3.301 \(\int \frac{1}{\sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a \sin (c+d x)+a}} \]

[Out]

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

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Rubi [A]  time = 0.0604187, antiderivative size = 34, normalized size of antiderivative = 1., 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} \[ -\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]),x]

[Out]

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

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

Mathematica [A]  time = 0.0641767, size = 34, normalized size = 1. \[ -\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a (\sin (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]),x]

[Out]

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

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Maple [A]  time = 0.102, size = 34, normalized size = 1. \begin{align*} -2\,{\frac{\cos \left ( dx+c \right ) }{d\sqrt{e\cos \left ( dx+c \right ) }\sqrt{a \left ( 1+\sin \left ( dx+c \right ) \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.58804, size = 176, normalized size = 5.18 \begin{align*} -\frac{2 \,{\left (\sqrt{a} \sqrt{e} - \frac{\sqrt{a} \sqrt{e} \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 )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{{\left (a e + \frac{a e \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{3}{2}} \sqrt{-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.43498, size = 107, normalized size = 3.15 \begin{align*} -\frac{2 \, \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{a d e \sin \left (d x + c\right ) + a d e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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