frac {\tan ^6(e+f x) \left (\frac {5 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+7\right )}{(\sec (e+f x) a+a)^3 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )^2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 440 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}-\frac {\int -\frac {a \tan ^4(e+f x) \left (\frac {19 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+15\right )}{(\sec (e+f x) a+a)^2 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{2 a^2}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {\int \frac {a \tan ^4(e+f x) \left (\frac {19 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+15\right )}{(\sec (e+f x) a+a)^2 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{2 a^2}+\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {\int \frac {\tan ^4(e+f x) \left (\frac {19 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+15\right )}{(\sec (e+f x) a+a)^2 \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{2 a}+\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {-\frac {\int \frac {6 a \tan ^2(e+f x) \left (\frac {21 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+19\right )}{(\sec (e+f x) a+a) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{3 a^2}-\frac {19 \tan ^3(e+f x)}{3 a (a \sec (e+f x)+a)^{3/2}}}{2 a}+\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {-\frac {2 \int \frac {\tan ^2(e+f x) \left (\frac {21 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+19\right )}{(\sec (e+f x) a+a) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a}-\frac {19 \tan ^3(e+f x)}{3 a (a \sec (e+f x)+a)^{3/2}}}{2 a}+\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {-\frac {2 \left (-\frac {\int \frac {2 a \left (\frac {22 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+21\right )}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a^2}-\frac {21 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}\right )}{a}-\frac {19 \tan ^3(e+f x)}{3 a (a \sec (e+f x)+a)^{3/2}}}{2 a}+\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {-\frac {2 \left (-\frac {2 \int \frac {\frac {22 a \tan ^2(e+f x)}{\sec (e+f x) a+a}+21}{\left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1\right ) \left (\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2\right )}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{a}-\frac {21 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}\right )}{a}-\frac {19 \tan ^3(e+f x)}{3 a (a \sec (e+f x)+a)^{3/2}}}{2 a}+\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {-\frac {2 \left (-\frac {2 \left (23 \int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+2}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )-\int \frac {1}{\frac {a \tan ^2(e+f x)}{\sec (e+f x) a+a}+1}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )\right )}{a}-\frac {21 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}\right )}{a}-\frac {19 \tan ^3(e+f x)}{3 a (a \sec (e+f x)+a)^{3/2}}}{2 a}+\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 a^3 c^5 \left (\frac {\frac {-\frac {2 \left (-\frac {2 \left (\frac {\arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a}}-\frac {23 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {2} \sqrt {a}}\right )}{a}-\frac {21 \tan (e+f x)}{a \sqrt {a \sec (e+f x)+a}}\right )}{a}-\frac {19 \tan ^3(e+f x)}{3 a (a \sec (e+f x)+a)^{3/2}}}{2 a}+\frac {3 \tan ^5(e+f x)}{2 a (a \sec (e+f x)+a)^{5/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )}}{2 a^2}+\frac {\tan ^7(e+f x)}{2 a^2 (a \sec (e+f x)+a)^{7/2} \left (\frac {a \tan ^2(e+f x)}{a \sec (e+f x)+a}+2\right )^2}\right )}{f}\) |
(2*a^3*c^5*(Tan[e + f*x]^7/(2*a^2*(a + a*Sec[e + f*x])^(7/2)*(2 + (a*Tan[e + f*x]^2)/(a + a*Sec[e + f*x]))^2) + ((3*Tan[e + f*x]^5)/(2*a*(a + a*Sec[ e + f*x])^(5/2)*(2 + (a*Tan[e + f*x]^2)/(a + a*Sec[e + f*x]))) + ((-19*Tan [e + f*x]^3)/(3*a*(a + a*Sec[e + f*x])^(3/2)) - (2*((-2*(ArcTan[(Sqrt[a]*T an[e + f*x])/Sqrt[a + a*Sec[e + f*x]]]/Sqrt[a] - (23*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*Sqrt[a])))/a - (21* Tan[e + f*x])/(a*Sqrt[a + a*Sec[e + f*x]])))/a)/(2*a))/(2*a^2)))/f
3.1.79.3.1 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[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[m, 1]
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 = 7.71 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(\frac {c^{5} \left (-6 \left (1-\cos \left (f x +e \right )\right )^{7} \csc \left (f x +e \right )^{7}+3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}}-9 \left (1-\cos \left (f x +e \right )\right )^{5} \csc \left (f x +e \right )^{5}-69 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right ) \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}}+82 \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-63 \csc \left (f x +e \right )+63 \cot \left (f x +e \right )\right ) \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}}{3 a^{3} f \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )}\) | \(304\) |
