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]

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

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Rubi [A]  time = 0.02, antiderivative size = 32, 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 = {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 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]

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]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\cos \relax (x)^{2} + 1}}{\cos \relax (x)^{4} + 2 \, \cos \relax (x)^{2} + 1}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (\cos \relax (x)^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

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)

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maple [B]  time = 2.34, size = 70, normalized size = 2.19 \[ -\frac {\sqrt {-\left (\sin ^{4}\relax (x )\right )+2 \left (\sin ^{2}\relax (x )\right )}\, \left (\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\left (\sin ^{2}\relax (x )\right )+2}\, \EllipticE \left (\cos \relax (x ), i\right )+\left (\sin ^{2}\relax (x )\right ) \cos \relax (x )\right )}{2 \sqrt {1-\left (\cos ^{4}\relax (x )\right )}\, \sin \relax (x ) \sqrt {1+\cos ^{2}\relax (x )}} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (\cos \relax (x)^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left ({\cos \relax (x)}^2+1\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^2 + 1)^(3/2),x)

[Out]

int(1/(cos(x)^2 + 1)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\cos ^{2}{\relax (x )} + 1\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((cos(x)**2 + 1)**(-3/2), x)

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