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

Optimal. Leaf size=108 \[ \frac {2 \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3119, 2653} \[ \frac {2 \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3119

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rubi steps

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

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Mathematica [C]  time = 6.23, size = 1408, normalized size = 13.04 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*b*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(c*e) + (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/
c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c
^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[
b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c
^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt
[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*e) + (b^2*
(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b
^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^
2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)
/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^
2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x
- ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])
)/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x
- ArcTan[c/b]]]))/(c*e) + (c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - A
rcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - Ar
cTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2
/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c
^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt
[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^
2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b
*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/e

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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.46, size = 720, normalized size = 6.67 \[ \frac {2 \left (-a +\sqrt {b^{2}+c^{2}}\right ) \sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a}{-a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {-\frac {\left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right ) \sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {\frac {\left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right ) \sqrt {b^{2}+c^{2}}}{-a +\sqrt {b^{2}+c^{2}}}}\, \sqrt {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right ) \left (\cos ^{2}\left (e x +d -\arctan \left (-b , c\right )\right )\right )+\left (\cos ^{2}\left (e x +d -\arctan \left (-b , c\right )\right )\right ) a}\, \left (\sqrt {b^{2}+c^{2}}\, \EllipticE \left (\sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )}{-a +\sqrt {b^{2}+c^{2}}}-\frac {a}{-a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {-a +\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right )-\sqrt {b^{2}+c^{2}}\, \EllipticF \left (\sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )}{-a +\sqrt {b^{2}+c^{2}}}-\frac {a}{-a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {-a +\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right )+\EllipticE \left (\sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )}{-a +\sqrt {b^{2}+c^{2}}}-\frac {a}{-a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {-a +\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) a -\EllipticF \left (\sqrt {-\frac {\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )}{-a +\sqrt {b^{2}+c^{2}}}-\frac {a}{-a +\sqrt {b^{2}+c^{2}}}}, \sqrt {-\frac {-a +\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) a \right )}{\sqrt {b^{2}+c^{2}}\, \sqrt {\left (\cos ^{2}\left (e x +d -\arctan \left (-b , c\right )\right )\right ) \left (\sqrt {b^{2}+c^{2}}\, \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a \right )}\, \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+a \sqrt {b^{2}+c^{2}}}{\sqrt {b^{2}+c^{2}}}}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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