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

Optimal. Leaf size=75 \[ \frac{2 \sqrt{a \cos (c+d x)+b \sin (c+d x)} E\left (\left .\frac{1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{d \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}}} \]

[Out]

(2*EllipticE[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(d*Sqrt[(a*Cos[c + d*x] + b
*Sin[c + d*x])/Sqrt[a^2 + b^2]])

________________________________________________________________________________________

Rubi [A]  time = 0.0296799, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3078, 2639} \[ \frac{2 \sqrt{a \cos (c+d x)+b \sin (c+d x)} E\left (\left .\frac{1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{d \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticE[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(d*Sqrt[(a*Cos[c + d*x] + b
*Sin[c + d*x])/Sqrt[a^2 + b^2]])

Rule 3078

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[c + d*x] +
b*Sin[c + d*x])^n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n, Int[Cos[c + d*x - ArcTan[a, b]]^n, x]
, x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) &&  !(GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0])

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

Mathematica [C]  time = 1.14195, size = 268, normalized size = 3.57 \[ \frac{\cos \left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right ) \left (\sqrt{\sin ^2\left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right )} \left (b \left (a^2+b^2\right ) \sin \left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right )-2 a \left (a^2+b^2\right ) \cos \left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right )+2 a^2 \sqrt{\frac{b^2}{a^2}+1} \sqrt{a \cos (c+d x)+b \sin (c+d x)} \sqrt{a \sqrt{\frac{b^2}{a^2}+1} \cos \left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right )}\right )-b \left (a^2+b^2\right ) \sin \left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right )\right )\right )}{b d \sqrt{\sin ^2\left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right )} \left (a \sqrt{\frac{b^2}{a^2}+1} \cos \left (-\tan ^{-1}\left (\frac{b}{a}\right )+c+d x\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[c + d*x - ArcTan[b/a]]*(-(b*(a^2 + b^2)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[b/a]]
^2]*Sin[c + d*x - ArcTan[b/a]]) + Sqrt[Sin[c + d*x - ArcTan[b/a]]^2]*(-2*a*(a^2 + b^2)*Cos[c + d*x - ArcTan[b/
a]] + 2*a^2*Sqrt[1 + b^2/a^2]*Sqrt[a*Sqrt[1 + b^2/a^2]*Cos[c + d*x - ArcTan[b/a]]]*Sqrt[a*Cos[c + d*x] + b*Sin
[c + d*x]] + b*(a^2 + b^2)*Sin[c + d*x - ArcTan[b/a]])))/(b*d*(a*Sqrt[1 + b^2/a^2]*Cos[c + d*x - ArcTan[b/a]])
^(3/2)*Sqrt[Sin[c + d*x - ArcTan[b/a]]^2])

________________________________________________________________________________________

Maple [A]  time = 1.204, size = 159, normalized size = 2.1 \begin{align*} -{\frac{1}{\cos \left ( dx+c-\arctan \left ( -a,b \right ) \right ) d}\sqrt{{a}^{2}+{b}^{2}}\sqrt{1+\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) }\sqrt{-2\,\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) +2}\sqrt{-\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1+\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1+\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{\sin \left ( dx+c-\arctan \left ( -a,b \right ) \right ) \sqrt{{a}^{2}+{b}^{2}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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