Integrand size = 18, antiderivative size = 98 \[ \int x^2 \sqrt {a+a \sin (c+d x)} \, dx=\frac {8 x \sqrt {a+a \sin (c+d x)}}{d^2}+\frac {16 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d^3}-\frac {2 x^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d} \] Output:
8*x*(a+a*sin(d*x+c))^(1/2)/d^2+16*cot(1/2*c+1/4*Pi+1/2*d*x)*(a+a*sin(d*x+c ))^(1/2)/d^3-2*x^2*cot(1/2*c+1/4*Pi+1/2*d*x)*(a+a*sin(d*x+c))^(1/2)/d
Time = 1.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int x^2 \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \left (\left (-8-4 d x+d^2 x^2\right ) \cos \left (\frac {1}{2} (c+d x)\right )-\left (-8+4 d x+d^2 x^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {a (1+\sin (c+d x))}}{d^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:
Integrate[x^2*Sqrt[a + a*Sin[c + d*x]],x]
Output:
(-2*((-8 - 4*d*x + d^2*x^2)*Cos[(c + d*x)/2] - (-8 + 4*d*x + d^2*x^2)*Sin[ (c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 0.50 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.15, 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, 3042, 3777, 25, 3042, 3118}
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.
\(\displaystyle \int x^2 \sqrt {a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^2 \sqrt {a \sin (c+d x)+a}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \int x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \int x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {4 \int x \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {4 \int x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {4 \left (\frac {2 \int -\sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}+\frac {2 x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )}{d}-\frac {2 x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {4 \left (\frac {2 x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {2 \int \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}\right )}{d}-\frac {2 x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {4 \left (\frac {2 x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {2 \int \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}\right )}{d}-\frac {2 x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {4 \left (\frac {4 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d^2}+\frac {2 x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )}{d}-\frac {2 x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
Input:
Int[x^2*Sqrt[a + a*Sin[c + d*x]],x]
Output:
Csc[c/2 + Pi/4 + (d*x)/2]*((-2*x^2*Cos[c/2 + Pi/4 + (d*x)/2])/d + (4*((4*C os[c/2 + Pi/4 + (d*x)/2])/d^2 + (2*x*Sin[c/2 + Pi/4 + (d*x)/2])/d))/d)*Sqr t[a + a*Sin[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 = 0.72 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {i \sqrt {2}\, \sqrt {-a \left (-2-2 \sin \left (d x +c \right )\right )}\, \left (-i d^{2} x^{2}+d^{2} x^{2} {\mathrm e}^{i \left (d x +c \right )}+4 i d x \,{\mathrm e}^{i \left (d x +c \right )}-4 d x +8 i-8 \,{\mathrm e}^{i \left (d x +c \right )}\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1+2 i {\mathrm e}^{i \left (d x +c \right )}\right ) d^{3}}\) | \(119\) |
Input:
int(x^2*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-I*2^(1/2)*(-a*(-2-2*sin(d*x+c)))^(1/2)/(exp(2*I*(d*x+c))-1+2*I*exp(I*(d*x +c)))*(-I*d^2*x^2+d^2*x^2*exp(I*(d*x+c))+4*I*d*x*exp(I*(d*x+c))-4*d*x+8*I- 8*exp(I*(d*x+c)))*(exp(I*(d*x+c))+I)/d^3
Exception generated. \[ \int x^2 \sqrt {a+a \sin (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x^2 \sqrt {a+a \sin (c+d x)} \, dx=\int x^{2} \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}\, dx \] Input:
integrate(x**2*(a+a*sin(d*x+c))**(1/2),x)
Output:
Integral(x**2*sqrt(a*(sin(c + d*x) + 1)), x)
\[ \int x^2 \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} x^{2} \,d x } \] Input:
integrate(x^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*sin(d*x + c) + a)*x^2, x)
Time = 0.34 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95 \[ \int x^2 \sqrt {a+a \sin (c+d x)} \, dx=2 \, \sqrt {2} \sqrt {a} {\left (\frac {4 \, x \cos \left (\frac {1}{4} \, \pi - \frac {1}{2} \, d x - \frac {1}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d^{2}} - \frac {{\left (d^{2} x^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 8 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{4} \, \pi - \frac {1}{2} \, d x - \frac {1}{2} \, c\right )}{d^{3}}\right )} \] Input:
integrate(x^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")
Output:
2*sqrt(2)*sqrt(a)*(4*x*cos(1/4*pi - 1/2*d*x - 1/2*c)*sgn(cos(-1/4*pi + 1/2 *d*x + 1/2*c))/d^2 - (d^2*x^2*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) - 8*sgn( cos(-1/4*pi + 1/2*d*x + 1/2*c)))*sin(1/4*pi - 1/2*d*x - 1/2*c)/d^3)
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.65 \[ \int x^2 \sqrt {a+a \sin (c+d x)} \, dx=\frac {2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (8\,\cos \left (c+d\,x\right )+4\,d\,x-d^2\,x^2\,\cos \left (c+d\,x\right )+4\,d\,x\,\sin \left (c+d\,x\right )\right )}{d^3\,\left (\sin \left (c+d\,x\right )+1\right )} \] Input:
int(x^2*(a + a*sin(c + d*x))^(1/2),x)
Output:
(2*(a*(sin(c + d*x) + 1))^(1/2)*(8*cos(c + d*x) + 4*d*x - d^2*x^2*cos(c + d*x) + 4*d*x*sin(c + d*x)))/(d^3*(sin(c + d*x) + 1))
\[ \int x^2 \sqrt {a+a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, x^{2}d x \right ) \] Input:
int(x^2*(a+a*sin(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(sin(c + d*x) + 1)*x**2,x)