3.4.8 \(\int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^{3/2}} \, dx\) [308]

Optimal. Leaf size=36 \[ -\frac {2 (e \cos (c+d x))^{3/2}}{3 d e (a+a \sin (c+d x))^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.05, 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 = {2750} \begin {gather*} -\frac {2 (e \cos (c+d x))^{3/2}}{3 d e (a \sin (c+d x)+a)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 49, normalized size = 1.36 \begin {gather*} -\frac {2 (e \cos (c+d x))^{3/2} \sqrt {a (1+\sin (c+d x))}}{3 a^2 d e (1+\sin (c+d x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 34, normalized size = 0.94

method	result	size
default	\(-\frac {2 \sqrt {e \cos \left (d x +c \right )}\, \cos \left (d x +c \right )}{3 d \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (27) = 54\).
time = 0.54, size = 126, normalized size = 3.50 \begin {gather*} -\frac {2 \, {\left (\sqrt {a} - \frac {\sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\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 )} e^{\frac {1}{2}}}{3 \, {\left (a^{2} + \frac {a^{2} \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 {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (27) = 54\).
time = 0.34, size = 104, normalized size = 2.89 \begin {gather*} \frac {2 \, {\left (\cos \left (d x + c\right ) e^{\frac {1}{2}} - e^{\frac {1}{2}} \sin \left (d x + c\right ) + e^{\frac {1}{2}}\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d - {\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e \cos {\left (c + d x \right )}}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [B]
time = 5.99, size = 82, normalized size = 2.28 \begin {gather*} -\frac {4\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (2\,\cos \left (c+d\,x\right )+\sin \left (2\,c+2\,d\,x\right )\right )}{3\,a^2\,d\,\left (15\,\sin \left (c+d\,x\right )-6\,\cos \left (2\,c+2\,d\,x\right )-\sin \left (3\,c+3\,d\,x\right )+10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*(e*cos(c + d*x))^(1/2)*(a*(sin(c + d*x) + 1))^(1/2)*(2*cos(c + d*x) + sin(2*c + 2*d*x)))/(3*a^2*d*(15*sin(
c + d*x) - 6*cos(2*c + 2*d*x) - sin(3*c + 3*d*x) + 10))

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