3.64 \(\int \frac{1}{(1+\cos ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{2} E\left (\left .x+\frac{\pi }{2}\right |-1\right )-\frac{\sin (x) \cos (x)}{2 \sqrt{\cos ^2(x)+1}} \]

[Out]

EllipticE[Pi/2 + x, -1]/2 - (Cos[x]*Sin[x])/(2*Sqrt[1 + Cos[x]^2])

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Rubi [A]  time = 0.0182579, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3184, 21, 3177} \[ \frac{1}{2} E\left (\left .x+\frac{\pi }{2}\right |-1\right )-\frac{\sin (x) \cos (x)}{2 \sqrt{\cos ^2(x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x]^2)^(-3/2),x]

[Out]

EllipticE[Pi/2 + x, -1]/2 - (Cos[x]*Sin[x])/(2*Sqrt[1 + Cos[x]^2])

Rule 3184

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
e + f*x]^2)^(p + 1))/(2*a*f*(p + 1)*(a + b)), x] + Dist[1/(2*a*(p + 1)*(a + b)), Int[(a + b*Sin[e + f*x]^2)^(p
+ 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ
[a + b, 0] && LtQ[p, -1]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3177

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[e + f*x, -(b/a)])/f, x]
/; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+\cos ^2(x)\right )^{3/2}} \, dx &=-\frac{\cos (x) \sin (x)}{2 \sqrt{1+\cos ^2(x)}}-\frac{1}{2} \int \frac{-1-\cos ^2(x)}{\sqrt{1+\cos ^2(x)}} \, dx\\ &=-\frac{\cos (x) \sin (x)}{2 \sqrt{1+\cos ^2(x)}}+\frac{1}{2} \int \sqrt{1+\cos ^2(x)} \, dx\\ &=\frac{1}{2} E\left (\left .\frac{\pi }{2}+x\right |-1\right )-\frac{\cos (x) \sin (x)}{2 \sqrt{1+\cos ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0599886, size = 35, normalized size = 1.09 \[ \frac{E\left (x\left |\frac{1}{2}\right .\right )}{\sqrt{2}}-\frac{\sin (2 x)}{2 \sqrt{2} \sqrt{\cos (2 x)+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x]^2)^(-3/2),x]

[Out]

EllipticE[x, 1/2]/Sqrt[2] - Sin[2*x]/(2*Sqrt[2]*Sqrt[3 + Cos[2*x]])

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Maple [B]  time = 1.058, size = 70, normalized size = 2.2 \begin{align*} -{\frac{1}{2\,\sin \left ( x \right ) }\sqrt{- \left ( \sin \left ( x \right ) \right ) ^{4}+2\, \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \sin \left ( x \right ) \right ) ^{2}+2}{\it EllipticE} \left ( \cos \left ( x \right ) ,i \right ) +\cos \left ( x \right ) \left ( \sin \left ( x \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{1- \left ( \cos \left ( x \right ) \right ) ^{4}}}}{\frac{1}{\sqrt{1+ \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cos(x)^2)^(3/2),x)

[Out]

-1/2*(-sin(x)^4+2*sin(x)^2)^(1/2)*((sin(x)^2)^(1/2)*(-sin(x)^2+2)^(1/2)*EllipticE(cos(x),I)+cos(x)*sin(x)^2)/(
1-cos(x)^4)^(1/2)/sin(x)/(1+cos(x)^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\cos \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((cos(x)^2 + 1)^(-3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\cos \left (x\right )^{2} + 1}}{\cos \left (x\right )^{4} + 2 \, \cos \left (x\right )^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^2)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(cos(x)^2 + 1)/(cos(x)^4 + 2*cos(x)^2 + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\cos \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cos(x)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((cos(x)^2 + 1)^(-3/2), x)