Integrand size = 27, antiderivative size = 241 \[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx=\frac {2 a^{5/2} c^3 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}} \]
2/35*a^2*(6*c+13*d)*(c+d*sec(f*x+e))^2*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2) +2/7*a^2*(c+d*sec(f*x+e))^3*tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+2/105*a^2* (72*c^3+486*c^2*d+378*c*d^2+104*d^3+d*(24*c^2+111*c*d+52*d^2)*sec(f*x+e))* tan(f*x+e)/f/(a+a*sec(f*x+e))^(1/2)+2*a^(5/2)*c^3*arctanh((a-a*sec(f*x+e)) ^(1/2)/a^(1/2))*tan(f*x+e)/f/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)
Time = 3.63 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.91 \[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx=\frac {a \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^3(e+f x) \sqrt {a (1+\sec (e+f x))} \left (420 \sqrt {2} c^3 \arcsin \left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right ) \cos ^{\frac {7}{2}}(e+f x)+2 \left (210 c^2 d+378 c d^2+164 d^3+9 \left (35 c^3+175 c^2 d+154 c d^2+52 d^3\right ) \cos (e+f x)+2 d \left (105 c^2+189 c d+52 d^2\right ) \cos (2 (e+f x))+105 c^3 \cos (3 (e+f x))+525 c^2 d \cos (3 (e+f x))+378 c d^2 \cos (3 (e+f x))+104 d^3 \cos (3 (e+f x))\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{420 f} \]
(a*Sec[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[a*(1 + Sec[e + f*x])]*(420*Sqrt[2] *c^3*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Cos[e + f*x]^(7/2) + 2*(210*c^2*d + 378*c*d^2 + 164*d^3 + 9*(35*c^3 + 175*c^2*d + 154*c*d^2 + 52*d^3)*Cos[e + f*x] + 2*d*(105*c^2 + 189*c*d + 52*d^2)*Cos[2*(e + f*x)] + 105*c^3*Cos[3*( e + f*x)] + 525*c^2*d*Cos[3*(e + f*x)] + 378*c*d^2*Cos[3*(e + f*x)] + 104* d^3*Cos[3*(e + f*x)])*Sin[(e + f*x)/2]))/(420*f)
Time = 0.41 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 4428, 27, 170, 27, 170, 27, 164, 73, 219}
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.
| \(\displaystyle \int (a \sec (e+f x)+a)^{3/2} (c+d \sec (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^{3/2} \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4428 |
| \(\displaystyle -\frac {a^2 \tan (e+f x) \int \frac {a \cos (e+f x) (\sec (e+f x)+1) (c+d \sec (e+f x))^3}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \int \frac {\cos (e+f x) (\sec (e+f x)+1) (c+d \sec (e+f x))^3}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \left (-\frac {2 \int -\frac {a \cos (e+f x) (c+d \sec (e+f x))^2 (7 c+(6 c+13 d) \sec (e+f x))}{2 \sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{7 a}-\frac {2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{7 a}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \left (\frac {1}{7} \int \frac {\cos (e+f x) (c+d \sec (e+f x))^2 (7 c+(6 c+13 d) \sec (e+f x))}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{7 a}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \left (\frac {1}{7} \left (-\frac {2 \int -\frac {a \cos (e+f x) (c+d \sec (e+f x)) \left (35 c^2+\left (24 c^2+111 d c+52 d^2\right ) \sec (e+f x)\right )}{2 \sqrt {a-a \sec (e+f x)}}d\sec (e+f x)}{5 a}-\frac {2 (6 c+13 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{5 a}\right )-\frac {2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{7 a}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\cos (e+f x) (c+d \sec (e+f x)) \left (35 c^2+\left (24 c^2+111 d c+52 d^2\right ) \sec (e+f x)\right )}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {2 (6 c+13 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{5 a}\right )-\frac {2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{7 a}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \left (\frac {1}{7} \left (\frac {1}{5} \left (35 c^3 \int \frac {\cos (e+f x)}{\sqrt {a-a \sec (e+f x)}}d\sec (e+f x)-\frac {2 \sqrt {a-a \sec (e+f x)} \left (d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)+2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )\right )}{3 a}\right )-\frac {2 (6 c+13 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{5 a}\right )-\frac {2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{7 