3.4.1 \(\int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}} \, dx\) [301]

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

[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 = 0.04, 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 = {2750} \begin {gather*} -\frac {2 \sqrt {e \cos (c+d x)}}{d e \sqrt {a \sin (c+d x)+a}} \end {gather*}

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 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 {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 = 0.06, size = 34, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {e \cos (c+d x)}}{d e \sqrt {a (1+\sin (c+d x))}} \end {gather*}

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 = 0.34, size = 34, normalized size = 1.00

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

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,method=_RETURNVERBOSE)

[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] Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (27) = 54\).
time = 0.56, size = 122, normalized size = 3.59 \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 )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )} e^{\left (-\frac {1}{2}\right )}}{{\left (a + \frac {a \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 {gather*}

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

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Fricas [A]
time = 0.35, size = 41, normalized size = 1.21 \begin {gather*} -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{a d e^{\frac {1}{2}} \sin \left (d x + c\right ) + a d e^{\frac {1}{2}}} \end {gather*}

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

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

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

[Out]

Timed out

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Mupad [B]
time = 5.66, size = 46, normalized size = 1.35 \begin {gather*} -\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 )} \end {gather*}

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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