\(\int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx\) [383]

Optimal result

Integrand size = 33, antiderivative size = 240 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=-\frac {2 (c \cos (d+e x)-a \sin (d+e x)) (b+a \cos (d+e x)+c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac {2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+a \cos (d+e x)+c \sin (d+e x))^2}{\left (a^2-b^2+c^2\right ) e \cos ^{\frac {3}{2}}(d+e x) \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \] Output:

-2*(c*cos(e*x+d)-a*sin(e*x+d))*(b+a*cos(e*x+d)+c*sin(e*x+d))/(a^2-b^2+c^2) 
/e/cos(e*x+d)^(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(3/2)-2*EllipticE(sin(1/ 
2*d+1/2*e*x-1/2*arctan(c,a)),2^(1/2)*((a^2+c^2)^(1/2)/(b+(a^2+c^2)^(1/2))) 
^(1/2))*(b+a*cos(e*x+d)+c*sin(e*x+d))^2/(a^2-b^2+c^2)/e/cos(e*x+d)^(3/2)/( 
(b+a*cos(e*x+d)+c*sin(e*x+d))/(b+(a^2+c^2)^(1/2)))^(1/2)/(a+b*sec(e*x+d)+c 
*tan(e*x+d))^(3/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 32.33 (sec) , antiderivative size = 54829, normalized size of antiderivative = 228.45 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[1/(Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2 
)),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3042, 3642, 3042, 3607, 3042, 3598, 3042, 3132}

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.

\(\displaystyle \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cos (d+e x)^{3/2} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}dx\)

\(\Big \downarrow \) 3642

\(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{3/2} \int \frac {1}{(b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}dx}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{3/2} \int \frac {1}{(b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}dx}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3607

\(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{3/2} \left (-\frac {\int \sqrt {b+a \cos (d+e x)+c \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}\right )}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{3/2} \left (-\frac {\int \sqrt {b+a \cos (d+e x)+c \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}\right )}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{3/2} \left (-\frac {\sqrt {a \cos (d+e x)+b+c \sin (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(a,c)\right )}{b+\sqrt {a^2+c^2}}}dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}\right )}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{3/2} \left (-\frac {\sqrt {a \cos (d+e x)+b+c \sin (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(a,c)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}\right )}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{3/2} \left (-\frac {2 \sqrt {a \cos (d+e x)+b+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}\right )}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}\)

Input:

Int[1/(Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)),x]
 

Output:

((b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)*((-2*(c*Cos[d + e*x] - a*Sin[ 
d + e*x]))/((a^2 - b^2 + c^2)*e*Sqrt[b + a*Cos[d + e*x] + c*Sin[d + e*x]]) 
 - (2*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[ 
a^2 + c^2])]*Sqrt[b + a*Cos[d + e*x] + c*Sin[d + e*x]])/((a^2 - b^2 + c^2) 
*e*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])))/(C 
os[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))
 

Defintions of rubi rules used

rule 3042

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

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3598

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ 
)]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + 
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]   Int[Sqrt[a/(a + S 
qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc 
Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] 
 && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]
 

rule 3607

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(-3/2), x_Symbol] :> Simp[2*((c*Cos[d + e*x] - b*Sin[d + e*x])/(e*(a^2 - b^ 
2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] + Simp[1/(a^2 - b^ 
2 - c^2)   Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{ 
a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
 

rule 3642

Int[cos[(d_.) + (e_.)*(x_)]^(n_)*((a_.) + (b_.)*sec[(d_.) + (e_.)*(x_)] + ( 
c_.)*tan[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[d + e*x]^n*((a + 
b*Sec[d + e*x] + c*Tan[d + e*x])^n/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n) 
   Int[(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d 
, e}, x] &&  !IntegerQ[n]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.25 (sec) , antiderivative size = 40867, normalized size of antiderivative = 170.28

method	result	size
default	\(\text {Expression too large to display}\)	\(40867\)

Input:

int(1/cos(e*x+d)^(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(3/2),x,method=_RETUR 
NVERBOSE)
 

Output:

result too large to display
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 1710, normalized size of antiderivative = 7.12 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/cos(e*x+d)^(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(3/2),x, algori 
thm="fricas")
 

Output:

