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

Optimal. Leaf size=68 \[ \frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e} \]

[Out]

(2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) + (2*b*(e*Sin[c + d*x])^(3/
2))/(3*d*e)

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Rubi [A]  time = 0.0489842, antiderivative size = 68, 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, 2640, 2639} \[ \frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) + (2*b*(e*Sin[c + d*x])^(3/
2))/(3*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 2640

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

Rule 2639

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

Rubi steps

\begin{align*} \int (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)} \, dx &=\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e}+a \int \sqrt{e \sin (c+d x)} \, dx\\ &=\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e}+\frac{\left (a \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{\sqrt{\sin (c+d x)}}\\ &=\frac{2 a E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e}\\ \end{align*}

Mathematica [A]  time = 0.097279, size = 60, normalized size = 0.88 \[ \frac{2 \sqrt{e \sin (c+d x)} \left (b \sin ^{\frac{3}{2}}(c+d x)-3 a E\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{3 d \sqrt{\sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*Sin[c + d*x]]*(-3*a*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + b*Sin[c + d*x]^(3/2)))/(3*d*Sqrt[Sin[c + d
*x]])

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Maple [A]  time = 1.609, size = 117, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ({\frac{2\,b}{3\,e} \left ( e\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{ae}{\cos \left ( dx+c \right ) }\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \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]

(2/3*b/e*(e*sin(d*x+c))^(3/2)-a*e*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*(2*EllipticE((1
-sin(d*x+c))^(1/2),1/2*2^(1/2))-EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2)))/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/
d

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

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

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