Integrand size = 25, antiderivative size = 164 \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {2 c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} f}-\frac {(43 c-3 d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(c-d) \tan (e+f x)}{4 f (a+a \sec (e+f x))^{5/2}}-\frac {(11 c-3 d) \tan (e+f x)}{16 a f (a+a \sec (e+f x))^{3/2}} \]
2*c*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/a^(5/2)/f-1/32*(43*c -3*d)*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)-1/4*(c-d)*tan(f*x+e)/f/(a+a*sec(f*x+e))^(5/2)-1/16*(11*c-3*d)* tan(f*x+e)/a/f/(a+a*sec(f*x+e))^(3/2)
Leaf count is larger than twice the leaf count of optimal. \(343\) vs. \(2(164)=328\).
Time = 7.47 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.09 \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {\left ((-43 c+3 d) \arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )+32 \sqrt {2} c \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sec ^{\frac {3}{2}}(e+f x) \sqrt {1+\sec (e+f x)} (c+d \sec (e+f x))}{4 f (d+c \cos (e+f x)) \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} (a (1+\sec (e+f x)))^{5/2}}+\frac {\cos ^5\left (\frac {1}{2} (e+f x)\right ) \sec ^2(e+f x) (c+d \sec (e+f x)) \left (\frac {1}{2} (-15 c+7 d) \sin \left (\frac {1}{2} (e+f x)\right )+\frac {1}{4} \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (19 c \sin \left (\frac {1}{2} (e+f x)\right )-11 d \sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {1}{2} \sec ^4\left (\frac {1}{2} (e+f x)\right ) \left (-c \sin \left (\frac {1}{2} (e+f x)\right )+d \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )}{f (d+c \cos (e+f x)) (a (1+\sec (e+f x)))^{5/2}} \]
(((-43*c + 3*d)*ArcSin[Tan[(e + f*x)/2]] + 32*Sqrt[2]*c*ArcTan[Tan[(e + f* x)/2]/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]])*Cos[(e + f*x)/2]^4*Sqrt[Cos[ e + f*x]/(1 + Cos[e + f*x])]*Sec[e + f*x]^(3/2)*Sqrt[1 + Sec[e + f*x]]*(c + d*Sec[e + f*x]))/(4*f*(d + c*Cos[e + f*x])*Sqrt[Sec[(e + f*x)/2]^2]*(a*( 1 + Sec[e + f*x]))^(5/2)) + (Cos[(e + f*x)/2]^5*Sec[e + f*x]^2*(c + d*Sec[ e + f*x])*(((-15*c + 7*d)*Sin[(e + f*x)/2])/2 + (Sec[(e + f*x)/2]^2*(19*c* Sin[(e + f*x)/2] - 11*d*Sin[(e + f*x)/2]))/4 + (Sec[(e + f*x)/2]^4*(-(c*Si n[(e + f*x)/2]) + d*Sin[(e + f*x)/2]))/2))/(f*(d + c*Cos[e + f*x])*(a*(1 + Sec[e + f*x]))^(5/2))
Time = 0.91 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4410, 27, 3042, 4410, 27, 3042, 4408, 3042, 4261, 216, 4282, 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.
| \(\displaystyle \int \frac {c+d \sec (e+f x)}{(a \sec (e+f x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4410 |
| \(\displaystyle -\frac {\int -\frac {8 a c-3 a (c-d) \sec (e+f x)}{2 (\sec (e+f x) a+a)^{3/2}}dx}{4 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {8 a c-3 a (c-d) \sec (e+f x)}{(\sec (e+f x) a+a)^{3/2}}dx}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {8 a c-3 a (c-d) \csc \left (e+f x+\frac {\pi }{2}\right )}{\left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 4410 |
| \(\displaystyle \frac {-\frac {\int -\frac {32 a^2 c-a^2 (11 c-3 d) \sec (e+f x)}{2 \sqrt {\sec (e+f x) a+a}}dx}{2 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {32 a^2 c-a^2 (11 c-3 d) \sec (e+f x)}{\sqrt {\sec (e+f x) a+a}}dx}{4 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {32 a^2 c-a^2 (11 c-3 d) \csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {\frac {32 a c \int \sqrt {\sec (e+f x) a+a}dx-a^2 (43 c-3 d) \int \frac {\sec (e+f x)}{\sqrt {\sec (e+f x) a+a}}dx}{4 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {32 a c \int \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx-a^2 (43 c-3 d) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {\frac {-\left (a^2 (43 c-3 d) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx\right )-\frac {64 a^2 c \int \frac {1}{\frac {a^2 \tan ^2(e+f x)}{\sec (e+f x) a+a}+a}d\left (-\frac {a \tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{f}}{4 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {64 a^{3/2} c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}-a^2 (43 c-3 d) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {\frac {\frac {2 a^2 (43 c-3 d) \int \frac {1}{\frac {a^2 \tan ^2(e+f x)}{\sec (e+f x) a+a}+2 a}d\left (-\frac {a \tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{f}+\frac {64 a^{3/2} c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}}{4 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {64 a^{3/2} c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}-\frac {\sqrt {2} a^{3/2} (43 c-3 d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{f}}{4 a^2}-\frac {a (11 c-3 d) \tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2}}}{8 a^2}-\frac {(c-d) \tan (e+f x)}{4 f (a \sec (e+f x)+a)^{5/2}}\) |
-1/4*((c - d)*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(5/2)) + (((64*a^(3/2) *c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (Sqrt[2]*a ^(3/2)*(43*c - 3*d)*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[ e + f*x]])])/f)/(4*a^2) - (a*(11*c - 3*d)*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(3/2)))/(8*a^2)
3.2.80.3.1 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[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c/a Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[(b*c - a*d)/a Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_)), x_Symbol] :> Simp[(-(b*c - a*d))*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(b*f*(2*m + 1))), x] + Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[a*c*(2*m + 1) - (b*c - a*d)*(m + 1)*Csc[e + f*x], x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(375\) vs. \(2(139)=278\).
