3.2.25 \(\int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx\) [125]

3.2.25.1 Optimal result
3.2.25.2 Mathematica [A] (verified)
3.2.25.3 Rubi [A] (verified)
3.2.25.4 Maple [B] (verified)
3.2.25.5 Fricas [C] (verification not implemented)
3.2.25.6 Sympy [F]
3.2.25.7 Maxima [F]
3.2.25.8 Giac [F]
3.2.25.9 Mupad [F(-1)]

3.2.25.1 Optimal result

Integrand size = 21, antiderivative size = 97 \[ \int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b d \sqrt {b \cos (c+d x)}}+\frac {2 b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {10 \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}} \]

output
2/7*b^2*sin(d*x+c)/d/(b*cos(d*x+c))^(7/2)+10/21*sin(d*x+c)/d/(b*cos(d*x+c) 
)^(3/2)+10/21*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(si 
n(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/b/d/(b*cos(d*x+c))^(1/2)
 
3.2.25.2 Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68 \[ \int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (5+3 \sec ^2(c+d x)\right ) \tan (c+d x)}{21 b d \sqrt {b \cos (c+d x)}} \]

input
Integrate[Sec[c + d*x]^3/(b*Cos[c + d*x])^(3/2),x]
 
output
(10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(5 + 3*Sec[c + d*x]^2 
)*Tan[c + d*x])/(21*b*d*Sqrt[b*Cos[c + d*x]])
 
3.2.25.3 Rubi [A] (verified)

Time = 0.46 (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, 3116, 3042, 3116, 3042, 3121, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 2030

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

\(\displaystyle b^3 \left (\frac {5 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b^2 d \sqrt {b \cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}\right )}{7 b^2}+\frac {2 \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}\right )\)

input
Int[Sec[c + d*x]^3/(b*Cos[c + d*x])^(3/2),x]
 
output
b^3*((2*Sin[c + d*x])/(7*b*d*(b*Cos[c + d*x])^(7/2)) + (5*((2*Sqrt[Cos[c + 
 d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*Sin[ 
c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2))))/(7*b^2))
 

3.2.25.3.1 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 3116
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1))   I 
nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && 
 IntegerQ[2*n]
 

rule 3120
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]
 

rule 3121
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) 
^n/Sin[c + d*x]^n   Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt 
Q[-1, n, 1] && IntegerQ[2*n]
 
3.2.25.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(397\) vs. \(2(109)=218\).

Time = 2.34 (sec) , antiderivative size = 398, normalized size of antiderivative = 4.10

method result size
default \(-\frac {2 \left (-40 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+60 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-30 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{21 b \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) \(398\)

input
int(sec(d*x+c)^3/(cos(d*x+c)*b)^(3/2),x,method=_RETURNVERBOSE)
 
output
-2/21*(-40*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*E 
llipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^6-40*cos(1/2*d*x+1 
/2*c)*sin(1/2*d*x+1/2*c)^6+60*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+ 
1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c) 
^4+40*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-30*(sin(1/2*d*x+1/2*c)^2)^(1 
/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)) 
*sin(1/2*d*x+1/2*c)^2-16*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+5*(sin(1/ 
2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d 
*x+1/2*c),2^(1/2)))/b*((2*cos(1/2*d*x+1/2*c)^2-1)*b*sin(1/2*d*x+1/2*c)^2)^ 
(1/2)/(-b*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/(2*cos(1/2* 
d*x+1/2*c)^2-1)^3/sin(1/2*d*x+1/2*c)/((2*cos(1/2*d*x+1/2*c)^2-1)*b)^(1/2)/ 
d
 
3.2.25.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.19 \[ \int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {-5 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )} {\left (5 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right )}{21 \, b^{2} d \cos \left (d x + c\right )^{4}} \]

input
integrate(sec(d*x+c)^3/(b*cos(d*x+c))^(3/2),x, algorithm="fricas")
 
output
1/21*(-5*I*sqrt(2)*sqrt(b)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d 
*x + c) + I*sin(d*x + c)) + 5*I*sqrt(2)*sqrt(b)*cos(d*x + c)^4*weierstrass 
PInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*sqrt(b*cos(d*x + c))*(5 
*cos(d*x + c)^2 + 3)*sin(d*x + c))/(b^2*d*cos(d*x + c)^4)
 
3.2.25.6 Sympy [F]

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

input
integrate(sec(d*x+c)**3/(b*cos(d*x+c))**(3/2),x)
 
output
Integral(sec(c + d*x)**3/(b*cos(c + d*x))**(3/2), x)
 
3.2.25.7 Maxima [F]

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

input
integrate(sec(d*x+c)^3/(b*cos(d*x+c))^(3/2),x, algorithm="maxima")
 
output
integrate(sec(d*x + c)^3/(b*cos(d*x + c))^(3/2), x)
 
3.2.25.8 Giac [F]

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

input
integrate(sec(d*x+c)^3/(b*cos(d*x+c))^(3/2),x, algorithm="giac")
 
output
integrate(sec(d*x + c)^3/(b*cos(d*x + c))^(3/2), x)
 
3.2.25.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]

input
int(1/(cos(c + d*x)^3*(b*cos(c + d*x))^(3/2)),x)
 
output
int(1/(cos(c + d*x)^3*(b*cos(c + d*x))^(3/2)), x)