\(\int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx\) [217]

Optimal result

Integrand size = 25, antiderivative size = 140 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}+\frac {22 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e} \]

[Out]

-22/15*a^3*(e*cos(d*x+c))^(3/2)/d/e-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+22/5*a^3*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2
*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2757, 2748, 2721, 2719} \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {22 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{3/2}}{35 d e}+\frac {22 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e} \]

[In]

Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3,x]

[Out]

(-22*a^3*(e*Cos[c + d*x])^(3/2))/(15*d*e) + (22*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[
Cos[c + d*x]]) - (2*a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2)/(7*d*e) - (22*(e*Cos[c + d*x])^(3/2)*(a^3
+ a^3*Sin[c + d*x]))/(35*d*e)

Rule 2719

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]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[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])

Rule 2757

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] + Dist[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] && IntegersQ[2*m, 2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}+\frac {1}{7} (11 a) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx \\ & = -\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}+\frac {1}{5} \left (11 a^2\right ) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x)) \, dx \\ & = -\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\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}+\frac {1}{5} \left (11 a^3\right ) \int \sqrt {e \cos (c+d x)} \, dx \\ & = -\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\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}+\frac {\left (11 a^3 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}} \\ & = -\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} \\ \end{align*}

Mathematica [C] (verified)

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 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=-\frac {16\ 2^{3/4} a^3 (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (1+\sin (c+d x))^{3/4}} \]

[In]

Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3,x]

[Out]

(-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))

Maple [A] (verified)

Time = 4.72 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.53

method	result	size
default	\(-\frac {2 a^{3} e \left (-240 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+480 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+200 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+126 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-231 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-440 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+125 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(214\)
parts	\(\frac {2 a^{3} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 a^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,e^{3}}-\frac {2 a^{3} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{d e}+\frac {12 a^{3} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e \left (4 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}\)	\(402\)

[In]

int((a+a*sin(d*x+c))^3*(e*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=\frac {231 i \, \sqrt {2} a^{3} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} a^{3} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{3} - 63 \, a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 140 \, a^{3} \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{105 \, d} \]

[In]

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

[Out]

1/105*(231*I*sqrt(2)*a^3*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x +
c))) - 231*I*sqrt(2)*a^3*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x +
c))) + 2*(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

Sympy [F(-1)]

Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]

[In]

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

[Out]

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