3.297 \(\int \frac {1}{\sqrt {1+\sec ^2(x)}} \, dx\)

Optimal. Leaf size=14 \[ \tan ^{-1}\left (\frac {\tan (x)}{\sqrt {\tan ^2(x)+2}}\right ) \]

[Out]

arctan(tan(x)/(2+tan(x)^2)^(1/2))

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Rubi [A]  time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4128, 377, 203} \[ \tan ^{-1}\left (\frac {\tan (x)}{\sqrt {\tan ^2(x)+2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + Sec[x]^2],x]

[Out]

ArcTan[Tan[x]/Sqrt[2 + Tan[x]^2]]

Rule 203

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

Rule 377

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 4128

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*} \int \frac {1}{\sqrt {1+\sec ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {2+x^2}} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\tan (x)}{\sqrt {2+\tan ^2(x)}}\right )\\ &=\tan ^{-1}\left (\frac {\tan (x)}{\sqrt {2+\tan ^2(x)}}\right )\\ \end {align*}

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Mathematica [B]  time = 0.03, size = 37, normalized size = 2.64 \[ \frac {\sin ^{-1}\left (\frac {\sin (x)}{\sqrt {2}}\right ) \sqrt {\cos (2 x)+3} \sec (x)}{\sqrt {2} \sqrt {\sec ^2(x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Sec[x]^2],x]

[Out]

(ArcSin[Sin[x]/Sqrt[2]]*Sqrt[3 + Cos[2*x]]*Sec[x])/(Sqrt[2]*Sqrt[1 + Sec[x]^2])

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fricas [B]  time = 0.57, size = 53, normalized size = 3.79 \[ \frac {1}{2} \, \arctan \left (\frac {\sqrt {\frac {\cos \relax (x)^{2} + 1}{\cos \relax (x)^{2}}} \cos \relax (x)^{3} \sin \relax (x) + \cos \relax (x) \sin \relax (x)}{\cos \relax (x)^{4} + \cos \relax (x)^{2} - 1}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sec(x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*arctan((sqrt((cos(x)^2 + 1)/cos(x)^2)*cos(x)^3*sin(x) + cos(x)*sin(x))/(cos(x)^4 + cos(x)^2 - 1)) - 1/2*ar
ctan(sin(x)/cos(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\sec \relax (x)^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sec(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(sec(x)^2 + 1), x)

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maple [C]  time = 1.10, size = 142, normalized size = 10.14 \[ \frac {\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (\sin ^{2}\relax (x )\right ) \sqrt {\frac {i \cos \relax (x )+1-i+\cos \relax (x )}{\cos \relax (x )+1}}\, \sqrt {-\frac {i \cos \relax (x )-\cos \relax (x )-1-i}{\cos \relax (x )+1}}\, \left (2 \EllipticPi \left (\frac {\left (-1\right )^{\frac {1}{4}} \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i, i\right ) \left (-1\right )^{\frac {3}{4}}+\sqrt {2}\, \EllipticF \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )-2 \EllipticPi \left (\frac {\left (-1\right )^{\frac {1}{4}} \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i, i\right ) \left (-1\right )^{\frac {1}{4}}\right )}{\sqrt {\frac {1+\cos ^{2}\relax (x )}{\cos \relax (x )^{2}}}\, \cos \relax (x ) \left (-1+\cos \relax (x )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+sec(x)^2)^(1/2),x)

[Out]

(-1/2+1/2*I)*sin(x)^2*((I*cos(x)+1-I+cos(x))/(cos(x)+1))^(1/2)*(-(I*cos(x)-cos(x)-1-I)/(cos(x)+1))^(1/2)*(2*El
lipticPi((-1)^(1/4)*(-1+cos(x))/sin(x),I,I)*(-1)^(3/4)+EllipticF((1/2+1/2*I)*2^(1/2)*(-1+cos(x))/sin(x),I)*2^(
1/2)-2*EllipticPi((-1)^(1/4)*(-1+cos(x))/sin(x),I,I)*(-1)^(1/4))/((1+cos(x)^2)/cos(x)^2)^(1/2)/cos(x)/(-1+cos(
x))

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maxima [B]  time = 0.58, size = 388, normalized size = 27.71 \[ -\frac {1}{2} \, \arctan \left (2 \, {\left (2 \, {\left (6 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 36 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 12 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 36 \, \sin \left (2 \, x\right )^{2} + 12 \, \cos \left (2 \, x\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ) + 6 \, \sin \left (2 \, x\right ), \cos \left (4 \, x\right ) + 6 \, \cos \left (2 \, x\right ) + 1\right )\right ), 2 \, {\left (2 \, {\left (6 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 36 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 12 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 36 \, \sin \left (2 \, x\right )^{2} + 12 \, \cos \left (2 \, x\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ) + 6 \, \sin \left (2 \, x\right ), \cos \left (4 \, x\right ) + 6 \, \cos \left (2 \, x\right ) + 1\right )\right ) + 8\right ) + \frac {1}{2} \, \arctan \left (2 \, {\left (2 \, {\left (6 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 36 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 12 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 36 \, \sin \left (2 \, x\right )^{2} + 12 \, \cos \left (2 \, x\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ) + 6 \, \sin \left (2 \, x\right ), \cos \left (4 \, x\right ) + 6 \, \cos \left (2 \, x\right ) + 1\right )\right ) + 2 \, \sin \left (2 \, x\right ), 2 \, {\left (2 \, {\left (6 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 36 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 12 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 36 \, \sin \left (2 \, x\right )^{2} + 12 \, \cos \left (2 \, x\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, x\right ) + 6 \, \sin \left (2 \, x\right ), \cos \left (4 \, x\right ) + 6 \, \cos \left (2 \, x\right ) + 1\right )\right ) + 2 \, \cos \left (2 \, x\right ) + 6\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sec(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/2*arctan2(2*(2*(6*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin(4*x)*sin(2*x) +
36*sin(2*x)^2 + 12*cos(2*x) + 1)^(1/4)*sin(1/2*arctan2(sin(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)), 2*
(2*(6*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin(4*x)*sin(2*x) + 36*sin(2*x)^2
+ 12*cos(2*x) + 1)^(1/4)*cos(1/2*arctan2(sin(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)) + 8) + 1/2*arctan2
(2*(2*(6*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin(4*x)*sin(2*x) + 36*sin(2*x)
^2 + 12*cos(2*x) + 1)^(1/4)*sin(1/2*arctan2(sin(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)) + 2*sin(2*x), 2
*(2*(6*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin(4*x)*sin(2*x) + 36*sin(2*x)^2
+ 12*cos(2*x) + 1)^(1/4)*cos(1/2*arctan2(sin(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)) + 2*cos(2*x) + 6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {\frac {1}{{\cos \relax (x)}^2}+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/cos(x)^2 + 1)^(1/2),x)

[Out]

int(1/(1/cos(x)^2 + 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\sec ^{2}{\relax (x )} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+sec(x)**2)**(1/2),x)

[Out]

Integral(1/sqrt(sec(x)**2 + 1), x)

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