3.316 \(\int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac {2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2671} \[ -\frac {2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

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

Rule 2671

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)/(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]

Rubi steps

\begin {align*} \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2 (e \cos (c+d x))^{5/2}}{5 d e (a+a \sin (c+d x))^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 49, normalized size = 1.36 \[ -\frac {2 \sqrt {a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2}}{5 a^3 d e (\sin (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

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

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fricas [B]  time = 0.67, size = 70, normalized size = 1.94 \[ -\frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (e \sin \left (d x + c\right ) - e\right )}}{5 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.19, size = 34, normalized size = 0.94 \[ -\frac {2 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right )}{5 d \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x)

[Out]

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

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maxima [B]  time = 0.96, size = 131, normalized size = 3.64 \[ -\frac {2 \, {\left (\sqrt {a} e^{\frac {3}{2}} - \frac {\sqrt {a} e^{\frac {3}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{5 \, {\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-2/5*(sqrt(a)*e^(3/2) - sqrt(a)*e^(3/2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(-sin(d*x + c)/(cos(d*x + c) + 1)
+ 1)^(3/2)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*d*(sin(
d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))

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mupad [B]  time = 6.57, size = 102, normalized size = 2.83 \[ -\frac {4\,e\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (c+d\,x\right )+2\,\cos \left (2\,c+2\,d\,x\right )+\sin \left (3\,c+3\,d\,x\right )+2\right )}{5\,a^3\,d\,\left (56\,\sin \left (c+d\,x\right )-28\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )-8\,\sin \left (3\,c+3\,d\,x\right )+35\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(c + d*x))^(3/2)/(a + a*sin(c + d*x))^(5/2),x)

[Out]

-(4*e*(e*cos(c + d*x))^(1/2)*(a*(sin(c + d*x) + 1))^(1/2)*(sin(c + d*x) + 2*cos(2*c + 2*d*x) + sin(3*c + 3*d*x
) + 2))/(5*a^3*d*(56*sin(c + d*x) - 28*cos(2*c + 2*d*x) + cos(4*c + 4*d*x) - 8*sin(3*c + 3*d*x) + 35))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))**(3/2)/(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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