Integrand size = 27, antiderivative size = 480 \[ \int \frac {(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx=-\frac {(c-d)^3 \tan (e+f x)}{4 a^2 f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^3 \tan (e+f x)}{16 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {(c-d)^2 (c+2 d) \tan (e+f x)}{2 a^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^3 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 (c-d)^3 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{16 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-d)^2 (c+2 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} \left (c^3-d^3\right ) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \]
-1/4*(c-d)^3*tan(f*x+e)/a^2/f/(1+sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2)-3/16 *(c-d)^3*tan(f*x+e)/a^2/f/(1+sec(f*x+e))/(a+a*sec(f*x+e))^(1/2)-1/2*(c-d)^ 2*(c+2*d)*tan(f*x+e)/a^2/f/(1+sec(f*x+e))/(a+a*sec(f*x+e))^(1/2)+2*c^3*arc tanh((a-a*sec(f*x+e))^(1/2)/a^(1/2))*tan(f*x+e)/a^(3/2)/f/(a-a*sec(f*x+e)) ^(1/2)/(a+a*sec(f*x+e))^(1/2)-3/32*(c-d)^3*arctanh(1/2*(a-a*sec(f*x+e))^(1 /2)*2^(1/2)/a^(1/2))*tan(f*x+e)/a^(3/2)/f*2^(1/2)/(a-a*sec(f*x+e))^(1/2)/( a+a*sec(f*x+e))^(1/2)-1/4*(c-d)^2*(c+2*d)*arctanh(1/2*(a-a*sec(f*x+e))^(1/ 2)*2^(1/2)/a^(1/2))*tan(f*x+e)/a^(3/2)/f*2^(1/2)/(a-a*sec(f*x+e))^(1/2)/(a +a*sec(f*x+e))^(1/2)-(c^3-d^3)*arctanh(1/2*(a-a*sec(f*x+e))^(1/2)*2^(1/2)/ a^(1/2))*2^(1/2)*tan(f*x+e)/a^(3/2)/f/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+ e))^(1/2)
Time = 9.68 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx=\frac {\left (\left (-43 c^3+9 c^2 d+15 c d^2+19 d^3\right ) \arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )+32 \sqrt {2} c^3 \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )\right ) \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {1+\sec (e+f x)} (c+d \sec (e+f x))^3}{4 f (d+c \cos (e+f x))^3 \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\sec (e+f x)} (a (1+\sec (e+f x)))^{5/2}}+\frac {\cos ^5\left (\frac {1}{2} (e+f x)\right ) (c+d \sec (e+f x))^3 \left (-\frac {3}{2} (-c+d)^2 (5 c+3 d) \sin \left (\frac {1}{2} (e+f x)\right )+\frac {1}{2} \sec ^4\left (\frac {1}{2} (e+f x)\right ) \left (-c^3 \sin \left (\frac {1}{2} (e+f x)\right )+3 c^2 d \sin \left (\frac {1}{2} (e+f x)\right )-3 c d^2 \sin \left (\frac {1}{2} (e+f x)\right )+d^3 \sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {1}{4} \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (19 c^3 \sin \left (\frac {1}{2} (e+f x)\right )-33 c^2 d \sin \left (\frac {1}{2} (e+f x)\right )+9 c d^2 \sin \left (\frac {1}{2} (e+f x)\right )+5 d^3 \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )}{f (d+c \cos (e+f x))^3 (a (1+\sec (e+f x)))^{5/2}} \]
(((-43*c^3 + 9*c^2*d + 15*c*d^2 + 19*d^3)*ArcSin[Tan[(e + f*x)/2]] + 32*Sq rt[2]*c^3*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])]*Sqrt[1 + Sec[e + f*x]]*(c + d*Sec[e + f*x])^3)/(4*f*(d + c*Cos[e + f*x])^3*Sqrt[Sec[(e + f* x)/2]^2]*Sqrt[Sec[e + f*x]]*(a*(1 + Sec[e + f*x]))^(5/2)) + (Cos[(e + f*x) /2]^5*(c + d*Sec[e + f*x])^3*((-3*(-c + d)^2*(5*c + 3*d)*Sin[(e + f*x)/2]) /2 + (Sec[(e + f*x)/2]^4*(-(c^3*Sin[(e + f*x)/2]) + 3*c^2*d*Sin[(e + f*x)/ 2] - 3*c*d^2*Sin[(e + f*x)/2] + d^3*Sin[(e + f*x)/2]))/2 + (Sec[(e + f*x)/ 2]^2*(19*c^3*Sin[(e + f*x)/2] - 33*c^2*d*Sin[(e + f*x)/2] + 9*c*d^2*Sin[(e + f*x)/2] + 5*d^3*Sin[(e + f*x)/2]))/4))/(f*(d + c*Cos[e + f*x])^3*(a*(1 + Sec[e + f*x]))^(5/2))
Time = 0.53 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 4428, 27, 198, 2009}
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.
