3.144 \(\int x^2 \sqrt{a+a \cos (c+d x)} \, dx\)

Optimal. Leaf size=88 \[ \frac{8 x \sqrt{a \cos (c+d x)+a}}{d^2}-\frac{16 \tan \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}}{d^3}+\frac{2 x^2 \tan \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}}{d} \]

[Out]

(8*x*Sqrt[a + a*Cos[c + d*x]])/d^2 - (16*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d^3 + (2*x^2*Sqrt[a + a*
Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d

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Rubi [A]  time = 0.112431, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3319, 3296, 2637} \[ \frac{8 x \sqrt{a \cos (c+d x)+a}}{d^2}-\frac{16 \tan \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}}{d^3}+\frac{2 x^2 \tan \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cos (c+d x)+a}}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + a*Cos[c + d*x]],x]

[Out]

(8*x*Sqrt[a + a*Cos[c + d*x]])/d^2 - (16*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d^3 + (2*x^2*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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^2 \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^2 \sin \left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{\pi }{4}+\frac{d x}{2}\right ) \, dx\\ &=\frac{2 x^2 \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}-\frac{\left (4 \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{c}{2}+\frac{d x}{2}\right ) \, dx}{d}\\ &=\frac{8 x \sqrt{a+a \cos (c+d x)}}{d^2}+\frac{2 x^2 \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}-\frac{\left (8 \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 \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \, dx}{d^2}\\ &=\frac{8 x \sqrt{a+a \cos (c+d x)}}{d^2}-\frac{16 \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{d^3}+\frac{2 x^2 \sqrt{a+a \cos (c+d x)} \tan \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.157771, size = 44, normalized size = 0.5 \[ \frac{2 \left (\left (d^2 x^2-8\right ) \tan \left (\frac{1}{2} (c+d x)\right )+4 d x\right ) \sqrt{a (\cos (c+d x)+1)}}{d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a + a*Cos[c + d*x]],x]

[Out]

(2*Sqrt[a*(1 + Cos[c + d*x])]*(4*d*x + (-8 + d^2*x^2)*Tan[(c + d*x)/2]))/d^3

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Maple [C]  time = 0.231, size = 105, normalized size = 1.2 \begin{align*}{\frac{-i\sqrt{2} \left ({d}^{2}{x}^{2}{{\rm e}^{i \left ( dx+c \right ) }}+4\,idx{{\rm e}^{i \left ( dx+c \right ) }}-{d}^{2}{x}^{2}+4\,idx-8\,{{\rm e}^{i \left ( dx+c \right ) }}+8 \right ) }{ \left ({{\rm e}^{i \left ( dx+c \right ) }}+1 \right ){d}^{3}}\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^2*(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^2*x^2*exp(I*(d*x+c))+4*I*d*x*e
xp(I*(d*x+c))-d^2*x^2+4*I*d*x-8*exp(I*(d*x+c))+8)/d^3

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Maxima [A]  time = 2.81163, size = 165, normalized size = 1.88 \begin{align*} \frac{2 \,{\left (\sqrt{2} \sqrt{a} c^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \,{\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} c +{\left (\sqrt{2}{\left (d x + c\right )}^{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4 \, \sqrt{2}{\left (d x + c\right )} \cos \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 8 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}\right )}}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt(2)*sqrt(a)*c^2*sin(1/2*d*x + 1/2*c) - 2*(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)*c + (sqrt(2)*(d*x + c)^2*sin(1/2*d*x + 1/2*c) + 4*sqrt(2)*(d*x + c)*cos(1/2*d*x + 1/2*c) -
8*sqrt(2)*sin(1/2*d*x + 1/2*c))*sqrt(a))/d^3

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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^2*(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^{2} \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**2*(a+a*cos(d*x+c))**(1/2),x)

[Out]

Integral(x**2*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^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cos(d*x + c) + a)*x^2, x)