\(\int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx\) [120]

Optimal result

Integrand size = 21, antiderivative size = 97 \[ \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx=\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 b^2 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}} \] Output:

14/15*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/b/d/cos(d*x+c)^(1/2)/(b*sec(d* 
x+c))^(1/2)+2/9*b^2*sin(d*x+c)/d/(b*sec(d*x+c))^(7/2)+14/45*sin(d*x+c)/d/( 
b*sec(d*x+c))^(3/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx=\frac {84 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) (33 \sin (c+d x)+5 \sin (3 (c+d x)))}{90 d \cos ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^{3/2}} \] Input:

Integrate[Cos[c + d*x]^3/(b*Sec[c + d*x])^(3/2),x]
 

Output:

(84*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^(3/2)*(33*Sin[c + d*x] + 5*Si 
n[3*(c + d*x)]))/(90*d*Cos[c + d*x]^(3/2)*(b*Sec[c + d*x])^(3/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.15, 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.

\(\displaystyle \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 2030

\(\displaystyle b^3 \int \frac {1}{\left (b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{9/2}}dx\)

\(\Big \downarrow \) 4256

\(\displaystyle b^3 \left (\frac {7 \int \frac {1}{(b \sec (c+d x))^{5/2}}dx}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle b^3 \left (\frac {7 \int \frac {1}{\left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\)

\(\Big \downarrow \) 4256

\(\displaystyle b^3 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {b \sec (c+d x)}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle b^3 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\)

\(\Big \downarrow \) 4258

\(\displaystyle b^3 \left (\frac {7 \left (\frac {3 \int \sqrt {\cos (c+d x)}dx}{5 b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle b^3 \left (\frac {7 \left (\frac {3 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle b^3 \left (\frac {7 \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 \sin (c+d x)}{9 b d (b \sec (c+d x))^{7/2}}\right )\)

Input:

Int[Cos[c + d*x]^3/(b*Sec[c + d*x])^(3/2),x]
 

Output:

b^3*((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))
 

Defintions of rubi rules used

rule 2030

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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]
 

rule 4256

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]
 

rule 4258

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 5.57 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.22

method	result	size
default	\(-\frac {2 \left (\sin \left (d x +c \right ) \left (-5 \cos \left (d x +c \right )^{4}-5 \cos \left (d x +c \right )^{3}-7 \cos \left (d x +c \right )^{2}-7 \cos \left (d x +c \right )-21\right )+21 i \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \left (2+\cos \left (d x +c \right )+\sec \left (d x +c \right )\right ) \operatorname {EllipticE}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right )-21 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \left (2+\cos \left (d x +c \right )+\sec \left (d x +c \right )\right )\right )}{45 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {b \sec \left (d x +c \right )}\, b}\)	\(215\)

Input:

int(cos(d*x+c)^3/(b*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2/45/d/(cos(d*x+c)+1)/(b*sec(d*x+c))^(1/2)/b*(sin(d*x+c)*(-5*cos(d*x+c)^4 
-5*cos(d*x+c)^3-7*cos(d*x+c)^2-7*cos(d*x+c)-21)+21*I*(1/(cos(d*x+c)+1))^(1 
/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(2+cos(d*x+c)+sec(d*x+c))*EllipticE( 
I*(cot(d*x+c)-csc(d*x+c)),I)-21*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(co 
s(d*x+c)+1))^(1/2)*(2+cos(d*x+c)+sec(d*x+c))*EllipticF(I*(cot(d*x+c)-csc(d 
*x+c)),I))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{4} + 7 \, \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) + 21 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{45 \, b^{2} d} \] Input:

integrate(cos(d*x+c)^3/(b*sec(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

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^2* 
d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**3/(b*sec(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{3}}{\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)^3/(b*sec(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate(cos(d*x + c)^3/(b*sec(d*x + c))^(3/2), x)
 

Giac [F]

\[ \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{3}}{\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)^3/(b*sec(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate(cos(d*x + c)^3/(b*sec(d*x + c))^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(cos(c + d*x)^3/(b/cos(c + d*x))^(3/2),x)
 

Output:

int(cos(c + d*x)^3/(b/cos(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {\cos ^3(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx=\frac {\sqrt {b}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{2}}d x \right )}{b^{2}} \] Input:

int(cos(d*x+c)^3/(b*sec(d*x+c))^(3/2),x)
 

Output:

(sqrt(b)*int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c + d*x)**2,x))/b**2