\(\int \sqrt {a+b \sec ^2(e+f x)} \, dx\) [385]

Optimal result

Integrand size = 16, antiderivative size = 79 \[ \int \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f} \]

[Out]

arctan(a^(1/2)*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)^(1/2))*a^(1/2)/f+arctanh(b^(1/2)*tan(f*x+e)/(a+b+b*tan(f*x+e)^2
)^(1/2))*b^(1/2)/f

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4213, 399, 223, 212, 385, 209} \[ \int \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{f}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{f} \]

[In]

Int[Sqrt[a + b*Sec[e + f*x]^2],x]

[Out]

(Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e +
f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 4213

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b+b x^2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f}+\frac {b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f} \\ & = \frac {\sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.59 \[ \int \sqrt {a+b \sec ^2(e+f x)} \, dx=-\frac {i \left (1+e^{2 i (e+f x)}\right ) \left (2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \left (-1+e^{2 i (e+f x)}\right )}{\sqrt {4 b e^{2 i (e+f x)}+a \left (1+e^{2 i (e+f x)}\right )^2}}\right )+\sqrt {a} \text {arctanh}\left (\frac {a+2 b+a e^{2 i (e+f x)}}{\sqrt {a} \sqrt {4 b e^{2 i (e+f x)}+a \left (1+e^{2 i (e+f x)}\right )^2}}\right )-\sqrt {a} \text {arctanh}\left (\frac {a+a e^{2 i (e+f x)}+2 b e^{2 i (e+f x)}}{\sqrt {a} \sqrt {4 b e^{2 i (e+f x)}+a \left (1+e^{2 i (e+f x)}\right )^2}}\right )\right ) \sqrt {a+b \sec ^2(e+f x)}}{2 \sqrt {4 b e^{2 i (e+f x)}+a \left (1+e^{2 i (e+f x)}\right )^2} f} \]

[In]

Integrate[Sqrt[a + b*Sec[e + f*x]^2],x]

[Out]

((-1/2*I)*(1 + E^((2*I)*(e + f*x)))*(2*Sqrt[b]*ArcTan[(Sqrt[b]*(-1 + E^((2*I)*(e + f*x))))/Sqrt[4*b*E^((2*I)*(
e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]] + Sqrt[a]*ArcTanh[(a + 2*b + a*E^((2*I)*(e + f*x)))/(Sqrt[a]*Sqrt[
4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])] - Sqrt[a]*ArcTanh[(a + a*E^((2*I)*(e + f*x)) + 2*b*
E^((2*I)*(e + f*x)))/(Sqrt[a]*Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2])])*Sqrt[a + b*Sec[
e + f*x]^2])/(Sqrt[4*b*E^((2*I)*(e + f*x)) + a*(1 + E^((2*I)*(e + f*x)))^2]*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(67)=134\).

Time = 4.09 (sec) , antiderivative size = 351, normalized size of antiderivative = 4.44

method	result	size
default	\(\frac {\sqrt {a +b \sec \left (f x +e \right )^{2}}\, \left (\sqrt {b}\, \ln \left (-\frac {4 \left (\sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \sqrt {b}\, \cos \left (f x +e \right )+\sqrt {b}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-\sin \left (f x +e \right ) a +a +b \right )}{\sin \left (f x +e \right )-1}\right ) \sqrt {-a}+\sqrt {b}\, \ln \left (\frac {4 \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \sqrt {b}\, \cos \left (f x +e \right )+4 \sqrt {b}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-4 \sin \left (f x +e \right ) a -4 a -4 b}{\sin \left (f x +e \right )+1}\right ) \sqrt {-a}+2 \ln \left (4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \cos \left (f x +e \right )+4 \sqrt {-a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}-4 \sin \left (f x +e \right ) a \right ) a \right ) \cos \left (f x +e \right )}{2 f \sqrt {-a}\, \left (1+\cos \left (f x +e \right )\right ) \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}}\)	\(351\)

[In]

int((a+b*sec(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/f/(-a)^(1/2)*(a+b*sec(f*x+e)^2)^(1/2)*(b^(1/2)*ln(-4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*
cos(f*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-sin(f*x+e)*a+a+b)/(sin(f*x+e)-1))*(-a)^(1/2)+b^
(1/2)*ln(4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(
f*x+e))^2)^(1/2)-sin(f*x+e)*a-a-b)/(sin(f*x+e)+1))*(-a)^(1/2)+2*ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x
+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a)*cos(f*x+e
)/(1+cos(f*x+e))/((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (67) = 134\).

