3.145 \(\int x \sqrt{a+a \cos (c+d x)} \, dx\)

Optimal. Leaf size=53 \[ \frac{4 \sqrt{a \cos (c+d x)+a}}{d^2}+\frac{2 x \tan \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}}{d} \]

[Out]

(4*Sqrt[a + a*Cos[c + d*x]])/d^2 + (2*x*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d

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Rubi [A]  time = 0.0608678, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3319, 3296, 2638} \[ \frac{4 \sqrt{a \cos (c+d x)+a}}{d^2}+\frac{2 x \tan \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}}{d} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + a*Cos[c + d*x]],x]

[Out]

(4*Sqrt[a + a*Cos[c + d*x]])/d^2 + (2*x*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d

Rule 3319

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])

Rule 3296

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 2638

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

Rubi steps

\begin{align*} \int x \sqrt{a+a \cos (c+d x)} \, dx &=\left (\sqrt{a+a \cos (c+d x)} \csc \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \int x \sin \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right ) \, dx\\ &=\frac{2 x \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}-\frac{\left (2 \sqrt{a+a \cos (c+d x)} \csc \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right )\right ) \int \sin \left (\frac{c}{2}+\frac{d x}{2}\right ) \, dx}{d}\\ &=\frac{4 \sqrt{a+a \cos (c+d x)}}{d^2}+\frac{2 x \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.128065, size = 34, normalized size = 0.64 \[ \frac{2 \left (d x \tan \left (\frac{1}{2} (c+d x)\right )+2\right ) \sqrt{a (\cos (c+d x)+1)}}{d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[a + a*Cos[c + d*x]],x]

[Out]

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

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Maple [C]  time = 0.218, size = 80, normalized size = 1.5 \begin{align*}{\frac{-i\sqrt{2} \left ( dx{{\rm e}^{i \left ( dx+c \right ) }}+2\,i{{\rm e}^{i \left ( dx+c \right ) }}-dx+2\,i \right ) }{ \left ({{\rm e}^{i \left ( dx+c \right ) }}+1 \right ){d}^{2}}\sqrt{a \left ({{\rm e}^{i \left ( dx+c \right ) }}+1 \right ) ^{2}{{\rm e}^{-i \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+cos(d*x+c)*a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 2.52002, size = 82, normalized size = 1.55 \begin{align*} -\frac{2 \,{\left (\sqrt{2} \sqrt{a} c \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) -{\left (\sqrt{2}{\left (d x + c\right )} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, \sqrt{2} \cos \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}\right )}}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(sqrt(2)*sqrt(a)*c*sin(1/2*d*x + 1/2*c) - (sqrt(2)*(d*x + c)*sin(1/2*d*x + 1/2*c) + 2*sqrt(2)*cos(1/2*d*x +
1/2*c))*sqrt(a))/d^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(a*(cos(c + d*x) + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cos \left (d x + c\right ) + a} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cos(d*x + c) + a)*x, x)