a}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \left (\frac {1}{7} \left (\frac {1}{5} \left (-\frac {70 c^3 \int \frac {1}{1-\frac {a-a \sec (e+f x)}{a}}d\sqrt {a-a \sec (e+f x)}}{a}-\frac {2 \sqrt {a-a \sec (e+f x)} \left (d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)+2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )\right )}{3 a}\right )-\frac {2 (6 c+13 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{5 a}\right )-\frac {2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{7 a}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^3 \tan (e+f x) \left (\frac {1}{7} \left (\frac {1}{5} \left (-\frac {70 c^3 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 \sqrt {a-a \sec (e+f x)} \left (d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)+2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )\right )}{3 a}\right )-\frac {2 (6 c+13 d) \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^2}{5 a}\right )-\frac {2 \sqrt {a-a \sec (e+f x)} (c+d \sec (e+f x))^3}{7 a}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}\) |
-((a^3*((-2*Sqrt[a - a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3)/(7*a) + ((-2* (6*c + 13*d)*Sqrt[a - a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2)/(5*a) + ((-7 0*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]])/Sqrt[a] - (2*Sqrt[a - a*S ec[e + f*x]]*(2*(36*c^3 + 243*c^2*d + 189*c*d^2 + 52*d^3) + d*(24*c^2 + 11 1*c*d + 52*d^2)*Sec[e + f*x]))/(3*a))/5)/7)*Tan[e + f*x])/(f*Sqrt[a - a*Se c[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))
3.2.54.3.1 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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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 = 6.62 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.23
| method | result | size |
| default | \(\frac {2 a \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (105 \,\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c^{3} \cos \left (f x +e \right )+105 \,\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c^{3}+105 \sin \left (f x +e \right ) c^{3}+525 \sin \left (f x +e \right ) c^{2} d +378 \sin \left (f x +e \right ) c \,d^{2}+104 \sin \left (f x +e \right ) d^{3}+105 c^{2} d \tan \left (f x +e \right )+189 c \,d^{2} \tan \left (f x +e \right )+52 d^{3} \tan \left (f x +e \right )+63 c \,d^{2} \tan \left (f x +e \right ) \sec \left (f x +e \right )+39 d^{3} \tan \left (f x +e \right ) \sec \left (f x +e \right )+15 d^{3} \tan \left (f x +e \right ) \sec \left (f x +e \right )^{2}\right )}{105 f \left (\cos \left (f x +e \right )+1\right )}\) | \(297\) |
| parts | \(\frac {2 c^{3} a \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )+\operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+\sin \left (f x +e \right )\right )}{f \left (\cos \left (f x +e \right )+1\right )}+\frac {2 d^{3} a \left (104 \cos \left (f x +e \right )^{3}+52 \cos \left (f x +e \right )^{2}+39 \cos \left (f x +e \right )+15\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \tan \left (f x +e \right ) \sec \left (f x +e \right )^{2}}{105 f \left (\cos \left (f x +e \right )+1\right )}+\frac {2 c^{2} d a \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (5 \sin \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f \left (\cos \left (f x +e \right )+1\right )}+\frac {6 c \,d^{2} a \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (6 \sin \left (f x +e \right )+3 \tan \left (f x +e \right )+\sec \left (f x +e \right ) \tan \left (f x +e \right )\right )}{5 f \left (\cos \left (f x +e \right )+1\right )}\) | \(348\) |
2/105*a/f*(a*(sec(f*x+e)+1))^(1/2)/(cos(f*x+e)+1)*(105*arctanh(sin(f*x+e)/ (cos(f*x+e)+1)/(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2))*(-cos(f*x+e)/(cos(f*x+e )+1))^(1/2)*c^3*cos(f*x+e)+105*arctanh(sin(f*x+e)/(cos(f*x+e)+1)/(-cos(f*x +e)/(cos(f*x+e)+1))^(1/2))*(-cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*c^3+105*sin( f*x+e)*c^3+525*sin(f*x+e)*c^2*d+378*sin(f*x+e)*c*d^2+104*sin(f*x+e)*d^3+10 5*c^2*d*tan(f*x+e)+189*c*d^2*tan(f*x+e)+52*d^3*tan(f*x+e)+63*c*d^2*tan(f*x +e)*sec(f*x+e)+39*d^3*tan(f*x+e)*sec(f*x+e)+15*d^3*tan(f*x+e)*sec(f*x+e)^2 )
