Integrand size = 18, antiderivative size = 120 \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=-\frac {96 \sqrt {a+a \sin (c+d x)}}{d^4}+\frac {12 x^2 \sqrt {a+a \sin (c+d x)}}{d^2}+\frac {48 x \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d^3}-\frac {2 x^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d} \]
-96*(a+a*sin(d*x+c))^(1/2)/d^4+12*x^2*(a+a*sin(d*x+c))^(1/2)/d^2+48*x*cot( 1/2*c+1/4*Pi+1/2*d*x)*(a+a*sin(d*x+c))^(1/2)/d^3-2*x^3*cot(1/2*c+1/4*Pi+1/ 2*d*x)*(a+a*sin(d*x+c))^(1/2)/d
Time = 2.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \left (\left (48-24 d x-6 d^2 x^2+d^3 x^3\right ) \cos \left (\frac {1}{2} (c+d x)\right )+\left (48+24 d x-6 d^2 x^2-d^3 x^3\right ) \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {a (1+\sin (c+d x))}}{d^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
(-2*((48 - 24*d*x - 6*d^2*x^2 + d^3*x^3)*Cos[(c + d*x)/2] + (48 + 24*d*x - 6*d^2*x^2 - d^3*x^3)*Sin[(c + d*x)/2])*Sqrt[a*(1 + Sin[c + d*x])])/(d^4*( Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 0.60 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {3042, 3800, 3042, 3777, 3042, 3777, 25, 3042, 3777, 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.
\(\displaystyle \int x^3 \sqrt {a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x^3 \sqrt {a \sin (c+d x)+a}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \int x^3 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \int x^3 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \int x^2 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \int x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \left (\frac {4 \int -x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}+\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 \int x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}\right )}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 \int x \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}\right )}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 \left (\frac {2 \int \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 x \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )}{d}\right )}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 \left (\frac {2 \int \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 x \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )}{d}\right )}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 \left (\frac {4 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d^2}-\frac {2 x \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )}{d}\right )}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\) |
Csc[c/2 + Pi/4 + (d*x)/2]*((-2*x^3*Cos[c/2 + Pi/4 + (d*x)/2])/d + (6*((2*x ^2*Sin[c/2 + Pi/4 + (d*x)/2])/d - (4*((-2*x*Cos[c/2 + Pi/4 + (d*x)/2])/d + (4*Sin[c/2 + Pi/4 + (d*x)/2])/d^2))/d))/d)*Sqrt[a + a*Sin[c + d*x]]
3.2.22.3.1 Defintions of rubi rules used
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 = 0.17 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {i \sqrt {2}\, \sqrt {-a \left (-2-2 \sin \left (d x +c \right )\right )}\, \left (-i x^{3} d^{3}+d^{3} x^{3} {\mathrm e}^{i \left (d x +c \right )}+6 i d^{2} x^{2} {\mathrm e}^{i \left (d x +c \right )}-6 d^{2} x^{2}+24 i d x -24 d x \,{\mathrm e}^{i \left (d x +c \right )}-48 i {\mathrm e}^{i \left (d x +c \right )}+48\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+2 i {\mathrm e}^{i \left (d x +c \right )}-1\right ) d^{4}}\) | \(145\) |
-I*2^(1/2)*(-a*(-2-2*sin(d*x+c)))^(1/2)/(exp(2*I*(d*x+c))+2*I*exp(I*(d*x+c ))-1)*(-I*x^3*d^3+d^3*x^3*exp(I*(d*x+c))+6*I*d^2*x^2*exp(I*(d*x+c))-6*d^2* x^2+24*I*d*x-24*d*x*exp(I*(d*x+c))-48*I*exp(I*(d*x+c))+48)*(exp(I*(d*x+c)) +I)/d^4
Exception generated. \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=\int x^{3} \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}\, dx \]
\[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} x^{3} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98 \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=2 \, \sqrt {2} \sqrt {a} {\left (\frac {6 \, {\left (d^{2} x^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 8 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \cos \left (\frac {1}{4} \, \pi - \frac {1}{2} \, d x - \frac {1}{2} \, c\right )}{d^{4}} - \frac {{\left (d^{3} x^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 24 \, d x \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{4} \, \pi - \frac {1}{2} \, d x - \frac {1}{2} \, c\right )}{d^{4}}\right )} \]
2*sqrt(2)*sqrt(a)*(6*(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)))*cos(1/4*pi - 1/2*d*x - 1/2*c)/d^4 - (d^3* x^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) - 24*d*x*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))*sin(1/4*pi - 1/2*d*x - 1/2*c)/d^4)
Time = 0.73 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.68 \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (48\,\sin \left (c+d\,x\right )-6\,d^2\,x^2+d^3\,x^3\,\cos \left (c+d\,x\right )-6\,d^2\,x^2\,\sin \left (c+d\,x\right )-24\,d\,x\,\cos \left (c+d\,x\right )+48\right )}{d^4\,\left (\sin \left (c+d\,x\right )+1\right )} \]