3.593 \(\int \sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx\)

Optimal. Leaf size=79 \[ \frac{\sqrt{a+(b \sin (e+f x)+c \cos (e+f x))^2} E\left (e+f x+\tan ^{-1}(b,c)|-\frac{b^2+c^2}{a}\right )}{f \sqrt{\frac{(b \sin (e+f x)+c \cos (e+f x))^2}{a}+1}} \]

[Out]

(EllipticE[e + f*x + ArcTan[b, c], -((b^2 + c^2)/a)]*Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2])/(f*Sqrt[1
+ (c*Cos[e + f*x] + b*Sin[e + f*x])^2/a])

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Rubi [F]  time = 0.698582, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \sqrt{a+(c \cos (e+f x)+b \sin (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2],x]

[Out]

((I/2)*Defer[Subst][Defer[Int][Sqrt[a + (c + b*x)^2/(1 + x^2)]/(I - x), x], x, Tan[e + f*x]])/f + ((I/2)*Defer
[Subst][Defer[Int][Sqrt[a + (c + b*x)^2/(1 + x^2)]/(I + x), x], x, Tan[e + f*x]])/f

Rubi steps

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

Mathematica [B]  time = 1.63885, size = 325, normalized size = 4.11 \[ -\frac{\left (\left (b^2-c^2\right ) \sin (2 (e+f x))+2 b c \cos (2 (e+f x))\right ) \sqrt{2 a+\left (c^2-b^2\right ) \cos (2 (e+f x))+b^2+2 b c \sin (2 (e+f x))+c^2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (b^2-c^2\right ) \cos (2 (e+f x))-2 b c \sin (2 (e+f x))+\sqrt{\left (b^2+c^2\right )^2}}{\sqrt{\left (b^2+c^2\right )^2}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{\left (b^2+c^2\right )^2}}{b^2+c^2+2 a+\sqrt{\left (b^2+c^2\right )^2}}\right )}{\sqrt{2} f \sqrt{\left (b^2+c^2\right )^2} \sqrt{\frac{\left (\left (b^2-c^2\right ) \sin (2 (e+f x))+2 b c \cos (2 (e+f x))\right )^2}{\left (b^2+c^2\right )^2}} \sqrt{\frac{2 a+\left (c^2-b^2\right ) \cos (2 (e+f x))+b^2+2 b c \sin (2 (e+f x))+c^2}{2 a+\sqrt{\left (b^2+c^2\right )^2}+b^2+c^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2],x]

[Out]

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

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Maple [B]  time = 1.51, size = 4061599, normalized size = 51412.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt((c*cos(f*x + e) + b*sin(f*x + e))^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{2 \, b c \cos \left (f x + e\right ) \sin \left (f x + e\right ) -{\left (b^{2} - c^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(2*b*c*cos(f*x + e)*sin(f*x + e) - (b^2 - c^2)*cos(f*x + e)^2 + b^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + (b*sin(e + f*x) + c*cos(e + f*x))**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt((c*cos(f*x + e) + b*sin(f*x + e))^2 + a), x)