Time = 0.31 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.00 \[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx=\left [\frac {105 \, {\left (a c^{3} \cos \left (f x + e\right )^{4} + a c^{3} \cos \left (f x + e\right )^{3}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) + 2 \, {\left (15 \, a d^{3} + {\left (105 \, a c^{3} + 525 \, a c^{2} d + 378 \, a c d^{2} + 104 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{2} d + 189 \, a c d^{2} + 52 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (21 \, a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{105 \, {\left (f \cos \left (f x + e\right )^{4} + f \cos \left (f x + e\right )^{3}\right )}}, -\frac {2 \, {\left (105 \, {\left (a c^{3} \cos \left (f x + e\right )^{4} + a c^{3} \cos \left (f x + e\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - {\left (15 \, a d^{3} + {\left (105 \, a c^{3} + 525 \, a c^{2} d + 378 \, a c d^{2} + 104 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{2} d + 189 \, a c d^{2} + 52 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (21 \, a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{105 \, {\left (f \cos \left (f x + e\right )^{4} + f \cos \left (f x + e\right )^{3}\right )}}\right ] \]
[1/105*(105*(a*c^3*cos(f*x + e)^4 + a*c^3*cos(f*x + e)^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*(15*a* d^3 + (105*a*c^3 + 525*a*c^2*d + 378*a*c*d^2 + 104*a*d^3)*cos(f*x + e)^3 + (105*a*c^2*d + 189*a*c*d^2 + 52*a*d^3)*cos(f*x + e)^2 + 3*(21*a*c*d^2 + 1 3*a*d^3)*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e ))/(f*cos(f*x + e)^4 + f*cos(f*x + e)^3), -2/105*(105*(a*c^3*cos(f*x + e)^ 4 + a*c^3*cos(f*x + e)^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))) - (15*a*d^3 + (105*a*c^3 + 525 *a*c^2*d + 378*a*c*d^2 + 104*a*d^3)*cos(f*x + e)^3 + (105*a*c^2*d + 189*a* c*d^2 + 52*a*d^3)*cos(f*x + e)^2 + 3*(21*a*c*d^2 + 13*a*d^3)*cos(f*x + e)) *sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4 + f*cos(f*x + e)^3)]
\[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx=\int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sec {\left (e + f x \right )}\right )^{3}\, dx \]
\[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sec \left (f x + e\right ) + c\right )}^{3} \,d x } \]
-1/210*(4*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*(7*(15*(a*c^3 + 3*a*c^2*d)*sin(6*f*x + 6*e) + 5*(9*a*c^3 + 33*a*c ^2*d + 18*a*c*d^2 + 4*a*d^3)*sin(4*f*x + 4*e) + (45*a*c^3 + 195*a*c^2*d + 144*a*c*d^2 + 52*a*d^3)*sin(2*f*x + 2*e))*cos(7/2*arctan2(sin(2*f*x + 2*e) , cos(2*f*x + 2*e) + 1)) - (105*a*c^3 + 525*a*c^2*d + 378*a*c*d^2 + 104*a* d^3 + 105*(a*c^3 + 3*a*c^2*d)*cos(6*f*x + 6*e) + 35*(9*a*c^3 + 33*a*c^2*d + 18*a*c*d^2 + 4*a*d^3)*cos(4*f*x + 4*e) + 7*(45*a*c^3 + 195*a*c^2*d + 144 *a*c*d^2 + 52*a*d^3)*cos(2*f*x + 2*e))*sin(7/2*arctan2(sin(2*f*x + 2*e), c os(2*f*x + 2*e) + 1)))*sqrt(a) + 105*((a*c^3*cos(2*f*x + 2*e)^4 + a*c^3*si n(2*f*x + 2*e)^4 + 4*a*c^3*cos(2*f*x + 2*e)^3 + 6*a*c^3*cos(2*f*x + 2*e)^2 + 4*a*c^3*cos(2*f*x + 2*e) + a*c^3 + 2*(a*c^3*cos(2*f*x + 2*e)^2 + 2*a*c^ 3*cos(2*f*x + 2*e) + a*c^3)*sin(2*f*x + 2*e)^2)*arctan2((cos(2*f*x + 2*e)^ 2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*sin(1/2*arctan2(sin (2*f*x + 2*e), cos(2*f*x + 2*e) + 1)), (cos(2*f*x + 2*e)^2 + sin(2*f*x + 2 *e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*cos(1/2*arctan2(sin(2*f*x + 2*e), co s(2*f*x + 2*e) + 1)) + 1) - (a*c^3*cos(2*f*x + 2*e)^4 + a*c^3*sin(2*f*x + 2*e)^4 + 4*a*c^3*cos(2*f*x + 2*e)^3 + 6*a*c^3*cos(2*f*x + 2*e)^2 + 4*a*c^3 *cos(2*f*x + 2*e) + a*c^3 + 2*(a*c^3*cos(2*f*x + 2*e)^2 + 2*a*c^3*cos(2*f* x + 2*e) + a*c^3)*sin(2*f*x + 2*e)^2)*arctan2((cos(2*f*x + 2*e)^2 + sin(2* f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*sin(1/2*arctan2(sin(2*f*x ...
\[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sec \left (f x + e\right ) + c\right )}^{3} \,d x } \]
Timed out. \[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^3 \,d x \]