\(\int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx\) [454]

Optimal result

Integrand size = 33, antiderivative size = 118 \[ \int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx=\frac {2 \sqrt {\cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sqrt {a+b \sec (d+e x)+c \tan (d+e x)}}{e \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3242, 3198, 2732} \[ \int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx=\frac {2 \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}} \]

[In]

Int[Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]],x]

[Out]

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

Rule 2732

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 3198

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[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
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[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 3242

Int[cos[(d_.) + (e_.)*(x_)]^(n_)*((a_.) + (b_.)*sec[(d_.) + (e_.)*(x_)] + (c_.)*tan[(d_.) + (e_.)*(x_)])^(n_),
x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)}\right ) \int \sqrt {b+a \cos (d+e x)+c \sin (d+e x)} \, dx}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}} \\ & = \frac {\left (\sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)}\right ) \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}{\sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}}} \\ & = \frac {2 \sqrt {\cos (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 ) \sqrt {a+b \sec (d+e x)+c \tan (d+e x)}}{e \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 32.80 (sec) , antiderivative size = 54676, normalized size of antiderivative = 463.36 \[ \int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx=\text {Result too large to show} \]

[In]

Integrate[Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]],x]

[Out]

Result too large to show

Maple [C] (verified)

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

Time = 24.27 (sec) , antiderivative size = 2660, normalized size of antiderivative = 22.54

method	result	size
risch	\(\text {Expression too large to display}\)	\(2660\)
default	\(\text {Expression too large to display}\)	\(12921\)

[In]

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

[Out]

I*(-I*exp(I*(e*x+d))^2*c+a*exp(I*(e*x+d))^2+2*b*exp(I*(e*x+d))+I*c+a)/e/(exp(I*(e*x+d))*(-I*exp(I*(e*x+d))^2*c
+a*exp(I*(e*x+d))^2+2*b*exp(I*(e*x+d))+I*c+a))^(1/2)*((exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2+1)/exp(RootOf(_Z^
2+1,index=1)*(e*x+d)))^(1/2)*(-(RootOf(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c-exp(RootOf(_Z^2
+1,index=1)*(e*x+d))^2*a-2*b*exp(RootOf(_Z^2+1,index=1)*(e*x+d))-RootOf(_Z^2+1,index=1)*c-a)/(exp(RootOf(_Z^2+
1,index=1)*(e*x+d))^2+1))^(1/2)*(-(RootOf(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c-exp(RootOf(_
Z^2+1,index=1)*(e*x+d))^2*a-2*b*exp(RootOf(_Z^2+1,index=1)*(e*x+d))-RootOf(_Z^2+1,index=1)*c-a)*(exp(RootOf(_Z
^2+1,index=1)*(e*x+d))^2+1))^(1/2)/(RootOf(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c-exp(RootOf(
_Z^2+1,index=1)*(e*x+d))^2*a-2*b*exp(RootOf(_Z^2+1,index=1)*(e*x+d))-RootOf(_Z^2+1,index=1)*c-a)*(exp(RootOf(_
Z^2+1,index=1)*(e*x+d))*(-RootOf(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c+exp(RootOf(_Z^2+1,ind
ex=1)*(e*x+d))^2*a+2*b*exp(RootOf(_Z^2+1,index=1)*(e*x+d))+RootOf(_Z^2+1,index=1)*c+a))^(1/2)/((-RootOf(_Z^2+1
,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c+exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*a+2*b*exp(RootOf(_Z^2+
1,index=1)*(e*x+d))+RootOf(_Z^2+1,index=1)*c+a)*(exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2+1))^(1/2)*2^(1/2)-I/e*(
-2*b*(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)*((exp(I*(e*x+d))+(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a))/(-b+(-a^2+b^2-c^2)^
(1/2))*(I*c-a))^(1/2)*((exp(I*(e*x+d))-(b+(-a^2+b^2-c^2)^(1/2))/(I*c-a))/(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)-(
b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)))^(1/2)*(-exp(I*(e*x+d))/(-b+(-a^2+b^2-c^2)^(1/2))*(I*c-a))^(1/2)/(-I*c*exp(I*
(e*x+d))^3+exp(I*(e*x+d))^3*a+I*c*exp(I*(e*x+d))+2*b*exp(I*(e*x+d))^2+a*exp(I*(e*x+d)))^(1/2)*EllipticF(((exp(
I*(e*x+d))+(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a))/(-b+(-a^2+b^2-c^2)^(1/2))*(I*c-a))^(1/2),(-(-b+(-a^2+b^2-c^2)^(1
/2))/(I*c-a)/(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)-(b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)))^(1/2))-(I*c+a)*(-2*(-I*exp
(I*(e*x+d))^2*c+a*exp(I*(e*x+d))^2+2*b*exp(I*(e*x+d))+I*c+a)/(I*c+a)/(exp(I*(e*x+d))*(-I*exp(I*(e*x+d))^2*c+a*
exp(I*(e*x+d))^2+2*b*exp(I*(e*x+d))+I*c+a))^(1/2)+2*((I*c-a)/(I*c+a)+(-2*I*c+2*a)/(I*c+a))*(-b+(-a^2+b^2-c^2)^
(1/2))/(I*c-a)*((exp(I*(e*x+d))+(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a))/(-b+(-a^2+b^2-c^2)^(1/2))*(I*c-a))^(1/2)*((
exp(I*(e*x+d))-(b+(-a^2+b^2-c^2)^(1/2))/(I*c-a))/(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)-(b+(-a^2+b^2-c^2)^(1/2))/
(I*c-a)))^(1/2)*(-exp(I*(e*x+d))/(-b+(-a^2+b^2-c^2)^(1/2))*(I*c-a))^(1/2)/(-I*c*exp(I*(e*x+d))^3+exp(I*(e*x+d)
)^3*a+I*c*exp(I*(e*x+d))+2*b*exp(I*(e*x+d))^2+a*exp(I*(e*x+d)))^(1/2)*((-(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)-(b+
(-a^2+b^2-c^2)^(1/2))/(I*c-a))*EllipticE(((exp(I*(e*x+d))+(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a))/(-b+(-a^2+b^2-c^2
)^(1/2))*(I*c-a))^(1/2),(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)/(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)-(b+(-a^2+b^2-c
^2)^(1/2))/(I*c-a)))^(1/2))+(b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)*EllipticF(((exp(I*(e*x+d))+(-b+(-a^2+b^2-c^2)^(1/
2))/(I*c-a))/(-b+(-a^2+b^2-c^2)^(1/2))*(I*c-a))^(1/2),(-(-b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)/(-(-b+(-a^2+b^2-c^2)
^(1/2))/(I*c-a)-(b+(-a^2+b^2-c^2)^(1/2))/(I*c-a)))^(1/2)))))*((exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2+1)/exp(Ro
otOf(_Z^2+1,index=1)*(e*x+d)))^(1/2)*(-(RootOf(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c-exp(Roo
tOf(_Z^2+1,index=1)*(e*x+d))^2*a-2*b*exp(RootOf(_Z^2+1,index=1)*(e*x+d))-RootOf(_Z^2+1,index=1)*c-a)/(exp(Root
Of(_Z^2+1,index=1)*(e*x+d))^2+1))^(1/2)*(-(RootOf(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c-exp(
RootOf(_Z^2+1,index=1)*(e*x+d))^2*a-2*b*exp(RootOf(_Z^2+1,index=1)*(e*x+d))-RootOf(_Z^2+1,index=1)*c-a)*(exp(R
ootOf(_Z^2+1,index=1)*(e*x+d))^2+1))^(1/2)/(RootOf(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c-exp
(RootOf(_Z^2+1,index=1)*(e*x+d))^2*a-2*b*exp(RootOf(_Z^2+1,index=1)*(e*x+d))-RootOf(_Z^2+1,index=1)*c-a)*(exp(
RootOf(_Z^2+1,index=1)*(e*x+d))*(-RootOf(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c+exp(RootOf(_Z
^2+1,index=1)*(e*x+d))^2*a+2*b*exp(RootOf(_Z^2+1,index=1)*(e*x+d))+RootOf(_Z^2+1,index=1)*c+a))^(1/2)/((-RootO
f(_Z^2+1,index=1)*exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*c+exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2*a+2*b*exp(Root
Of(_Z^2+1,index=1)*(e*x+d))+RootOf(_Z^2+1,index=1)*c+a)*(exp(RootOf(_Z^2+1,index=1)*(e*x+d))^2+1))^(1/2)*2^(1/
2)