-2/3*((-I*a*b^2 + b^2*c + (-I*a^2*b + a*b*c)*cos(e*x + d) + (-I*a*b*c + b* 
c^2)*sin(e*x + d))*sqrt(1/2*a - 1/2*I*c)*weierstrassPInverse(-4/3*(3*a^4 - 
 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 
 + 2*a^2*c^2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 - 27*a*b*c^4 - 9*I*b*c^5 + 
2*I*(9*a^2*b + 4*b^3)*c^3 - 6*(3*a^3*b - 4*a*b^3)*c^2 + 3*I*(9*a^4*b - 8*a 
^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*a*b + 2*I*b*c + 3*( 
a^2 + c^2)*cos(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2)) + 
(I*a*b^2 + b^2*c + (I*a^2*b + a*b*c)*cos(e*x + d) + (I*a*b*c + b*c^2)*sin( 
e*x + d))*sqrt(1/2*a + 1/2*I*c)*weierstrassPInverse(-4/3*(3*a^4 - 4*a^2*b^ 
2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*c^4 - 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2* 
c^2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 - 27*a*b*c^4 + 9*I*b*c^5 - 2*I*(9*a^ 
2*b + 4*b^3)*c^3 - 6*(3*a^3*b - 4*a*b^3)*c^2 - 3*I*(9*a^4*b - 8*a^2*b^3)*c 
)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*a*b - 2*I*b*c + 3*(a^2 + c^2 
)*cos(e*x + d) - 3*(I*a^2 + I*c^2)*sin(e*x + d))/(a^2 + c^2)) + 3*(I*a^2*b 
 + I*b*c^2 + (I*a^3 + I*a*c^2)*cos(e*x + d) + (I*a^2*c + I*c^3)*sin(e*x + 
d))*sqrt(1/2*a - 1/2*I*c)*weierstrassZeta(-4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2* 
c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4) 
, 8/27*(9*a^5*b - 8*a^3*b^3 - 27*a*b*c^4 - 9*I*b*c^5 + 2*I*(9*a^2*b + 4*b^ 
3)*c^3 - 6*(3*a^3*b - 4*a*b^3)*c^2 + 3*I*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3 
*a^4*c^2 + 3*a^2*c^4 + c^6), weierstrassPInverse(-4/3*(3*a^4 - 4*a^2*b^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/cos(e*x+d)**(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \cos \left (e x + d\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/cos(e*x+d)^(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(3/2),x, algori 
thm="maxima")
 

Output:

integrate(1/((b*sec(e*x + d) + c*tan(e*x + d) + a)^(3/2)*cos(e*x + d)^(3/2 
)), x)
 

Giac [F]

\[ \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \cos \left (e x + d\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/cos(e*x+d)^(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(3/2),x, algori 
thm="giac")
 

Output:

integrate(1/((b*sec(e*x + d) + c*tan(e*x + d) + a)^(3/2)*cos(e*x + d)^(3/2 
)), x)
 

Mupad [F(-1)]

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

int(1/(cos(d + e*x)^(3/2)*(a + c*tan(d + e*x) + b/cos(d + e*x))^(3/2)),x)
 

Output:

int(1/(cos(d + e*x)^(3/2)*(a + c*tan(d + e*x) + b/cos(d + e*x))^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=\int \frac {\sqrt {\sec \left (e x +d \right ) b +\tan \left (e x +d \right ) c +a}\, \sqrt {\cos \left (e x +d \right )}}{\cos \left (e x +d \right )^{2} \sec \left (e x +d \right )^{2} b^{2}+2 \cos \left (e x +d \right )^{2} \sec \left (e x +d \right ) \tan \left (e x +d \right ) b c +2 \cos \left (e x +d \right )^{2} \sec \left (e x +d \right ) a b +\cos \left (e x +d \right )^{2} \tan \left (e x +d \right )^{2} c^{2}+2 \cos \left (e x +d \right )^{2} \tan \left (e x +d \right ) a c +\cos \left (e x +d \right )^{2} a^{2}}d x \] Input:

int(1/cos(e*x+d)^(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(3/2),x)
 

Output:

int((sqrt(sec(d + e*x)*b + tan(d + e*x)*c + a)*sqrt(cos(d + e*x)))/(cos(d 
+ e*x)**2*sec(d + e*x)**2*b**2 + 2*cos(d + e*x)**2*sec(d + e*x)*tan(d + e* 
x)*b*c + 2*cos(d + e*x)**2*sec(d + e*x)*a*b + cos(d + e*x)**2*tan(d + e*x) 
**2*c**2 + 2*cos(d + e*x)**2*tan(d + e*x)*a*c + cos(d + e*x)**2*a**2),x)