Integrand size = 28, antiderivative size = 36 \[ \int (a+a \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 a^3 c^4 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}} \] Output:
2/7*a^3*c^4*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)
Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(36)=72\).
Time = 0.88 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.03 \[ \int (a+a \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \sqrt {c-c \sin (e+f x)}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:
Integrate[(a + a*Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]],x]
Output:
(2*a^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*Sqrt[c - c*Sin[e + f*x]])/( 7*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))
Time = 0.36 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3215, 3042, 3152}
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.
\(\displaystyle \int (a \sin (e+f x)+a)^3 \sqrt {c-c \sin (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (e+f x)+a)^3 \sqrt {c-c \sin (e+f x)}dx\) |
\(\Big \downarrow \) 3215 |
\(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^{5/2}}dx\) |
\(\Big \downarrow \) 3152 |
\(\displaystyle \frac {2 a^3 c^4 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}\) |
Input:
Int[(a + a*Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]],x]
Output:
(2*a^3*c^4*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^(7/2))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^(m - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 2.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {2 \left (-1+\sin \left (f x +e \right )\right ) c \left (1+\sin \left (f x +e \right )\right )^{4} a^{3}}{7 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(49\) |
parts | \(-\frac {2 a^{3} \left (-1+\sin \left (f x +e \right )\right ) \left (1+\sin \left (f x +e \right )\right ) c}{\cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (-1+\sin \left (f x +e \right )\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (5 \sin \left (f x +e \right )^{3}-6 \sin \left (f x +e \right )^{2}+8 \sin \left (f x +e \right )-16\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (-1+\sin \left (f x +e \right )\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (-2+\sin \left (f x +e \right )\right )}{\cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a^{3} \left (-1+\sin \left (f x +e \right )\right ) c \left (1+\sin \left (f x +e \right )\right ) \left (3 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+8\right )}{5 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(244\) |
Input:
int((a+sin(f*x+e)*a)^3*(c-c*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/7*(-1+sin(f*x+e))*c*(1+sin(f*x+e))^4*a^3/cos(f*x+e)/(c-c*sin(f*x+e))^(1 /2)/f
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (32) = 64\).
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 3.92 \[ \int (a+a \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 \, {\left (a^{3} \cos \left (f x + e\right )^{4} - 3 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + 8 \, a^{3} - {\left (a^{3} \cos \left (f x + e\right )^{3} + 4 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} \cos \left (f x + e\right ) - 8 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{7 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
2/7*(a^3*cos(f*x + e)^4 - 3*a^3*cos(f*x + e)^3 - 8*a^3*cos(f*x + e)^2 + 4* a^3*cos(f*x + e) + 8*a^3 - (a^3*cos(f*x + e)^3 + 4*a^3*cos(f*x + e)^2 - 4* a^3*cos(f*x + e) - 8*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f *x + e) - f*sin(f*x + e) + f)
\[ \int (a+a \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)} \, dx=a^{3} \left (\int 3 \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int 3 \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\, dx + \int \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\, dx + \int \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx\right ) \] Input:
integrate((a+a*sin(f*x+e))**3*(c-c*sin(f*x+e))**(1/2),x)
Output:
a**3*(Integral(3*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x), x) + Integral(3*s qrt(-c*sin(e + f*x) + c)*sin(e + f*x)**2, x) + Integral(sqrt(-c*sin(e + f* x) + c)*sin(e + f*x)**3, x) + Integral(sqrt(-c*sin(e + f*x) + c), x))
\[ \int (a+a \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {-c \sin \left (f x + e\right ) + c} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^3*sqrt(-c*sin(f*x + e) + c), x)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (32) = 64\).
Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.64 \[ \int (a+a \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)} \, dx=-\frac {\sqrt {2} {\left (35 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 21 \, a^{3} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 7 \, a^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + a^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{28 \, f} \] Input:
integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
-1/28*sqrt(2)*(35*a^3*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(sin(-1/4*pi + 1/2 *f*x + 1/2*e)) + 21*a^3*cos(-3/4*pi + 3/2*f*x + 3/2*e)*sgn(sin(-1/4*pi + 1 /2*f*x + 1/2*e)) + 7*a^3*cos(-5/4*pi + 5/2*f*x + 5/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + a^3*cos(-7/4*pi + 7/2*f*x + 7/2*e)*sgn(sin(-1/4*pi + 1 /2*f*x + 1/2*e)))*sqrt(c)/f
Timed out. \[ \int (a+a \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c-c\,\sin \left (e+f\,x\right )} \,d x \] Input:
int((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(1/2),x)
Output:
int((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^(1/2), x)
\[ \int (a+a \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)} \, dx=\sqrt {c}\, a^{3} \left (\int \sqrt {-\sin \left (f x +e \right )+1}d x +\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x +3 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right )+3 \left (\int \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right )\right ) \] Input:
int((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(1/2),x)
Output:
sqrt(c)*a**3*(int(sqrt( - sin(e + f*x) + 1),x) + int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3,x) + 3*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2, x) + 3*int(sqrt( - sin(e + f*x) + 1)*sin(e + f*x),x))