Fricas [C] (verification not implemented)

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

Time = 0.14 (sec) , antiderivative size = 1371, normalized size of antiderivative = 11.62 \[ \int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*(sqrt(2)*(-I*a*b + b*c)*sqrt(a - 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
)) + sqrt(2)*(I*a*b + b*c)*sqrt(a + 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*sqrt(2)*(-I*a^2 - I*c^2)*sqrt(a - 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^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*sqrt(2)*(I*a^2 + I*c^
2)*sqrt(a + 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), weierstrassPIn
verse(-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))))/((a^2 + c^2)*e)

Sympy [F]

\[ \int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx=\int \sqrt {a + b \sec {\left (d + e x \right )} + c \tan {\left (d + e x \right )}} \sqrt {\cos {\left (d + e x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(a + b*sec(d + e*x) + c*tan(d + e*x))*sqrt(cos(d + e*x)), x)

Maxima [F]

\[ \int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx=\int { \sqrt {b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a} \sqrt {\cos \left (e x + d\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*sec(e*x + d) + c*tan(e*x + d) + a)*sqrt(cos(e*x + d)), x)

Giac [F]

\[ \int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx=\int { \sqrt {b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a} \sqrt {\cos \left (e x + d\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*sec(e*x + d) + c*tan(e*x + d) + a)*sqrt(cos(e*x + d)), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\cos (d+e x)} \sqrt {a+b \sec (d+e x)+c \tan (d+e x)} \, dx=\int \sqrt {\cos \left (d+e\,x\right )}\,\sqrt {a+c\,\mathrm {tan}\left (d+e\,x\right )+\frac {b}{\cos \left (d+e\,x\right )}} \,d x \]

[In]

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

[Out]

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