Integrand size = 18, antiderivative size = 88 \[ \int x^2 \sqrt {a+a \cos (c+d x)} \, dx=\frac {8 x \sqrt {a+a \cos (c+d x)}}{d^2}-\frac {16 \sqrt {a+a \cos (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3}+\frac {2 x^2 \sqrt {a+a \cos (c+d x)} \tan \left (\frac {c}{2}+\frac {d x}{2}\right )}{d} \] Output:
8*x*(a+a*cos(d*x+c))^(1/2)/d^2-16*(a+a*cos(d*x+c))^(1/2)*tan(1/2*d*x+1/2*c )/d^3+2*x^2*(a+a*cos(d*x+c))^(1/2)*tan(1/2*d*x+1/2*c)/d
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.50 \[ \int x^2 \sqrt {a+a \cos (c+d x)} \, dx=\frac {2 \sqrt {a (1+\cos (c+d x))} \left (4 d x+\left (-8+d^2 x^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d^3} \] Input:
Integrate[x^2*Sqrt[a + a*Cos[c + d*x]],x]
Output:
(2*Sqrt[a*(1 + Cos[c + d*x])]*(4*d*x + (-8 + d^2*x^2)*Tan[(c + d*x)/2]))/d ^3
Time = 0.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3800, 3042, 3777, 25, 3042, 3777, 3042, 3117}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 \sqrt {a \cos (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^2 \sqrt {a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \int x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \int x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\frac {4 \int -x \sin \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}+\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 \int x \sin \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 \int x \sin \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 \left (\frac {2 \int \cos \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}-\frac {2 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}\right )}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 \left (\frac {2 \int \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{2}\right )dx}{d}-\frac {2 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}\right )}{d}\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cos (c+d x)+a} \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 \left (\frac {4 \sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2}-\frac {2 x \cos \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}\right )}{d}\right )\) |
Input:
Int[x^2*Sqrt[a + a*Cos[c + d*x]],x]
Output:
Sqrt[a + a*Cos[c + d*x]]*Sec[c/2 + (d*x)/2]*((2*x^2*Sin[c/2 + (d*x)/2])/d - (4*((-2*x*Cos[c/2 + (d*x)/2])/d + (4*Sin[c/2 + (d*x)/2])/d^2))/d)
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(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])
Result contains complex when optimal does not.
Time = 1.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.19
method | result | size |
risch | \(-\frac {i \sqrt {2}\, \sqrt {a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2} {\mathrm e}^{-i \left (d x +c \right )}}\, \left (d^{2} x^{2} {\mathrm e}^{i \left (d x +c \right )}+4 i d x \,{\mathrm e}^{i \left (d x +c \right )}-x^{2} d^{2}+4 i d x -8 \,{\mathrm e}^{i \left (d x +c \right )}+8\right )}{\left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) d^{3}}\) | \(105\) |
Input:
int(x^2*(a+a*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-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*exp(I*(d*x+c))-x^2*d^2+4*I*d*x-8*exp(I* (d*x+c))+8)/d^3
Exception generated. \[ \int x^2 \sqrt {a+a \cos (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*(a+a*cos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x^2 \sqrt {a+a \cos (c+d x)} \, dx=\int x^{2} \sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}\, dx \] Input:
integrate(x**2*(a+a*cos(d*x+c))**(1/2),x)
Output:
Integral(x**2*sqrt(a*(cos(c + d*x) + 1)), x)
Time = 0.16 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.39 \[ \int x^2 \sqrt {a+a \cos (c+d x)} \, dx=\frac {2 \, {\left (\sqrt {2} \sqrt {a} c^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, {\left (\sqrt {2} {\left (d x + c\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, \sqrt {2} \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a} c + {\left (\sqrt {2} {\left (d x + c\right )}^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, \sqrt {2} {\left (d x + c\right )} \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}\right )}}{d^{3}} \] Input:
integrate(x^2*(a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")
Output:
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
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.88 \[ \int x^2 \sqrt {a+a \cos (c+d x)} \, dx=2 \, \sqrt {2} \sqrt {a} {\left (\frac {4 \, x \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d^{2}} + \frac {{\left (d^{2} x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 8 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d^{3}}\right )} \] Input:
integrate(x^2*(a+a*cos(d*x+c))^(1/2),x, algorithm="giac")
Output:
2*sqrt(2)*sqrt(a)*(4*x*cos(1/2*d*x + 1/2*c)*sgn(cos(1/2*d*x + 1/2*c))/d^2 + (d^2*x^2*sgn(cos(1/2*d*x + 1/2*c)) - 8*sgn(cos(1/2*d*x + 1/2*c)))*sin(1/ 2*d*x + 1/2*c)/d^3)
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int x^2 \sqrt {a+a \cos (c+d x)} \, dx=\frac {2\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}\,\left (4\,d\,x-8\,\sin \left (c+d\,x\right )+d^2\,x^2\,\sin \left (c+d\,x\right )+4\,d\,x\,\cos \left (c+d\,x\right )\right )}{d^3\,\left (\cos \left (c+d\,x\right )+1\right )} \] Input:
int(x^2*(a + a*cos(c + d*x))^(1/2),x)
Output:
(2*(a*(cos(c + d*x) + 1))^(1/2)*(4*d*x - 8*sin(c + d*x) + d^2*x^2*sin(c + d*x) + 4*d*x*cos(c + d*x)))/(d^3*(cos(c + d*x) + 1))
\[ \int x^2 \sqrt {a+a \cos (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\cos \left (d x +c \right )+1}\, x^{2}d x \right ) \] Input:
int(x^2*(a+a*cos(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(cos(c + d*x) + 1)*x**2,x)