Integrand size = 25, antiderivative size = 115 \[ \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{9/2} \, dx=\frac {2 b^4 (5 A+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 d}+\frac {2 b^3 (5 A+7 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 A b^2 (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d} \]
2/21*b^3*(5*A+7*C)*(b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+2/21*b^4*(5*A+7*C)*(c os(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)*(b*sec(d*x+c))^(1/2)/d+2/7*A*b^2*(b*sec(d*x+c)) ^(5/2)*tan(d*x+c)/d
Time = 1.82 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.68 \[ \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{9/2} \, dx=\frac {b^2 (b \sec (c+d x))^{5/2} \left (2 (5 A+7 C) \cos ^{\frac {5}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+(5 A+7 C) \sin (2 (c+d x))+6 A \tan (c+d x)\right )}{21 d} \]
(b^2*(b*Sec[c + d*x])^(5/2)*(2*(5*A + 7*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C)*Sin[2*(c + d*x)] + 6*A*Tan[c + d*x]))/(21*d)
Time = 0.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3717, 3042, 4534, 3042, 4255, 3042, 4258, 3042, 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.
| \(\displaystyle \int (b \sec (c+d x))^{9/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3717 |
| \(\displaystyle b^2 \int (b \sec (c+d x))^{5/2} \left (A \sec ^2(c+d x)+C\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \int \left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A \csc \left (c+d x+\frac {\pi }{2}\right )^2+C\right )dx\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle b^2 \left (\frac {1}{7} (5 A+7 C) \int (b \sec (c+d x))^{5/2}dx+\frac {2 A \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \left (\frac {1}{7} (5 A+7 C) \int \left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx+\frac {2 A \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle b^2 \left (\frac {1}{7} (5 A+7 C) \left (\frac {1}{3} b^2 \int \sqrt {b \sec (c+d x)}dx+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 A \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \left (\frac {1}{7} (5 A+7 C) \left (\frac {1}{3} b^2 \int \sqrt {b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 A \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}\right )\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle b^2 \left (\frac {1}{7} (5 A+7 C) \left (\frac {1}{3} b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 A \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \left (\frac {1}{7} (5 A+7 C) \left (\frac {1}{3} b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 A \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle b^2 \left (\frac {1}{7} (5 A+7 C) \left (\frac {2 b^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}\right )+\frac {2 A \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}\right )\) |
b^2*(((5*A + 7*C)*((2*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqr t[b*Sec[c + d*x]])/(3*d) + (2*b*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) ))/7 + (2*A*(b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d))
3.1.26.3.1 Defintions of rubi rules used
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[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Csc[e + f*x])^(m - n*p )*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
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_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Result contains complex when optimal does not.
Time = 72.99 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(-\frac {2 b^{4} \sqrt {b \sec \left (d x +c \right )}\, \left (5 i \cos \left (d x +c \right ) A F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+7 i \cos \left (d x +c \right ) C F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+5 i A F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+7 i C F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-5 A \tan \left (d x +c \right )-7 \tan \left (d x +c \right ) C -3 \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right ) A \right )}{21 d}\) | \(291\) |
| parts | \(-\frac {2 i A \sqrt {b \sec \left (d x +c \right )}\, b^{4} \left (5 \cos \left (d x +c \right ) F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+5 F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+5 i \tan \left (d x +c \right )+3 i \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )\right )}{21 d}-\frac {2 C \,b^{4} \sqrt {b \sec \left (d x +c \right )}\, \left (i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\tan \left (d x +c \right )\right )}{3 d}\) | \(307\) |
-2/21*b^4/d*(b*sec(d*x+c))^(1/2)*(5*I*cos(d*x+c)*A*EllipticF(I*(csc(d*x+c) -cot(d*x+c)),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2) +7*I*cos(d*x+c)*C*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)*(1/(1+cos(d*x+c)) )^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+5*I*A*EllipticF(I*(csc(d*x+c)-co t(d*x+c)),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+7* I*C*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)*(1/(1+cos(d*x+c)))^(1/2)*(cos(d *x+c)/(1+cos(d*x+c)))^(1/2)-5*A*tan(d*x+c)-7*tan(d*x+c)*C-3*tan(d*x+c)*sec (d*x+c)^2*A)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.23 \[ \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{9/2} \, dx=\frac {-i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} b^{\frac {9}{2}} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} b^{\frac {9}{2}} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left ({\left (5 \, A + 7 \, C\right )} b^{4} \cos \left (d x + c\right )^{2} + 3 \, A b^{4}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{21 \, d \cos \left (d x + c\right )^{3}} \]
1/21*(-I*sqrt(2)*(5*A + 7*C)*b^(9/2)*cos(d*x + c)^3*weierstrassPInverse(-4 , 0, cos(d*x + c) + I*sin(d*x + c)) + I*sqrt(2)*(5*A + 7*C)*b^(9/2)*cos(d* x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*((5 *A + 7*C)*b^4*cos(d*x + c)^2 + 3*A*b^4)*sqrt(b/cos(d*x + c))*sin(d*x + c)) /(d*cos(d*x + c)^3)
Timed out. \[ \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{9/2} \, dx=\text {Timed out} \]
\[ \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{9/2} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {9}{2}} \,d x } \]
\[ \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{9/2} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {9}{2}} \,d x } \]
Timed out. \[ \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{9/2} \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{9/2} \,d x \]