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

Optimal. Leaf size=13 \[ \sqrt {\cos ^2(x)+1} \sec (x) \]

[Out]

sec(x)*(1+cos(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {264} \[ \sqrt {\cos ^2(x)+1} \sec (x) \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x]*Tan[x])/Sqrt[1 + Cos[x]^2],x]

[Out]

Sqrt[1 + Cos[x]^2]*Sec[x]

Rule 264

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 13, normalized size = 1.00 \[ \sqrt {\cos ^2(x)+1} \sec (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x]*Tan[x])/Sqrt[1 + Cos[x]^2],x]

[Out]

Sqrt[1 + Cos[x]^2]*Sec[x]

________________________________________________________________________________________

fricas [A]  time = 0.67, size = 16, normalized size = 1.23 \[ \frac {\sqrt {\cos \relax (x)^{2} + 1} + \cos \relax (x)}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(cos(x)^2 + 1) + cos(x))/cos(x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 21, normalized size = 1.62 \[ -\frac {2}{{\left (\sqrt {\cos \relax (x)^{2} + 1} - \cos \relax (x)\right )}^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/((sqrt(cos(x)^2 + 1) - cos(x))^2 - 1)

________________________________________________________________________________________

maple [B]  time = 0.11, size = 25, normalized size = 1.92 \[ \frac {1+\sec ^{2}\relax (x )}{\sqrt {\frac {1+\sec ^{2}\relax (x )}{\sec \relax (x )^{2}}}\, \sec \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)*tan(x)/(1+cos(x)^2)^(1/2),x)

[Out]

1/((1+sec(x)^2)/sec(x)^2)^(1/2)/sec(x)*(1+sec(x)^2)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 13, normalized size = 1.00 \[ \frac {\sqrt {\cos \relax (x)^{2} + 1}}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(cos(x)^2 + 1)/cos(x)

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 13, normalized size = 1.00 \[ \frac {\sqrt {{\cos \relax (x)}^2+1}}{\cos \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/(cos(x)*(cos(x)^2 + 1)^(1/2)),x)

[Out]

(cos(x)^2 + 1)^(1/2)/cos(x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tan(x)*sec(x)/sqrt(cos(x)**2 + 1), x)

________________________________________________________________________________________