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

Optimal. Leaf size=66 \[ \frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)}}{d e} \]

[Out]

(2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (2*b*Sqrt[e*Sin[c + d*x]]
)/(d*e)

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Rubi [A]  time = 0.0509949, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2669, 2642, 2641} \[ \frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)}}{d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (2*b*Sqrt[e*Sin[c + d*x]]
)/(d*e)

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{a+b \cos (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx &=\frac{2 b \sqrt{e \sin (c+d x)}}{d e}+a \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx\\ &=\frac{2 b \sqrt{e \sin (c+d x)}}{d e}+\frac{\left (a \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{\sqrt{e \sin (c+d x)}}\\ &=\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)}}{d e}\\ \end{align*}

Mathematica [A]  time = 0.201985, size = 54, normalized size = 0.82 \[ \frac{2 \left (b \sin (c+d x)-a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{d \sqrt{e \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(a*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]]) + b*Sin[c + d*x]))/(d*Sqrt[e*Sin[c + d*x]])

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Maple [A]  time = 1.162, size = 92, normalized size = 1.4 \begin{align*} -{\frac{1}{d\cos \left ( dx+c \right ) } \left ( a\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ) -2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) b \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*cos(d*x + c) + a)*sqrt(e*sin(d*x + c))/(e*sin(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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