Integrand size = 25, antiderivative size = 140 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}+\frac {22 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e} \] Output:
-22/15*a^3*(e*cos(d*x+c))^(3/2)/d/e+22/5*a^3*(e*cos(d*x+c))^(1/2)*Elliptic E(sin(1/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1/2)-2/7*a*(e*cos(d*x+c))^(3/2 )*(a+a*sin(d*x+c))^2/d/e-22/35*(e*cos(d*x+c))^(3/2)*(a^3+a^3*sin(d*x+c))/d /e
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.47 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=-\frac {16\ 2^{3/4} a^3 (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (1+\sin (c+d x))^{3/4}} \] Input:
Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3,x]
Output:
(-16*2^(3/4)*a^3*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-11/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 + Sin[c + d*x])^(3/4))
Time = 0.67 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3157, 3042, 3157, 3042, 3148, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {11}{7} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^2dx-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{7} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^2dx-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)dx-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)dx-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (a \int \sqrt {e \cos (c+d x)}dx-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (a \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\) |
Input:
Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3,x]
Output:
(-2*a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2)/(7*d*e) + (11*a*((7*a *((-2*a*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (2*a*Sqrt[e*Cos[c + d*x]]*Ellipt icE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])))/5 - (2*(e*Cos[c + d*x])^(3/2 )*(a^2 + a^2*Sin[c + d*x]))/(5*d*e)))/7
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
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 + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Time = 0.94 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.53
| method | result | size |
| default | \(-\frac {2 a^{3} e \left (-240 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+504 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+480 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-504 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+200 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+126 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-231 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-440 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+125 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(214\) |
| parts | \(\frac {2 a^{3} \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, e \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}+\frac {2 a^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,e^{3}}-\frac {2 a^{3} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{d e}+\frac {12 a^{3} \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, e \left (4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}\) | \(402\) |
Input:
int((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-2/105/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a^3*e*(-240* sin(1/2*d*x+1/2*c)^9+504*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+480*sin(1 /2*d*x+1/2*c)^7-504*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+200*sin(1/2*d* x+1/2*c)^5+126*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-231*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c ),2^(1/2))-440*sin(1/2*d*x+1/2*c)^3+125*sin(1/2*d*x+1/2*c))/d
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=-\frac {2 \, {\left (-231 i \, \sqrt {\frac {1}{2}} a^{3} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {\frac {1}{2}} a^{3} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (15 \, a^{3} \cos \left (d x + c\right )^{3} - 63 \, a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 140 \, a^{3} \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{105 \, d} \] Input:
integrate((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-2/105*(-231*I*sqrt(1/2)*a^3*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPIn verse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*I*sqrt(1/2)*a^3*sqrt(e) *weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d* x + c))) - (15*a^3*cos(d*x + c)^3 - 63*a^3*cos(d*x + c)*sin(d*x + c) - 140 *a^3*cos(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(1/2)*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
integrate(sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + a)^3, x)
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
integrate(sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + a)^3, x)
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^3,x)
Output:
int((e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^3, x)
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=\frac {\sqrt {e}\, a^{3} \left (-2 \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )+\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) d +\left (\int \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )^{3}d x \right ) d +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )^{2}d x \right ) d \right )}{d} \] Input:
int((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^3,x)
Output:
(sqrt(e)*a**3*( - 2*sqrt(cos(c + d*x))*cos(c + d*x) + int(sqrt(cos(c + d*x )),x)*d + int(sqrt(cos(c + d*x))*sin(c + d*x)**3,x)*d + 3*int(sqrt(cos(c + d*x))*sin(c + d*x)**2,x)*d))/d