\(\int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx\) [317]

Optimal result

Integrand size = 27, antiderivative size = 76 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 (e \cos (c+d x))^{3/2}}{7 d e (a+a \sin (c+d x))^{5/2}}-\frac {4 (e \cos (c+d x))^{3/2}}{21 a d e (a+a \sin (c+d x))^{3/2}} \]

[Out]

-2/7*(e*cos(d*x+c))^(3/2)/d/e/(a+a*sin(d*x+c))^(5/2)-4/21*(e*cos(d*x+c))^(3/2)/a/d/e/(a+a*sin(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {4 (e \cos (c+d x))^{3/2}}{21 a d e (a \sin (c+d x)+a)^{3/2}}-\frac {2 (e \cos (c+d x))^{3/2}}{7 d e (a \sin (c+d x)+a)^{5/2}} \]

[In]

Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(e*Cos[c + d*x])^(3/2))/(7*d*e*(a + a*Sin[c + d*x])^(5/2)) - (4*(e*Cos[c + d*x])^(3/2))/(21*a*d*e*(a + a*S
in[c + d*x])^(3/2))

Rule 2750

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

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), 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] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e \cos (c+d x))^{3/2}}{7 d e (a+a \sin (c+d x))^{5/2}}+\frac {2 \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{3/2}} \, dx}{7 a} \\ & = -\frac {2 (e \cos (c+d x))^{3/2}}{7 d e (a+a \sin (c+d x))^{5/2}}-\frac {4 (e \cos (c+d x))^{3/2}}{21 a d e (a+a \sin (c+d x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 (e \cos (c+d x))^{3/2} \sqrt {a (1+\sin (c+d x))} (5+2 \sin (c+d x))}{21 a^3 d e (1+\sin (c+d x))^3} \]

[In]

Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(e*Cos[c + d*x])^(3/2)*Sqrt[a*(1 + Sin[c + d*x])]*(5 + 2*Sin[c + d*x]))/(21*a^3*d*e*(1 + Sin[c + d*x])^3)

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.51

method	result	size
default	\(\frac {2 \left (2 \left (\cos ^{2}\left (d x +c \right )\right )+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )+5 \cos \left (d x +c \right )-3 \sin \left (d x +c \right )+3\right ) \sqrt {e \cos \left (d x +c \right )}}{21 d \left (\cos ^{2}\left (d x +c \right )-\cos \left (d x +c \right ) \sin \left (d x +c \right )-\cos \left (d x +c \right )-2 \sin \left (d x +c \right )-2\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, a^{2}}\)	\(115\)

[In]

int((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/21/d*(2*cos(d*x+c)^2+2*cos(d*x+c)*sin(d*x+c)+5*cos(d*x+c)-3*sin(d*x+c)+3)*(e*cos(d*x+c))^(1/2)/(cos(d*x+c)^2
-cos(d*x+c)*sin(d*x+c)-cos(d*x+c)-2*sin(d*x+c)-2)/(a*(1+sin(d*x+c)))^(1/2)/a^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (64) = 128\).

Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (2 \, \cos \left (d x + c\right )^{2} + {\left (2 \, \cos \left (d x + c\right ) - 3\right )} \sin \left (d x + c\right ) + 5 \, \cos \left (d x + c\right ) + 3\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{21 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

2/21*sqrt(e*cos(d*x + c))*(2*cos(d*x + c)^2 + (2*cos(d*x + c) - 3)*sin(d*x + c) + 5*cos(d*x + c) + 3)*sqrt(a*s
in(d*x + c) + a)/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(
d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))

Sympy [F]

\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {e \cos {\left (c + d x \right )}}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((e*cos(d*x+c))**(1/2)/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Integral(sqrt(e*cos(c + d*x))/(a*(sin(c + d*x) + 1))**(5/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (64) = 128\).

Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (5 \, \sqrt {a} \sqrt {e} + \frac {4 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} \sqrt {-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{21 \, {\left (a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}}} \]

[In]

integrate((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-2/21*(5*sqrt(a)*sqrt(e) + 4*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 4*sqrt(a)*sqrt(e)*sin(d*x + c)^
3/(cos(d*x + c) + 1)^3 - 5*sqrt(a)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*sqrt(-sin(d*x + c)/(cos(d*x +
c) + 1) + 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^2/((a^3 + 2*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a
^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2))

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 6.52 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\sqrt {-e\,\left (2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}\,\left (-58\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+18\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2+26\,\sin \left (2\,c+2\,d\,x\right )-\sin \left (4\,c+4\,d\,x\right )+20\right )}{21\,a^3\,d\,\left (240\,{\sin \left (c+d\,x\right )}^2+210\,\sin \left (c+d\,x\right )-20\,{\sin \left (2\,c+2\,d\,x\right )}^2-45\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (5\,c+5\,d\,x\right )+16\right )} \]

[In]

int((e*cos(c + d*x))^(1/2)/(a + a*sin(c + d*x))^(5/2),x)

[Out]

-(8*(a*(sin(c + d*x) + 1))^(1/2)*(-e*(2*sin(c/2 + (d*x)/2)^2 - 1))^(1/2)*(26*sin(2*c + 2*d*x) - sin(4*c + 4*d*
x) - 58*sin(c/2 + (d*x)/2)^2 + 18*sin((3*c)/2 + (3*d*x)/2)^2 + 20))/(21*a^3*d*(210*sin(c + d*x) - 45*sin(3*c +
3*d*x) + sin(5*c + 5*d*x) - 20*sin(2*c + 2*d*x)^2 + 240*sin(c + d*x)^2 + 16))