\(\int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx\) [259]

Optimal result

Integrand size = 25, antiderivative size = 118 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}-\frac {4 e \sqrt {e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{21 d \left (a^3+a^3 \sin (c+d x)\right )} \]

[Out]

-2/21*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/
2)/a^3/d/(e*cos(d*x+c))^(1/2)-4/7*e*(e*cos(d*x+c))^(1/2)/a/d/(a+a*sin(d*x+c))^2+2/21*e*(e*cos(d*x+c))^(1/2)/d/
(a^3+a^3*sin(d*x+c))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2759, 2762, 2721, 2720} \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sqrt {e \cos (c+d x)}}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {4 e \sqrt {e \cos (c+d x)}}{7 a d (a \sin (c+d x)+a)^2} \]

[In]

Int[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^3,x]

[Out]

(-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a^3*d*Sqrt[e*Cos[c + d*x]]) - (4*e*Sqrt[e*Cos[c + d*
x]])/(7*a*d*(a + a*Sin[c + d*x])^2) + (2*e*Sqrt[e*Cos[c + d*x]])/(21*d*(a^3 + a^3*Sin[c + d*x]))

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 2762

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*((g*Cos[e
+ f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*Sin[e + f*x]))), x] + Dist[p/(a*(p - 1)), Int[(g*Cos[e + f*x])^p, x], x
] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] &&  !GeQ[p, 1] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 e \sqrt {e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}-\frac {e^2 \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))} \, dx}{7 a^2} \\ & = -\frac {4 e \sqrt {e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{21 a^3} \\ & = -\frac {4 e \sqrt {e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {\left (e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^3 \sqrt {e \cos (c+d x)}} \\ & = -\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}-\frac {4 e \sqrt {e \cos (c+d x)}}{7 a d (a+a \sin (c+d x))^2}+\frac {2 e \sqrt {e \cos (c+d x)}}{21 d \left (a^3+a^3 \sin (c+d x)\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.56 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx=-\frac {(e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {11}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{5\ 2^{3/4} a^3 d e (1+\sin (c+d x))^{5/4}} \]

[In]

Integrate[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^3,x]

[Out]

-1/5*((e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[5/4, 11/4, 9/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^3*d*e*(1 + Si
n[c + d*x])^(5/4))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(130)=260\).

Time = 5.24 (sec) , antiderivative size = 401, normalized size of antiderivative = 3.40

method	result	size
default	\(\frac {2 \left (8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-28 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-22 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\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}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+28 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{2}}{21 \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) a^{3} \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}\)	\(401\)

[In]

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

[Out]

2/21/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/a^3/sin(1/2*d*x+1/2*c)/(-2*sin(
1/2*d*x+1/2*c)^2*e+e)^(1/2)*(8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d
*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^6+8*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*(sin(1/2*d*x+1/2*c)^2)^
(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-8*sin(1/2*d*
x+1/2*c)^4*cos(1/2*d*x+1/2*c)+6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*
d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2-28*sin(1/2*d*x+1/2*c)^5-22*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-
(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+28*sin(1/2
*d*x+1/2*c)^3+5*sin(1/2*d*x+1/2*c))*e^2/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 = 171, normalized size of antiderivative = 1.45 \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx=-\frac {{\left (-i \, \sqrt {2} e \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} e \sin \left (d x + c\right ) + 2 i \, \sqrt {2} e\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (i \, \sqrt {2} e \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} e \sin \left (d x + c\right ) - 2 i \, \sqrt {2} e\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {e \cos \left (d x + c\right )} {\left (e \sin \left (d x + c\right ) - 5 \, e\right )}}{21 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]

[In]

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

[Out]

-1/21*((-I*sqrt(2)*e*cos(d*x + c)^2 + 2*I*sqrt(2)*e*sin(d*x + c) + 2*I*sqrt(2)*e)*sqrt(e)*weierstrassPInverse(
-4, 0, cos(d*x + c) + I*sin(d*x + c)) + (I*sqrt(2)*e*cos(d*x + c)^2 - 2*I*sqrt(2)*e*sin(d*x + c) - 2*I*sqrt(2)
*e)*sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*sqrt(e*cos(d*x + c))*(e*sin(d*x + c)
- 5*e))/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x + c) - 2*a^3*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]

[In]

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

[Out]

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