Integrand size = 21, antiderivative size = 100 \[ \int \cos ^7(c+d x) (b \sec (c+d x))^{5/2} \, dx=\frac {14 b^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 b^6 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b^4 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}} \]
2/9*b^6*sin(d*x+c)/d/(b*sec(d*x+c))^(7/2)+14/45*b^4*sin(d*x+c)/d/(b*sec(d* x+c))^(3/2)+14/15*b^3*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elli pticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1/2)/(b*sec(d*x+c))^(1/2)
Time = 0.51 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.74 \[ \int \cos ^7(c+d x) (b \sec (c+d x))^{5/2} \, dx=\frac {b^2 \sqrt {b \sec (c+d x)} \left (84 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^2(c+d x) (33 \sin (c+d x)+5 \sin (3 (c+d x)))\right )}{90 d} \]
(b^2*Sqrt[b*Sec[c + d*x]]*(84*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^2*(33*Sin[c + d*x] + 5*Sin[3*(c + d*x)])))/(90*d)
Time = 0.49 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 2030, 4256, 3042, 4256, 3042, 4258, 3042, 3119}
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.
\(\displaystyle \int \cos ^7(c+d x) (b \sec (c+d x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{\csc \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle b^7 \int \frac {1}{\left (b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{9/2}}dx\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle b^7 \left (\frac {7 \int \frac {1}{(b \sec (c+d x))^{5/2}}dx}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b^7 \left (\frac {7 \int \frac {1}{\left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle b^7 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {b \sec (c+d x)}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b^7 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle b^7 \left (\frac {7 \left (\frac {3 \int \sqrt {\cos (c+d x)}dx}{5 b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b^7 \left (\frac {7 \left (\frac {3 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle b^7 \left (\frac {7 \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
b^7*((2*Sin[c + d*x])/(9*b*d*(b*Sec[c + d*x])^(7/2)) + (7*((6*EllipticE[(c + d*x)/2, 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*Sin[ c + d*x])/(5*b*d*(b*Sec[c + d*x])^(3/2))))/(9*b^2))
3.1.99.3.1 Defintions of rubi rules used
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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]
Result contains complex when optimal does not.
Time = 4.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.87
\[-\frac {2 \sqrt {b \sec \left (d x +c \right )}\, b^{2} \left (i \left (-21 \cos \left (d x +c \right )-21\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )+i \left (21 \cos \left (d x +c \right )+21\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )+\left (5 \cos \left (d x +c \right )^{5}+2 \cos \left (d x +c \right )^{3}+14 \cos \left (d x +c \right )-21\right ) \cot \left (d x +c \right )\right )}{45 d}\]
-2/45/d*(b*sec(d*x+c))^(1/2)*b^2*(I*(-21*cos(d*x+c)-21)*(1/(cos(d*x+c)+1)) ^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(I*(-cot(d*x+c)+csc(d*x+ c)),I)+I*(21*cos(d*x+c)+21)*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+ c)+1))^(1/2)*EllipticF(I*(-cot(d*x+c)+csc(d*x+c)),I)+(5*cos(d*x+c)^5+2*cos (d*x+c)^3+14*cos(d*x+c)-21)*cot(d*x+c))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11 \[ \int \cos ^7(c+d x) (b \sec (c+d x))^{5/2} \, dx=\frac {21 i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} b^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (5 \, b^{2} \cos \left (d x + c\right )^{4} + 7 \, b^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45 \, d} \]
1/45*(21*I*sqrt(2)*b^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I*sqrt(2)*b^(5/2)*weierstrassZeta( -4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(5*b ^2*cos(d*x + c)^4 + 7*b^2*cos(d*x + c)^2)*sqrt(b/cos(d*x + c))*sin(d*x + c ))/d
Timed out. \[ \int \cos ^7(c+d x) (b \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cos ^7(c+d x) (b \sec (c+d x))^{5/2} \, dx=\int { \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{7} \,d x } \]
\[ \int \cos ^7(c+d x) (b \sec (c+d x))^{5/2} \, dx=\int { \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{7} \,d x } \]
Timed out. \[ \int \cos ^7(c+d x) (b \sec (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^7\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]