3.439 \(\int \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]

[Out]

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

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Rubi [A]  time = 0.0381648, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {3112} \[ -\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]

[Out]

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

Rule 3112

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Simp[(-2*(c*Cos[d
+ e*x] - b*Sin[d + e*x]))/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]), x] /; FreeQ[{a, b, c, d, e}, x] && E
qQ[a^2 - b^2 - c^2, 0]

Rubi steps

\begin{align*} \int \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx &=-\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}\\ \end{align*}

Mathematica [C]  time = 21.5246, size = 11415, normalized size = 200.26 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]

[Out]

Result too large to show

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Maple [B]  time = 1.706, size = 117, normalized size = 2.1 \begin{align*} 2\,{\frac{\sqrt{{b}^{2}+{c}^{2}} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -1 \right ) \left ( 1+\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) }{\cos \left ( ex+d-\arctan \left ( -b,c \right ) \right ) e}{\frac{1}{\sqrt{{\frac{{b}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) +{c}^{2}\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -{b}^{2}-{c}^{2}}{\sqrt{{b}^{2}+{c}^{2}}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.86122, size = 201, normalized size = 3.53 \begin{align*} \frac{2 \,{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt{b^{2} + c^{2}}\right )} \sqrt{b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) - \sqrt{b^{2} + c^{2}}}}{c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*cos(e*x + d) + c*sin(e*x + d) + sqrt(b^2 + c^2))*sqrt(b*cos(e*x + d) + c*sin(e*x + d) - sqrt(b^2 + c^2))/
(c*e*cos(e*x + d) - b*e*sin(e*x + d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b*cos(d + e*x) + c*sin(d + e*x) - sqrt(b**2 + c**2)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError