Integrand size = 43, antiderivative size = 202 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (6 a A b+3 a^2 B+5 b^2 B+10 a b C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (14 a b B+7 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 \left (4 A b^2+14 a b B+a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a (4 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{7 d} \]
2/5*(6*A*a*b+3*B*a^2+5*B*b^2+10*C*a*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))/d+2/21*(14*B*a*b+7*b^2* (A+3*C)+a^2*(5*A+7*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell ipticF(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/35*a*(4*A*b+7*B*a)*cos(d*x+c)^(3/2) *sin(d*x+c)/d+2/21*(4*A*b^2+14*B*a*b+a^2*(5*A+7*C))*sin(d*x+c)*cos(d*x+c)^ (1/2)/d+2/7*A*(b+a*cos(d*x+c))^2*sin(d*x+c)*cos(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.80 (sec) , antiderivative size = 2361, normalized size of antiderivative = 11.69 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]
Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Cot[c])/(5*d) + ((2 3*a^2*A + 28*A*b^2 + 56*a*b*B + 28*a^2*C)*Cos[d*x]*Sin[c])/(21*d) + (2*a*( 2*A*b + a*B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (a^2*A*Cos[3*d*x]*Sin[3*c])/(7*d ) + ((23*a^2*A + 28*A*b^2 + 56*a*b*B + 28*a^2*C)*Cos[c]*Sin[d*x])/(21*d) + (2*a*(2*A*b + a*B)*Cos[2*c]*Sin[2*d*x])/(5*d) + (a^2*A*Cos[3*c]*Sin[3*d*x ])/(7*d)))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (20*a^2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2} , {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - Ar cTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]) ]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^2*(A + 2 *C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^2 *Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcT an[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x] ^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(S qrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - Ar cTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a*b*B*Cos[c + d*x]^4*Csc[c]*H ypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b...
Time = 1.24 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.372, Rules used = {3042, 4600, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 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.
| \(\displaystyle \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{7/2} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b)^2 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\sqrt {\cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^2 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{7} \int \frac {(b+a \cos (c+d x)) \left ((4 A b+7 a B) \cos ^2(c+d x)+(5 a A+7 b B+7 a C) \cos (c+d x)+b (A+7 C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {(b+a \cos (c+d x)) \left ((4 A b+7 a B) \cos ^2(c+d x)+(5 a A+7 b B+7 a C) \cos (c+d x)+b (A+7 C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left ((4 A b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(5 a A+7 b B+7 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+b (A+7 C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {5 (A+7 C) b^2+5 \left ((5 A+7 C) a^2+14 b B a+4 A b^2\right ) \cos ^2(c+d x)+7 \left (3 B a^2+6 A b a+10 b C a+5 b^2 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {5 (A+7 C) b^2+5 \left ((5 A+7 C) a^2+14 b B a+4 A b^2\right ) \cos ^2(c+d x)+7 \left (3 B a^2+6 A b a+10 b C a+5 b^2 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {5 (A+7 C) b^2+5 \left ((5 A+7 C) a^2+14 b B a+4 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (3 B a^2+6 A b a+10 b C a+5 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {5 \left ((5 A+7 C) a^2+14 b B a+7 b^2 (A+3 C)\right )+21 \left (3 B a^2+6 A b a+10 b C a+5 b^2 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{3 d}\right )+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {5 \left ((5 A+7 C) a^2+14 b B a+7 b^2 (A+3 C)\right )+21 \left (3 B a^2+6 A b a+10 b C a+5 b^2 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{3 d}\right )+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {5 \left ((5 A+7 C) a^2+14 b B a+7 b^2 (A+3 C)\right )+21 \left (3 B a^2+6 A b a+10 b C a+5 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{3 d}\right )+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 \left (3 a^2 B+6 a A b+10 a b C+5 b^2 B\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{3 d}\right )+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 \left (3 a^2 B+6 a A b+10 a b C+5 b^2 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{3 d}\right )+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^2 B+6 a A b+10 a b C+5 b^2 B\right )}{d}\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{3 d}\right )+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+14 a b B+4 A b^2\right )}{3 d}+\frac {1}{3} \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right )}{d}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^2 B+6 a A b+10 a b C+5 b^2 B\right )}{d}\right )\right )+\frac {2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\) |
(2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2*a *(4*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (((42*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE[(c + d*x)/2, 2])/d + (10*(14*a*b* B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d)/3 + ( 10*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x]) /(3*d))/5)/7
3.13.100.3.1 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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x _)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) *(x_)]^2), x_Symbol] :> Simp[d^(m + 2) Int[(b + a*Cos[e + f*x])^m*(d*Cos[ e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr eeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(705\) vs. \(2(238)=476\).