Time = 0.49 (sec) , antiderivative size = 1227, normalized size of antiderivative = 15.53 \[ \int \sqrt {a+b \sec ^2(e+f x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sec(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

[1/8*(sqrt(-a)*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^
2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(
f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f
*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)
*sin(f*x + e)) + 2*sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)
*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/
cos(f*x + e)^4))/f, 1/8*(4*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*c
os(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))) + sqrt(-a)*log(128*a^4*cos(f*x +
e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b +
70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 - 8*(16*a^3*cos(f*x + e)^
7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2
- b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)))/f, -1/4*(sqrt(a)*arc
tan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt(
(a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2
)*sin(f*x + e))) - sqrt(b)*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)
*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/
cos(f*x + e)^4))/f, -1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a
*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a
*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e))) - 2*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*c
os(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e))))/
f]

Sympy [F]

\[ \int \sqrt {a+b \sec ^2(e+f x)} \, dx=\int \sqrt {a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]

[In]

integrate((a+b*sec(f*x+e)**2)**(1/2),x)

[Out]

Integral(sqrt(a + b*sec(e + f*x)**2), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.81 (sec) , antiderivative size = 3227, normalized size of antiderivative = 40.85 \[ \int \sqrt {a+b \sec ^2(e+f x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sec(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*(2*sqrt(a)*b^(3/2)*arctan2(a*sin(2*f*x + 2*e) + (a^2*cos(4*f*x + 4*e)^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2
+ 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*
b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*
cos(2*f*x + 2*e))^(1/4)*sqrt(a)*sin(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x
+ 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a)), a*cos(2*f*x + 2*e) + (a^2*cos(4*f*x + 4*e)^2 + a^2*sin(4*f*x + 4
*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(a^
2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + 4*
(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*sqrt(a)*cos(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e
), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a)) + a + 2*b) + a^(3/2)*sqrt(b)*arctan2(2*(a^2*cos(4*f
*x + 4*e)^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x
+ 4*e)*sin(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*
x + 2*e))*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*sqrt(a)*sin(1/2*arctan2(a*sin(4*f*x + 4*e
) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a)), 2*(a^2*cos(4*f*x +
4*e)^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e
)*sin(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2
*e))*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*sqrt(a)*cos(1/2*arctan2(a*sin(4*f*x + 4*e) + 2
*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos(2*f*x + 2*e) + a)) + 4*a + 4*b) + a*b*log(((
a + b)*sqrt((16*b^2*cos(2*f*x + 2*e)^4 + 16*b^2*sin(2*f*x + 2*e)^4 + (a^2 + 2*a*b + b^2)*abs(2*e^(2*I*f*x + 2*
I*e) + 2)^4 - 64*b^2*cos(2*f*x + 2*e)^3 + 96*b^2*cos(2*f*x + 2*e)^2 - 8*((a*b + b^2)*cos(2*f*x + 2*e)^2 - (a*b
+ b^2)*sin(2*f*x + 2*e)^2 + a*b + b^2 - 2*(a*b + b^2)*cos(2*f*x + 2*e))*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2 - 64
*b^2*cos(2*f*x + 2*e) + 32*(b^2*cos(2*f*x + 2*e)^2 - 2*b^2*cos(2*f*x + 2*e) + b^2)*sin(2*f*x + 2*e)^2 + 16*b^2
)/(a^2 + 2*a*b + b^2))*cos(1/2*arctan2(8*(b*cos(2*f*x + 2*e) - b)*sin(2*f*x + 2*e)/((a + b)*abs(2*e^(2*I*f*x +
