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

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

[Out]

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

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Rubi [A]  time = 0.0681025, 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} \[ \frac{2 \sqrt{a \sin (c+d x)+a}}{d e \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + a*Sin[c + d*x]])/(d*e*Sqrt[e*Cos[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{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx &=\frac{2 \sqrt{a+a \sin (c+d x)}}{d e \sqrt{e \cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.116662, size = 34, normalized size = 1. \[ \frac{2 \sqrt{a (\sin (c+d x)+1)}}{d e \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.123, size = 34, normalized size = 1. \begin{align*} 2\,{\frac{\cos \left ( dx+c \right ) \sqrt{a \left ( 1+\sin \left ( dx+c \right ) \right ) }}{d \left ( e\cos \left ( dx+c \right ) \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.5534, size = 177, normalized size = 5.21 \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 (e^{2} + \frac{e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 2.56197, size = 95, normalized size = 2.79 \begin{align*} \frac{2 \, \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{d e^{2} \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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