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

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Rubi [A]  time = 0.054289, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 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 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 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*}

Mathematica [A]  time = 0.047725, 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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Maple [B]  time = 1.055, size = 101, normalized size = 2.4 \begin{align*}{\frac{1}{3\,\sin \left ( x \right ) }\sqrt{ \left ( 1+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( -\cos \left ( x \right ) \left ( \sin \left ( x \right ) \right ) ^{4}+2\,\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \sin \left ( x \right ) \right ) ^{2}+2}{\it EllipticF} \left ( \cos \left ( x \right ) ,i \right ) -6\,\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 ) +2\,\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+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*cos(x)*sin(x)^2)/(1-cos(x)^4)^(1/2)/sin(x)/(1+co
s(x)^2)^(1/2)

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

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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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+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{\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+cos(x)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((cos(x)^2 + 1)^(3/2), x)