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

Optimal. Leaf size=55 \[ -\frac {2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, 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*}

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Mathematica [C]  time = 32.71, size = 11586, normalized size = 210.65 \[ \text {Result too large to show} \]

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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fricas [A]  time = 0.91, size = 80, normalized size = 1.45 \[ \frac {2 \, \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}} {\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) - \sqrt {b^{2} + c^{2}}\right )}}{c e \cos \left (e x + d\right ) - b e \sin \left (e x + d\right )} \]

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*sqrt(b*cos(e*x + d) + c*sin(e*x + d) + sqrt(b^2 + c^2))*(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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Simp
lification assuming b near 0sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argum
ent Value

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maple [B]  time = 0.36, size = 113, normalized size = 2.05 \[ \frac {2 \left (1+\sin \left (e x +d -\arctan \left (-b , c\right )\right )\right ) \sqrt {b^{2}+c^{2}}\, \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right )}{\cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+b^{2}+c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e} \]

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*(1+sin(e*x+d-arctan(-b,c)))*(b^2+c^2)^(1/2)*(sin(e*x+d-arctan(-b,c))-1)/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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )+\sqrt {b^2+c^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} + \sqrt {b^{2} + c^{2}}}\, dx \]

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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