| parts | \(\text {Expression too large to display}\) | \(1225\) |
1/3*c^5/a^3/f*(-6*(1-cos(f*x+e))^7*csc(f*x+e)^7+3*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)^(3/2)-9*(1-cos(f*x+e))^5*csc(f*x+e)^5-69*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)^(3/2)+82*(1-cos(f*x+e))^3*csc(f*x+e)^3-63*csc(f*x+e)+63 *cot(f*x+e))*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)/(-cot(f*x+e)+c sc(f*x+e)-1)/(-cot(f*x+e)+csc(f*x+e)+1)
Time = 1.69 (sec) , antiderivative size = 742, normalized size of antiderivative = 2.85 \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx=\left [\frac {69 \, \sqrt {2} {\left (a c^{5} \cos \left (f x + e\right )^{4} + 3 \, a c^{5} \cos \left (f x + e\right )^{3} + 3 \, a c^{5} \cos \left (f x + e\right )^{2} + a c^{5} \cos \left (f x + e\right )\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 ) - 6 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + 3 \, c^{5} \cos \left (f x + e\right )^{2} + c^{5} \cos \left (f x + e\right )\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 (37 \, c^{5} \cos \left (f x + e\right )^{3} + 70 \, c^{5} \cos \left (f x + e\right )^{2} + 20 \, c^{5} \cos \left (f x + e\right ) - c^{5}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{6 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f \cos \left (f x + e\right )\right )}}, -\frac {6 \, {\left (c^{5} \cos \left (f x + e\right )^{4} + 3 \, c^{5} \cos \left (f x + e\right )^{3} + 3 \, c^{5} \cos \left (f x + e\right )^{2} + c^{5} \cos \left (f x + e\right )\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 (37 \, c^{5} \cos \left (f x + e\right )^{3} + 70 \, c^{5} \cos \left (f x + e\right )^{2} + 20 \, c^{5} \cos \left (f x + e\right ) - c^{5}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - \frac {69 \, \sqrt {2} {\left (a c^{5} \cos \left (f x + e\right )^{4} + 3 \, a c^{5} \cos \left (f x + e\right )^{3} + 3 \, a c^{5} \cos \left (f x + e\right )^{2} + a c^{5} \cos \left (f x + e\right )\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}}}{3 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} + 3 \, a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f \cos \left (f x + e\right )\right )}}\right ] \]
[1/6*(69*sqrt(2)*(a*c^5*cos(f*x + e)^4 + 3*a*c^5*cos(f*x + e)^3 + 3*a*c^5* cos(f*x + e)^2 + a*c^5*cos(f*x + e))*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)) - 6*(c^5*cos(f*x + e)^4 + 3*c^5*cos(f*x + e)^3 + 3*c^5*cos(f*x + e)^2 + c^5* cos(f*x + e))*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*(37*c^5*cos(f*x + e)^3 + 70*c^5*cos(f*x + e)^2 + 2 0*c^5*cos(f*x + e) - c^5)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^4 + 3*a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e )^2 + a^3*f*cos(f*x + e)), -1/3*(6*(c^5*cos(f*x + e)^4 + 3*c^5*cos(f*x + e )^3 + 3*c^5*cos(f*x + e)^2 + c^5*cos(f*x + e))*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*(37*c ^5*cos(f*x + e)^3 + 70*c^5*cos(f*x + e)^2 + 20*c^5*cos(f*x + e) - c^5)*sqr t((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) - 69*sqrt(2)*(a*c^5*cos( f*x + e)^4 + 3*a*c^5*cos(f*x + e)^3 + 3*a*c^5*cos(f*x + e)^2 + a*c^5*cos(f *x + e))*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^3*f*cos(f*x + e)^4 + 3*a^3*f*cos(f* x + e)^3 + 3*a^3*f*cos(f*x + e)^2 + a^3*f*cos(f*x + e))]
\[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx=- c^{5} \left (\int \frac {5 \sec {\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {10 \sec ^{2}{\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {10 \sec ^{3}{\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {5 \sec ^{4}{\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx + \int \frac {\sec ^{5}{\left (e + f x \right )}}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx + \int \left (- \frac {1}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\right )\, dx\right ) \]
-c**5*(Integral(5*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(-10*sec(e + f*x)**2/(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*sq rt(a*sec(e + f*x) + a)), x) + Integral(10*sec(e + f*x)**3/(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(-5*sec(e + f*x)**4/(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)**5/(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 (-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))
Timed out. \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {(c-c \sec (e+f x))^5}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^5}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]