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

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Rubi [A]  time = 0.06, antiderivative size = 34, 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} \[ -\frac {2 \sqrt {e \cos (c+d x)}}{d e \sqrt {a \sin (c+d x)+a}} \]

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*}

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Mathematica [A]  time = 0.07, size = 34, normalized size = 1.00 \[ -\frac {2 \sqrt {e \cos (c+d x)}}{d e \sqrt {a (\sin (c+d x)+1)}} \]

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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fricas [A]  time = 0.82, size = 41, normalized size = 1.21 \[ -\frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{a d e \sin \left (d x + c\right ) + a d e} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]

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)

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maple [A]  time = 0.17, size = 34, normalized size = 1.00 \[ -\frac {2 \cos \left (d x +c \right )}{d \sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+\sin \left (d x +c \right )\right )}} \]

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 = 0.81, size = 130, normalized size = 3.82 \[ -\frac {2 \, {\left (\sqrt {a} \sqrt {e} - \frac {\sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{{\left (a e + \frac {a e \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} \sqrt {-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \]

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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mupad [B]  time = 5.66, size = 46, normalized size = 1.35 \[ -\frac {2\,\cos \left (c+d\,x\right )\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}}{a\,d\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (\sin \left (c+d\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sqrt {e \cos {\left (c + d x \right )}}}\, dx \]

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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