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

Optimal. Leaf size=43 \[ -\frac {2}{3} F\left (\left .x+\frac {\pi }{2}\right |-1\right )+2 E\left (\left .x+\frac {\pi }{2}\right |-1\right )+\frac {1}{3} \sin (x) \cos (x) \sqrt {\cos ^2(x)+1} \]

[Out]

-2*(sin(x)^2)^(1/2)/sin(x)*EllipticE(cos(x),I)+2/3*(sin(x)^2)^(1/2)/sin(x)*EllipticF(cos(x),I)+1/3*cos(x)*sin(
x)*(1+cos(x)^2)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3180, 3172, 3177, 3182} \[ -\frac {2}{3} F\left (\left .x+\frac {\pi }{2}\right |-1\right )+2 E\left (\left .x+\frac {\pi }{2}\right |-1\right )+\frac {1}{3} \sin (x) \cos (x) \sqrt {\cos ^2(x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cos[x]^2)^(3/2),x]

[Out]

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

Rule 3172

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[
B/b, Int[Sqrt[a + b*Sin[e + f*x]^2], x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /;
FreeQ[{a, b, e, f, 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 3180

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*f*p), x] + Dist[1/(2*p), Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*
a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && GtQ[p, 1]

Rule 3182

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1*EllipticF[e + f*x, -(b/a)])/(Sqrt[a]*
f), x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rubi steps

\begin {align*} \int \left (1+\cos ^2(x)\right )^{3/2} \, dx &=\frac {1}{3} \cos (x) \sqrt {1+\cos ^2(x)} \sin (x)+\frac {1}{3} \int \frac {4+6 \cos ^2(x)}{\sqrt {1+\cos ^2(x)}} \, dx\\ &=\frac {1}{3} \cos (x) \sqrt {1+\cos ^2(x)} \sin (x)-\frac {2}{3} \int \frac {1}{\sqrt {1+\cos ^2(x)}} \, dx+2 \int \sqrt {1+\cos ^2(x)} \, dx\\ &=2 E\left (\left .\frac {\pi }{2}+x\right |-1\right )-\frac {2}{3} F\left (\left .\frac {\pi }{2}+x\right |-1\right )+\frac {1}{3} \cos (x) \sqrt {1+\cos ^2(x)} \sin (x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 39, normalized size = 0.91 \[ \frac {-4 F\left (x\left |\frac {1}{2}\right .\right )+24 E\left (x\left |\frac {1}{2}\right .\right )+\sin (2 x) \sqrt {\cos (2 x)+3}}{6 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cos[x]^2)^(3/2),x]

[Out]

(24*EllipticE[x, 1/2] - 4*EllipticF[x, 1/2] + Sqrt[3 + Cos[2*x]]*Sin[2*x])/(6*Sqrt[2])

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cos \relax (x)^{2} + 1\right )}^{\frac {3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((cos(x)^2 + 1)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+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 = 101, normalized size = 2.35 \[ \frac {\sqrt {\left (1+\cos ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )}\, \left (-\cos \relax (x ) \left (\sin ^{4}\relax (x )\right )+2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\left (\sin ^{2}\relax (x )\right )+2}\, \EllipticF \left (\cos \relax (x ), i\right )-6 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\left (\sin ^{2}\relax (x )\right )+2}\, \EllipticE \left (\cos \relax (x ), i\right )+2 \left (\sin ^{2}\relax (x )\right ) \cos \relax (x )\right )}{3 \sqrt {1-\left (\cos ^{4}\relax (x )\right )}\, \sin \relax (x ) \sqrt {1+\cos ^{2}\relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((1+cos(x)^2)*sin(x)^2)^(1/2)*(-cos(x)*sin(x)^4+2*(sin(x)^2)^(1/2)*(-sin(x)^2+2)^(1/2)*EllipticF(cos(x),I)
-6*(sin(x)^2)^(1/2)*(-sin(x)^2+2)^(1/2)*EllipticE(cos(x),I)+2*sin(x)^2*cos(x))/(1-cos(x)^4)^(1/2)/sin(x)/(1+co
s(x)^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+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 = -0.02 \[ \int {\left ({\cos \relax (x)}^2+1\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)^2 + 1)^(3/2),x)

[Out]

int((cos(x)^2 + 1)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((cos(x)**2 + 1)**(3/2), x)

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