\(\int \frac {(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx\) [237]

Optimal result

Integrand size = 25, antiderivative size = 66 \[ \int \frac {(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx=\frac {2 e \sqrt {e \cos (c+d x)}}{a d}+\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{a d \sqrt {e \cos (c+d x)}} \]

[Out]

2*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
/d/(e*cos(d*x+c))^(1/2)+2*e*(e*cos(d*x+c))^(1/2)/a/d

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2761, 2721, 2720} \[ \int \frac {(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx=\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{a d \sqrt {e \cos (c+d x)}}+\frac {2 e \sqrt {e \cos (c+d x)}}{a d} \]

[In]

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

[Out]

(2*e*Sqrt[e*Cos[c + d*x]])/(a*d) + (2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(a*d*Sqrt[e*Cos[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 2761

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 e \sqrt {e \cos (c+d x)}}{a d}+\frac {e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{a} \\ & = \frac {2 e \sqrt {e \cos (c+d x)}}{a d}+\frac {\left (e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{a \sqrt {e \cos (c+d x)}} \\ & = \frac {2 e \sqrt {e \cos (c+d x)}}{a d}+\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{a d \sqrt {e \cos (c+d x)}} \\ \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 = 1.00 \[ \int \frac {(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx=-\frac {2 \sqrt [4]{2} (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{5 a d e (1+\sin (c+d x))^{5/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {2 e^{2} \left (\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(110\)
risch	\(\frac {\sqrt {2}\, e \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}}{d a}+\frac {2 \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {i \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, F\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right ) e \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}\, \sqrt {e \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right ) {\mathrm e}^{i \left (d x +c \right )}}}{d \sqrt {{\mathrm e}^{3 i \left (d x +c \right )} e +{\mathrm e}^{i \left (d x +c \right )} e}\, a \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}\)	\(217\)

[In]

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

[Out]

-2/a/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^2*((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))+2*sin(1/2*d*x+1/2*c)^3-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 = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11 \[ \int \frac {(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx=\frac {-i \, \sqrt {2} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {e \cos \left (d x + c\right )} e}{a d} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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