Integrand size = 21, antiderivative size = 95 \[ \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}} \]
2/9*b^3*sin(d*x+c)/d/(b*sec(d*x+c))^(7/2)+14/45*b*sin(d*x+c)/d/(b*sec(d*x+ c))^(3/2)+14/15*(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/cos(d*x+c)^(1/2)/(b*sec(d*x+c))^(1/2)
Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\frac {\frac {336 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)}}+4 \cos (c+d x) (33 \sin (c+d x)+5 \sin (3 (c+d x)))}{360 d \sqrt {b \sec (c+d x)}} \]
((336*EllipticE[(c + d*x)/2, 2])/Sqrt[Cos[c + d*x]] + 4*Cos[c + d*x]*(33*S in[c + d*x] + 5*Sin[3*(c + d*x)]))/(360*d*Sqrt[b*Sec[c + d*x]])
Time = 0.47 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.18, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^4 \sqrt {b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle b^4 \int \frac {1}{\left (b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle b^4 \left (\frac {7 \int \frac {1}{(b \sec (c+d x))^{5/2}}dx}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^4 \left (\frac {7 \int \frac {1}{\left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle b^4 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {b \sec (c+d x)}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^4 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle b^4 \left (\frac {7 \left (\frac {3 \int \sqrt {\cos (c+d x)}dx}{5 b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^4 \left (\frac {7 \left (\frac {3 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle b^4 \left (\frac {7 \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\) |
b^4*((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))
3.2.10.3.1 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 = 0.58 (sec) , antiderivative size = 451, normalized size of antiderivative = 4.75
| method | result | size |
| default | \(-\frac {2 \left (21 i \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\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}}\, \cos \left (d x +c \right )-21 i \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\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}}\, \cos \left (d x +c \right )-5 \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+42 i \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 )-42 i \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 )-5 \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+21 i \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 ) \sec \left (d x +c \right )-21 i \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 ) \sec \left (d x +c \right )-7 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-7 \cos \left (d x +c \right ) \sin \left (d x +c \right )-21 \sin \left (d x +c \right )\right )}{45 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {b \sec \left (d x +c \right )}}\) | \(451\) |
-2/45/d/(cos(d*x+c)+1)/(b*sec(d*x+c))^(1/2)*(21*I*(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) *cos(d*x+c)-21*I*(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)*cos(d*x+c)-5*cos(d*x+c)^4*sin(d* x+c)+42*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Ellip ticF(I*(-cot(d*x+c)+csc(d*x+c)),I)-42*I*(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)-5*cos(d*x +c)^3*sin(d*x+c)+21*I*(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)*sec(d*x+c)-21*I*(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)+c sc(d*x+c)),I)*sec(d*x+c)-7*cos(d*x+c)^2*sin(d*x+c)-7*cos(d*x+c)*sin(d*x+c) -21*sin(d*x+c))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{4} + 7 \, \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) + 21 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{45 \, b d} \]
1/45*(2*(5*cos(d*x + c)^4 + 7*cos(d*x + c)^2)*sqrt(b/cos(d*x + c))*sin(d*x + c) + 21*I*sqrt(2)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4 , 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I*sqrt(2)*sqrt(b)*weierstrassZet a(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/(b*d)
\[ \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )}}{\sqrt {b \sec {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4}}{\sqrt {b \sec \left (d x + c\right )}} \,d x } \]
\[ \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4}}{\sqrt {b \sec \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}} \,d x \]