Time = 2.89 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.29
| method | result | size |
| default | \(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (2 c \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-2 d \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-32 c \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )-11 c \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+3 d \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+43 c \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )-3 d \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 a^{3} f}\) | \(376\) |
| parts | \(\frac {c \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-2 \left (1-\cos \left (f x +e \right )\right )^{3} \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \csc \left (f x +e \right )^{3}+32 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right )+13 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-43 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 f \,a^{3}}+\frac {d \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (2 \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{\frac {3}{2}} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-3 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )\right )}{32 f \,a^{3}}\) | \(441\) |
-1/32/a^3/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*c*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*(-co t(f*x+e)+csc(f*x+e))-2*d*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*(-cot(f*x +e)+csc(f*x+e))-32*c*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*c*((1-cos(f*x+e))^2*csc(f*x+e)^2-1 )^(1/2)*(-cot(f*x+e)+csc(f*x+e))+3*d*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/ 2)*(-cot(f*x+e)+csc(f*x+e))+43*c*ln(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e))^ 2*csc(f*x+e)^2-1)^(1/2))-3*d*ln(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e))^2*cs c(f*x+e)^2-1)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (139) = 278\).
Time = 3.61 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.09 \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\left [\frac {\sqrt {2} {\left ({\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right ) + 43 \, c - 3 \, d\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 ) - 64 \, {\left (c \cos \left (f x + e\right )^{3} + 3 \, c \cos \left (f x + e\right )^{2} + 3 \, c \cos \left (f x + e\right ) + c\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 ({\left (15 \, c - 7 \, d\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c - 3 \, d\right )} \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 )}{64 \, {\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 {\sqrt {2} {\left ({\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (43 \, c - 3 \, d\right )} \cos \left (f x + e\right ) + 43 \, c - 3 \, d\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 ) - 64 \, {\left (c \cos \left (f x + e\right )^{3} + 3 \, c \cos \left (f x + e\right )^{2} + 3 \, c \cos \left (f x + e\right ) + c\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 ({\left (15 \, c - 7 \, d\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c - 3 \, d\right )} \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 )}{32 \, {\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 ] \]
[1/64*(sqrt(2)*((43*c - 3*d)*cos(f*x + e)^3 + 3*(43*c - 3*d)*cos(f*x + e)^ 2 + 3*(43*c - 3*d)*cos(f*x + e) + 43*c - 3*d)*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)) - 64*(c*cos(f*x + e)^3 + 3*c*cos(f*x + e)^2 + 3*c*cos(f*x + e) + c) *sqrt(-a)*log((2*a*cos(f*x + e)^2 + 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/c os(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e) + 1)) - 4*((15*c - 7*d)*cos(f*x + e)^2 + (11*c - 3*d)*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), 1/32*(sqrt(2)*((43* c - 3*d)*cos(f*x + e)^3 + 3*(43*c - 3*d)*cos(f*x + e)^2 + 3*(43*c - 3*d)*c os(f*x + e) + 43*c - 3*d)*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))) - 64*(c*cos(f*x + e)^3 + 3*c*cos(f*x + e)^2 + 3*c*cos(f*x + e) + c)*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*((15*c - 7*d)*cos(f*x + e)^2 + (11*c - 3*d)*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)]
\[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {c + d \sec {\left (e + f x \right )}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {c+d \sec (e+f x)}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {c+\frac {d}{\cos \left (e+f\,x\right )}}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]