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]

Sqrt[1 + Cos[x]^2]*Sec[x]

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Rubi [A]  time = 0.0792028, antiderivative size = 13, normalized size of antiderivative = 1., 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 = 0.017403, size = 13, normalized size = 1. \[ \sqrt{\cos ^2(x)+1} \sec (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + Cos[x]^2]*Sec[x]

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Maple [B]  time = 0.033, size = 25, normalized size = 1.9 \begin{align*}{\frac{1+ \left ( \sec \left ( x \right ) \right ) ^{2}}{\sec \left ( x \right ) }{\frac{1}{\sqrt{{\frac{1+ \left ( \sec \left ( x \right ) \right ) ^{2}}{ \left ( \sec \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}

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)

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Maxima [A]  time = 1.45455, size = 18, normalized size = 1.38 \begin{align*} \frac{\sqrt{\cos \left (x\right )^{2} + 1}}{\cos \left (x\right )} \end{align*}

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)

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Fricas [A]  time = 2.41079, size = 51, normalized size = 3.92 \begin{align*} \frac{\sqrt{\cos \left (x\right )^{2} + 1} + \cos \left (x\right )}{\cos \left (x\right )} \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan{\left (x \right )} \sec{\left (x \right )}}{\sqrt{\cos ^{2}{\left (x \right )} + 1}}\, dx \end{align*}

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)

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Giac [A]  time = 1.11641, size = 28, normalized size = 2.15 \begin{align*} -\frac{2}{{\left (\sqrt{\cos \left (x\right )^{2} + 1} - \cos \left (x\right )\right )}^{2} - 1} \end{align*}

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)