3.2.52 \(\int x \sqrt {a+a \cos (x)} \, dx\) [152]

Optimal. Leaf size=32 \[ 4 \sqrt {a+a \cos (x)}+2 x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right ) \]

[Out]

4*(a+a*cos(x))^(1/2)+2*x*(a+a*cos(x))^(1/2)*tan(1/2*x)

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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3400, 3377, 2718} \begin {gather*} 4 \sqrt {a \cos (x)+a}+2 x \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + a*Cos[x]],x]

[Out]

4*Sqrt[a + a*Cos[x]] + 2*x*Sqrt[a + a*Cos[x]]*Tan[x/2]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3400

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(2*a)^IntPart[n]
*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])), Int[(c + d*x)^m*Sin[e/2
+ a*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rubi steps

\begin {align*} \int x \sqrt {a+a \cos (x)} \, dx &=\left (\sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x \cos \left (\frac {x}{2}\right ) \, dx\\ &=2 x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )-\left (2 \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \sin \left (\frac {x}{2}\right ) \, dx\\ &=4 \sqrt {a+a \cos (x)}+2 x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.69 \begin {gather*} 2 \sqrt {a (1+\cos (x))} \left (2+x \tan \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[a + a*Cos[x]],x]

[Out]

2*Sqrt[a*(1 + Cos[x])]*(2 + x*Tan[x/2])

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Maple [C] Result contains complex when optimal does not.
time = 0.06, size = 55, normalized size = 1.72

method	result	size
risch	\(-\frac {i \sqrt {2}\, \sqrt {a \left ({\mathrm e}^{i x}+1\right )^{2} {\mathrm e}^{-i x}}\, \left (2 i {\mathrm e}^{i x}+x \,{\mathrm e}^{i x}+2 i-x \right )}{{\mathrm e}^{i x}+1}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+a*cos(x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-I*2^(1/2)*(a*(exp(I*x)+1)^2*exp(-I*x))^(1/2)/(exp(I*x)+1)*(2*I*exp(I*x)+x*exp(I*x)+2*I-x)

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Maxima [A]
time = 0.51, size = 24, normalized size = 0.75 \begin {gather*} 2 \, {\left (\sqrt {2} x \sin \left (\frac {1}{2} \, x\right ) + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt(2)*x*sin(1/2*x) + 2*sqrt(2)*cos(1/2*x))*sqrt(a)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {a \left (\cos {\left (x \right )} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(a*(cos(x) + 1)), x)

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Giac [A]
time = 0.43, size = 31, normalized size = 0.97 \begin {gather*} 2 \, \sqrt {2} {\left (x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) + 2 \, \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2)*(x*sgn(cos(1/2*x))*sin(1/2*x) + 2*cos(1/2*x)*sgn(cos(1/2*x)))*sqrt(a)

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Mupad [B]
time = 0.31, size = 50, normalized size = 1.56 \begin {gather*} \frac {2\,\sqrt {a}\,\sqrt {\cos \left (x\right )+1}\,\left (x\,\cos \left (x\right )+\cos \left (x\right )\,2{}\mathrm {i}-2\,\sin \left (x\right )-x+x\,\sin \left (x\right )\,1{}\mathrm {i}+2{}\mathrm {i}\right )}{\cos \left (x\right )\,1{}\mathrm {i}-\sin \left (x\right )+1{}\mathrm {i}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + a*cos(x))^(1/2),x)

[Out]

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

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