\(\int x^3 \sqrt {a+a \sin (c+d x)} \, dx\) [122]

Optimal result

Integrand size = 18, antiderivative size = 120 \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=-\frac {96 \sqrt {a+a \sin (c+d x)}}{d^4}+\frac {12 x^2 \sqrt {a+a \sin (c+d x)}}{d^2}+\frac {48 x \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d^3}-\frac {2 x^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \sqrt {a+a \sin (c+d x)}}{d} \] Output:

-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
 

Mathematica [A] (verified)

Time = 2.99 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \left (\left (48-24 d x-6 d^2 x^2+d^3 x^3\right ) \cos \left (\frac {1}{2} (c+d x)\right )+\left (48+24 d x-6 d^2 x^2-d^3 x^3\right ) \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {a (1+\sin (c+d x))}}{d^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:

Integrate[x^3*Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

(-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]))
 

Rubi [A] (verified)

Time = 0.64 (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

\(\displaystyle \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (c+d x)+a} \left (\frac {6 \left (\frac {2 x^2 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 \left (\frac {4 \sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d^2}-\frac {2 x \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )}{d}\right )}{d}-\frac {2 x^3 \cos \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}\right )\)

Input:

Int[x^3*Sqrt[a + a*Sin[c + d*x]],x]
 

Output:

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]]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3118

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

rule 3777

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]
 

rule 3800

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])
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21

method	result	size
risch	\(-\frac {i \sqrt {2}\, \sqrt {-a \left (-2-2 \sin \left (d x +c \right )\right )}\, \left (-i x^{3} d^{3}+d^{3} x^{3} {\mathrm e}^{i \left (d x +c \right )}+6 i d^{2} x^{2} {\mathrm e}^{i \left (d x +c \right )}-6 x^{2} d^{2}+24 i d x -24 d x \,{\mathrm e}^{i \left (d x +c \right )}-48 i {\mathrm e}^{i \left (d x +c \right )}+48\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1+2 i {\mathrm e}^{i \left (d x +c \right )}\right ) d^{4}}\)	\(145\)

Input:

int(x^3*(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*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*x^2* 
d^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
 

Fricas [F(-2)]

Exception generated. \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(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)
 

Sympy [F]

\[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=\int x^{3} \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}\, dx \] Input:

integrate(x**3*(a+a*sin(d*x+c))**(1/2),x)
 

Output:

Integral(x**3*sqrt(a*(sin(c + d*x) + 1)), x)
 

Maxima [F]

\[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} x^{3} \,d x } \] Input:

integrate(x^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(a*sin(d*x + c) + a)*x^3, x)
 

Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98 \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=2 \, \sqrt {2} \sqrt {a} {\left (\frac {6 \, {\left (d^{2} x^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 8 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \cos \left (\frac {1}{4} \, \pi - \frac {1}{2} \, d x - \frac {1}{2} \, c\right )}{d^{4}} - \frac {{\left (d^{3} x^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 24 \, d x \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{4} \, \pi - \frac {1}{2} \, d x - \frac {1}{2} \, c\right )}{d^{4}}\right )} \] Input:

integrate(x^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

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)
 

Mupad [B] (verification not implemented)

Time = 36.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.68 \[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (48\,\sin \left (c+d\,x\right )-6\,d^2\,x^2+d^3\,x^3\,\cos \left (c+d\,x\right )-6\,d^2\,x^2\,\sin \left (c+d\,x\right )-24\,d\,x\,\cos \left (c+d\,x\right )+48\right )}{d^4\,\left (\sin \left (c+d\,x\right )+1\right )} \] Input:

int(x^3*(a + a*sin(c + d*x))^(1/2),x)
 

Output:

-(2*(a*(sin(c + d*x) + 1))^(1/2)*(48*sin(c + d*x) - 6*d^2*x^2 + d^3*x^3*co 
s(c + d*x) - 6*d^2*x^2*sin(c + d*x) - 24*d*x*cos(c + d*x) + 48))/(d^4*(sin 
(c + d*x) + 1))
 

Reduce [F]

\[ \int x^3 \sqrt {a+a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, x^{3}d x \right ) \] Input:

int(x^3*(a+a*sin(d*x+c))^(1/2),x)
 

Output:

sqrt(a)*int(sqrt(sin(c + d*x) + 1)*x**3,x)