| \(\displaystyle \int \frac {(c+d \sec (e+f x))^3}{(a \sec (e+f x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4428 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {\cos (e+f x) (c+d \sec (e+f x))^3}{a^3 (\sec (e+f x)+1)^3 \sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\tan (e+f x) \int \frac {\cos (e+f x) (c+d \sec (e+f x))^3}{(\sec (e+f x)+1)^3 \sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 198 |
| \(\displaystyle -\frac {\tan (e+f x) \int \left (\frac {\cos (e+f x) c^3}{\sqrt {a-a \sec (e+f x)}}+\frac {d^3-c^3}{(\sec (e+f x)+1) \sqrt {a-a \sec (e+f x)}}-\frac {(c-d)^2 (c+2 d)}{(\sec (e+f x)+1)^2 \sqrt {a-a \sec (e+f x)}}-\frac {(c-d)^3}{(\sec (e+f x)+1)^3 \sqrt {a-a \sec (e+f x)}}\right )d\sec (e+f x)}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\tan (e+f x) \left (\frac {\sqrt {2} \left (c^3-d^3\right ) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {2 c^3 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {3 (c-d)^3 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} \sqrt {a}}+\frac {(c-d)^2 (c+2 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} \sqrt {a}}+\frac {3 (c-d)^3 \sqrt {a-a \sec (e+f x)}}{16 a (\sec (e+f x)+1)}+\frac {(c-d)^2 (c+2 d) \sqrt {a-a \sec (e+f x)}}{2 a (\sec (e+f x)+1)}+\frac {(c-d)^3 \sqrt {a-a \sec (e+f x)}}{4 a (\sec (e+f x)+1)^2}\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((((-2*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]])/Sqrt[a] + (3*(c - d )^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*Sqrt[ a]) + ((c - d)^2*(c + 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[ a])])/(2*Sqrt[2]*Sqrt[a]) + (Sqrt[2]*(c^3 - d^3)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])])/Sqrt[a] + ((c - d)^3*Sqrt[a - a*Sec[e + f*x]]) /(4*a*(1 + Sec[e + f*x])^2) + (3*(c - d)^3*Sqrt[a - a*Sec[e + f*x]])/(16*a *(1 + Sec[e + f*x])) + ((c - d)^2*(c + 2*d)*Sqrt[a - a*Sec[e + f*x]])/(2*a *(1 + Sec[e + f*x])))*Tan[e + f*x])/(a*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))
3.2.78.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_))^(q_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && IntegersQ[p, q]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[a^2*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])) Subst[Int[(a + b*x)^(m - 1/2)*((c + d *x)^n/(x*Sqrt[a - b*x])), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0 ] && IntegerQ[m - 1/2]
Time = 4.86 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.43
| 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 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, c^{3} \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-6 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, c^{2} d \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}+6 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, c \,d^{2} \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-2 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d^{3} \left (1-\cos \left (f x +e \right )\right )^{3} \csc \left (f x +e \right )^{3}-32 c^{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 )-13 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, c^{3} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+15 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, c^{2} d \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+9 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, c \,d^{2} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-11 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d^{3} \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+43 c^{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 )-9 c^{2} 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 )-15 c \,d^{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 )-19 d^{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 a^{3} f}\) | \(688\) |
| parts | \(\text {Expression too large to display}\) | \(840\) |