Time = 438.84 (sec) , antiderivative size = 706, normalized size of antiderivative = 3.50
| method | result | size |
| default | \(-\frac {2 \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}}\, \left (240 A \,a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-360 A \,a^{2}-336 a A b -168 B \,a^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (280 A \,a^{2}+336 a A b +140 A \,b^{2}+168 B \,a^{2}+280 B a b +140 C \,a^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-80 A \,a^{2}-84 a A b -70 A \,b^{2}-42 B \,a^{2}-140 B a b -70 C \,a^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+25 A \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, a^{2}+35 A \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, b^{2}-126 A \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, a b +70 B \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, a b -63 B \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, a^{2}-105 B \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, b^{2}+35 C \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, a^{2}+105 C \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, b^{2}-210 C \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, a b \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}\) | \(706\) |
int(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
-2/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*A*a^2* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(-360*A*a^2-336*A*a*b-168*B*a^2)*s in(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(280*A*a^2+336*A*a*b+140*A*b^2+168* B*a^2+280*B*a*b+140*C*a^2)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-80*A* a^2-84*A*a*b-70*A*b^2-42*B*a^2-140*B*a*b-70*C*a^2)*sin(1/2*d*x+1/2*c)^2*co s(1/2*d*x+1/2*c)+25*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1 /2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^2+35*A*EllipticF(cos(1/2 *d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2- 1)^(1/2)*b^2-126*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2* c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a*b+70*B*EllipticF(cos(1/2*d* x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^ (1/2)*a*b-63*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2 )^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^2-105*B*EllipticE(cos(1/2*d*x+1 /2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/ 2)*b^2+35*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^( 1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^2+105*C*EllipticF(cos(1/2*d*x+1/2* c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)* b^2-210*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a*b)/(-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)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.39 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (15 \, A a^{2} \cos \left (d x + c\right )^{2} + 5 \, {\left (5 \, A + 7 \, C\right )} a^{2} + 70 \, B a b + 35 \, A b^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{2} + 14 i \, B a b + 7 i \, {\left (A + 3 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{2} - 14 i \, B a b - 7 i \, {\left (A + 3 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-3 i \, B a^{2} - 2 i \, {\left (3 \, A + 5 \, C\right )} a b - 5 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (3 i \, B a^{2} + 2 i \, {\left (3 \, A + 5 \, C\right )} a b + 5 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{105 \, d} \]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
1/105*(2*(15*A*a^2*cos(d*x + c)^2 + 5*(5*A + 7*C)*a^2 + 70*B*a*b + 35*A*b^ 2 + 21*(B*a^2 + 2*A*a*b)*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - 5 *sqrt(2)*(I*(5*A + 7*C)*a^2 + 14*I*B*a*b + 7*I*(A + 3*C)*b^2)*weierstrassP Inverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*sqrt(2)*(-I*(5*A + 7*C)* a^2 - 14*I*B*a*b - 7*I*(A + 3*C)*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*sqrt(2)*(-3*I*B*a^2 - 2*I*(3*A + 5*C)*a*b - 5*I *B*b^2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I *sin(d*x + c))) - 21*sqrt(2)*(3*I*B*a^2 + 2*I*(3*A + 5*C)*a*b + 5*I*B*b^2) *weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d* x + c))))/d
Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^2*c os(d*x + c)^(7/2), x)
\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^2*c os(d*x + c)^(7/2), x)
Time = 19.27 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.50 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {C\,a^2\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,A\,b^2\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}+\frac {2\,B\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,B\,a\,b\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {4\,C\,a\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {2\,A\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,A\,a\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(C*a^2*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2 , 2))/3))/d + (2*A*b^2*(cos(c + d*x)^(1/2)*sin(c + d*x) + ellipticF(c/2 + (d*x)/2, 2)))/(3*d) + (2*B*b^2*ellipticE(c/2 + (d*x)/2, 2))/d + (2*C*b^2*e llipticF(c/2 + (d*x)/2, 2))/d + (2*B*a*b*((2*cos(c + d*x)^(1/2)*sin(c + d* x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (4*C*a*b*ellipticE(c/2 + ( d*x)/2, 2))/d - (2*A*a^2*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9 /4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (2*B*a^2*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7* d*(sin(c + d*x)^2)^(1/2)) - (4*A*a*b*cos(c + d*x)^(7/2)*sin(c + d*x)*hyper geom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2))