2*I*e) + 2)^2), ((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2 - 4*b*cos(2*f*x + 2*e)^2 + 4*b*sin(2*f*x + 2*e)^2 +
8*b*cos(2*f*x + 2*e) - 4*b)/((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2)))^2 + (a + b)*sqrt((16*b^2*cos(2*f*x +
2*e)^4 + 16*b^2*sin(2*f*x + 2*e)^4 + (a^2 + 2*a*b + b^2)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^4 - 64*b^2*cos(2*f*x
+ 2*e)^3 + 96*b^2*cos(2*f*x + 2*e)^2 - 8*((a*b + b^2)*cos(2*f*x + 2*e)^2 - (a*b + b^2)*sin(2*f*x + 2*e)^2 + a*
b + b^2 - 2*(a*b + b^2)*cos(2*f*x + 2*e))*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2 - 64*b^2*cos(2*f*x + 2*e) + 32*(b^2
*cos(2*f*x + 2*e)^2 - 2*b^2*cos(2*f*x + 2*e) + b^2)*sin(2*f*x + 2*e)^2 + 16*b^2)/(a^2 + 2*a*b + b^2))*sin(1/2*
arctan2(8*(b*cos(2*f*x + 2*e) - b)*sin(2*f*x + 2*e)/((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2), ((a + b)*abs(2
*e^(2*I*f*x + 2*I*e) + 2)^2 - 4*b*cos(2*f*x + 2*e)^2 + 4*b*sin(2*f*x + 2*e)^2 + 8*b*cos(2*f*x + 2*e) - 4*b)/((
a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2)))^2 + 4*b*cos(2*f*x + 2*e)^2 + 4*b*sin(2*f*x + 2*e)^2 - 4*sqrt(a*b +
b^2)*((16*b^2*cos(2*f*x + 2*e)^4 + 16*b^2*sin(2*f*x + 2*e)^4 + (a^2 + 2*a*b + b^2)*abs(2*e^(2*I*f*x + 2*I*e) +
2)^4 - 64*b^2*cos(2*f*x + 2*e)^3 + 96*b^2*cos(2*f*x + 2*e)^2 - 8*((a*b + b^2)*cos(2*f*x + 2*e)^2 - (a*b + b^2
)*sin(2*f*x + 2*e)^2 + a*b + b^2 - 2*(a*b + b^2)*cos(2*f*x + 2*e))*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2 - 64*b^2*c
os(2*f*x + 2*e) + 32*(b^2*cos(2*f*x + 2*e)^2 - 2*b^2*cos(2*f*x + 2*e) + b^2)*sin(2*f*x + 2*e)^2 + 16*b^2)/(a^2
+ 2*a*b + b^2))^(1/4)*cos(1/2*arctan2(8*(b*cos(2*f*x + 2*e) - b)*sin(2*f*x + 2*e)/((a + b)*abs(2*e^(2*I*f*x +
2*I*e) + 2)^2), ((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2 - 4*b*cos(2*f*x + 2*e)^2 + 4*b*sin(2*f*x + 2*e)^2 +
8*b*cos(2*f*x + 2*e) - 4*b)/((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2)))*sin(2*f*x + 2*e) - 4*(sqrt(a*b + b^2
)*cos(2*f*x + 2*e) - sqrt(a*b + b^2))*((16*b^2*cos(2*f*x + 2*e)^4 + 16*b^2*sin(2*f*x + 2*e)^4 + (a^2 + 2*a*b +
b^2)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^4 - 64*b^2*cos(2*f*x + 2*e)^3 + 96*b^2*cos(2*f*x + 2*e)^2 - 8*((a*b + b^2
)*cos(2*f*x + 2*e)^2 - (a*b + b^2)*sin(2*f*x + 2*e)^2 + a*b + b^2 - 2*(a*b + b^2)*cos(2*f*x + 2*e))*abs(2*e^(2
*I*f*x + 2*I*e) + 2)^2 - 64*b^2*cos(2*f*x + 2*e) + 32*(b^2*cos(2*f*x + 2*e)^2 - 2*b^2*cos(2*f*x + 2*e) + b^2)*
sin(2*f*x + 2*e)^2 + 16*b^2)/(a^2 + 2*a*b + b^2))^(1/4)*sin(1/2*arctan2(8*(b*cos(2*f*x + 2*e) - b)*sin(2*f*x +
2*e)/((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2), ((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2 - 4*b*cos(2*f*x +
2*e)^2 + 4*b*sin(2*f*x + 2*e)^2 + 8*b*cos(2*f*x + 2*e) - 4*b)/((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2))) - 8
*b*cos(2*f*x + 2*e) + 4*b)/((a + b)*abs(2*e^(2*I*f*x + 2*I*e) + 2)^2)) - (a^(3/2) + 2*sqrt(a)*b)*sqrt(b)*arcta
n2(2*a*sin(2*f*x + 2*e) + 2*(a^2*cos(4*f*x + 4*e)^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2 + 4*a*b + 4*b^2)*cos(2*f
*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*b^2)*sin(2*f*x + 2*e)^2 +
a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*cos(2*f*x + 2*e))^(1/4)*s
qrt(a)*sin(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e) + 2*(a + 2*b)*cos
(2*f*x + 2*e) + a)), 2*a*cos(2*f*x + 2*e) + 2*(a^2*cos(4*f*x + 4*e)^2 + a^2*sin(4*f*x + 4*e)^2 + 4*(a^2 + 4*a*
b + 4*b^2)*cos(2*f*x + 2*e)^2 + 4*(a^2 + 2*a*b)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(a^2 + 4*a*b + 4*b^2)*si
n(2*f*x + 2*e)^2 + a^2 + 2*(a^2 + 2*(a^2 + 2*a*b)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + 4*(a^2 + 2*a*b)*cos(2*f
*x + 2*e))^(1/4)*sqrt(a)*cos(1/2*arctan2(a*sin(4*f*x + 4*e) + 2*(a + 2*b)*sin(2*f*x + 2*e), a*cos(4*f*x + 4*e)
+ 2*(a + 2*b)*cos(2*f*x + 2*e) + a)) + 2*a + 4*b))/(a*sqrt(b)*f)

Giac [F]

\[ \int \sqrt {a+b \sec ^2(e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )^{2} + a} \,d x } \]

[In]

integrate((a+b*sec(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(f*x + e)^2 + a), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \sec ^2(e+f x)} \, dx=\int \sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}} \,d x \]

[In]

int((a + b/cos(e + f*x)^2)^(1/2),x)

[Out]

int((a + b/cos(e + f*x)^2)^(1/2), x)