-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*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*c^3*(1 -cos(f*x+e))^3*csc(f*x+e)^3-6*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*c^2* d*(1-cos(f*x+e))^3*csc(f*x+e)^3+6*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)* c*d^2*(1-cos(f*x+e))^3*csc(f*x+e)^3-2*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1 /2)*d^3*(1-cos(f*x+e))^3*csc(f*x+e)^3-32*c^3*2^(1/2)*arctanh(2^(1/2)/((1-c os(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)))-13*((1-cos(f* x+e))^2*csc(f*x+e)^2-1)^(1/2)*c^3*(-cot(f*x+e)+csc(f*x+e))+15*((1-cos(f*x+ e))^2*csc(f*x+e)^2-1)^(1/2)*c^2*d*(-cot(f*x+e)+csc(f*x+e))+9*((1-cos(f*x+e ))^2*csc(f*x+e)^2-1)^(1/2)*c*d^2*(-cot(f*x+e)+csc(f*x+e))-11*((1-cos(f*x+e ))^2*csc(f*x+e)^2-1)^(1/2)*d^3*(-cot(f*x+e)+csc(f*x+e))+43*c^3*ln(csc(f*x+ e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))-9*c^2*d*ln(csc(f*x+ e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))-15*c*d^2*ln(csc(f*x +e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))-19*d^3*ln(csc(f*x+ e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)))
Time = 30.82 (sec) , antiderivative size = 880, normalized size of antiderivative = 1.83 \[ \int \frac {(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx=\left [\frac {\sqrt {2} {\left ({\left (43 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} - 19 \, d^{3}\right )} \cos \left (f x + e\right )^{3} + 43 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} - 19 \, d^{3} + 3 \, {\left (43 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} - 19 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (43 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} - 19 \, d^{3}\right )} \cos \left (f x + e\right )\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^{3} \cos \left (f x + e\right )^{3} + 3 \, c^{3} \cos \left (f x + e\right )^{2} + 3 \, 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 ) - 4 \, {\left (3 \, {\left (5 \, c^{3} - 7 \, c^{2} d - c d^{2} + 3 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} + 13 \, d^{3}\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} - 9 \, c^{2} d - 15 \, c d^{2} - 19 \, d^{3}\right )} \cos \left (f x + e\right )^{3} + 43 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} - 19 \, d^{3} + 3 \, {\left (43 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} - 19 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (43 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} - 19 \, d^{3}\right )} \cos \left (f x + e\right )\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^{3} \cos \left (f x + e\right )^{3} + 3 \, c^{3} \cos \left (f x + e\right )^{2} + 3 \, 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 ) - 2 \, {\left (3 \, {\left (5 \, c^{3} - 7 \, c^{2} d - c d^{2} + 3 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (11 \, c^{3} - 9 \, c^{2} d - 15 \, c d^{2} + 13 \, d^{3}\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 - 9*c^2*d - 15*c*d^2 - 19*d^3)*cos(f*x + e)^3 + 43 *c^3 - 9*c^2*d - 15*c*d^2 - 19*d^3 + 3*(43*c^3 - 9*c^2*d - 15*c*d^2 - 19*d ^3)*cos(f*x + e)^2 + 3*(43*c^3 - 9*c^2*d - 15*c*d^2 - 19*d^3)*cos(f*x + e) )*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)/(c os(f*x + e)^2 + 2*cos(f*x + e) + 1)) - 64*(c^3*cos(f*x + e)^3 + 3*c^3*cos( f*x + e)^2 + 3*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)) - 4*(3*(5*c^3 - 7*c^2*d - c*d ^2 + 3*d^3)*cos(f*x + e)^2 + (11*c^3 - 9*c^2*d - 15*c*d^2 + 13*d^3)*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 - 9*c^2*d - 15*c*d^2 - 19*d^3)*cos(f*x + e)^3 + 43*c^3 - 9*c^2*d - 15*c*d^2 - 19*d^3 + 3*(43*c^3 - 9*c^2*d - 15*c*d^2 - 19*d^3)*cos (f*x + e)^2 + 3*(43*c^3 - 9*c^2*d - 15*c*d^2 - 19*d^3)*cos(f*x + e))*sqrt( a)*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sq rt(a)*sin(f*x + e))) - 64*(c^3*cos(f*x + e)^3 + 3*c^3*cos(f*x + e)^2 + 3*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))) - 2*(3*(5*c^3 - 7*c^2*d - c*d^2 + 3*d^3)*cos(f*x + e)^2 + (11*c^3 - 9*c^2*d - 15*c*d^2 + 13*d^3)*cos(f*x...
\[ \int \frac {(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{3}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{5/2}} \, dx=\int \frac {{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^3}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]