Integrand size = 33, antiderivative size = 211 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {4 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (41 A+42 B) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (11 A+7 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)} \]
2/7*a*A*(a+a*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/35*(11*A+7*B)*( a^3+a^3*sec(d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+4/105*a^3*(41*A+42*B)*si n(d*x+c)/d/sec(d*x+c)^(1/2)+4/5*a^3*(7*A+9*B)*(cos(1/2*d*x+1/2*c)^2)^(1/2) /cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2) *sec(d*x+c)^(1/2)/d+4/21*a^3*(13*A+21*B)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos( 1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec( d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.71 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.92 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (40 (13 A+21 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-56 i (7 A+9 B) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (168 i (7 A+9 B)+5 (107 A+84 B) \sin (c+d x)+42 (3 A+B) \sin (2 (c+d x))+15 A \sin (3 (c+d x)))\right )}{210 d} \]
(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(40*(13*A + 21*B)*Sqrt[Cos [c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(7*A + 9*B)*E^(I*(c + d*x))* Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*( c + d*x))] + Cos[c + d*x]*((168*I)*(7*A + 9*B) + 5*(107*A + 84*B)*Sin[c + d*x] + 42*(3*A + B)*Sin[2*(c + d*x)] + 15*A*Sin[3*(c + d*x)])))/(210*d*E^( I*d*x))
Time = 1.24 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 4505, 27, 3042, 4505, 3042, 4484, 27, 3042, 4274, 3042, 4258, 3042, 3119, 3120}
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.
\(\displaystyle \int \frac {(a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx\) |
\(\Big \downarrow \) 4505 |
\(\displaystyle \frac {2}{7} \int \frac {(\sec (c+d x) a+a)^2 (a (11 A+7 B)+a (A+7 B) \sec (c+d x))}{2 \sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \int \frac {(\sec (c+d x) a+a)^2 (a (11 A+7 B)+a (A+7 B) \sec (c+d x))}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (11 A+7 B)+a (A+7 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4505 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(\sec (c+d x) a+a) \left ((41 A+42 B) a^2+(8 A+21 B) \sec (c+d x) a^2\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((41 A+42 B) a^2+(8 A+21 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4484 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}-\frac {2}{3} \int -\frac {21 (7 A+9 B) a^3+5 (13 A+21 B) \sec (c+d x) a^3}{2 \sqrt {\sec (c+d x)}}dx\right )+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {21 (7 A+9 B) a^3+5 (13 A+21 B) \sec (c+d x) a^3}{\sqrt {\sec (c+d x)}}dx+\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {21 (7 A+9 B) a^3+5 (13 A+21 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (21 a^3 (7 A+9 B) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+5 a^3 (13 A+21 B) \int \sqrt {\sec (c+d x)}dx\right )+\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (21 a^3 (7 A+9 B) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 a^3 (13 A+21 B) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (5 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (5 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (5 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {42 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {1}{7} \left (\frac {2 (11 A+7 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2}{5} \left (\frac {2 a^3 (41 A+42 B) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{3} \left (\frac {10 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {42 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {2 a A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
(2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2 *(11*A + 7*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/ 2)) + (2*(((42*a^3*(7*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2 ]*Sqrt[Sec[c + d*x]])/d + (10*a^3*(13*A + 21*B)*Sqrt[Cos[c + d*x]]*Ellipti cF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d)/3 + (2*a^3*(41*A + 42*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])))/5)/7
3.2.98.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[n, -1]
Time = 24.85 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.82
method | result | size |
default | \(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (120 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-432 A -84 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (602 A +294 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-208 A -126 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+65 A \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 A \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+105 B \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-189 B \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(385\) |
parts | \(\text {Expression too large to display}\) | \(888\) |
-4/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(120*A* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(-432*A-84*B)*sin(1/2*d*x+1/2*c)^6 *cos(1/2*d*x+1/2*c)+(602*A+294*B)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+ (-208*A-126*B)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+65*A*(2*sin(1/2*d*x +1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2* c),2^(1/2))-147*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^ (1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+105*B*(2*sin(1/2*d*x+1/2*c)^2- 1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2) )-189*B*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Elli pticE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/ 2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00 \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (13 \, A + 21 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (13 \, A + 21 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (7 \, A + 9 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (7 \, A + 9 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (15 \, A a^{3} \cos \left (d x + c\right )^{3} + 21 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 5 \, {\left (26 \, A + 21 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d} \]
-2/105*(5*I*sqrt(2)*(13*A + 21*B)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*I*sqrt(2)*(13*A + 21*B)*a^3*weierstrassPInverse( -4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*I*sqrt(2)*(7*A + 9*B)*a^3*weier strassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c) )) + 21*I*sqrt(2)*(7*A + 9*B)*a^3*weierstrassZeta(-4, 0, weierstrassPInver se(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (15*A*a^3*cos(d*x + c)^3 + 21* (3*A + B)*a^3*cos(d*x + c)^2 + 5*(26*A + 21*B)*a^3*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
\[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=a^{3} \left (\int \frac {A}{\sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 A}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {A}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {B}{\sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 B}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 B}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int B \sqrt {\sec {\left (c + d x \right )}}\, dx\right ) \]
a**3*(Integral(A/sec(c + d*x)**(7/2), x) + Integral(3*A/sec(c + d*x)**(5/2 ), x) + Integral(3*A/sec(c + d*x)**(3/2), x) + Integral(A/sqrt(sec(c + d*x )), x) + Integral(B/sec(c + d*x)**(5/2), x) + Integral(3*B/sec(c + d*x)**( 3/2), x) + Integral(3*B/sqrt(sec(c + d*x)), x) + Integral(B*sqrt(sec(c + d *x)), x))
\[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]