3.434 \(\int \frac{1}{\sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx\)

Optimal. Leaf size=88 \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )+\sqrt{b^2+c^2}}}\right )}{e \sqrt [4]{b^2+c^2}} \]

[Out]

(Sqrt[2]*ArcTanh[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^
2]*Cos[d + e*x - ArcTan[b, c]]])])/((b^2 + c^2)^(1/4)*e)

________________________________________________________________________________________

Rubi [A]  time = 0.119567, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {3115, 2649, 206} \[ \frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )+\sqrt{b^2+c^2}}}\right )}{e \sqrt [4]{b^2+c^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]

[Out]

(Sqrt[2]*ArcTanh[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^
2]*Cos[d + e*x - ArcTan[b, c]]])])/((b^2 + c^2)^(1/4)*e)

Rule 3115

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Int[1/Sqrt[a +
Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx &=\int \frac{1}{\sqrt{\sqrt{b^2+c^2}+\sqrt{b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}} \, dx\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{2 \sqrt{b^2+c^2}-x^2} \, dx,x,-\frac{\sqrt{b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt{\sqrt{b^2+c^2}+\sqrt{b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{e}\\ &=\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt [4]{b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt{2} \sqrt{\sqrt{b^2+c^2}+\sqrt{b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{\sqrt [4]{b^2+c^2} e}\\ \end{align*}

Mathematica [C]  time = 33.8649, size = 63264, normalized size = 718.91 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 1.432, size = 172, normalized size = 2. \begin{align*} -{\frac{ \left ( 1+\sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) \right ) \sqrt{2}}{\cos \left ( ex+d-\arctan \left ( -b,c \right ) \right ) e}\sqrt{-\sqrt{{b}^{2}+{c}^{2}} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -1 \right ) }{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-\sqrt{{b}^{2}+{c}^{2}} \left ( \sin \left ( ex+d-\arctan \left ( -b,c \right ) \right ) -1 \right ) }{\frac{1}{\sqrt [4]{{b}^{2}+{c}^{2}}}}} \right ){\frac{1}{\sqrt [4]{{b}^{2}+{c}^{2}}}}{\frac{1}{\sqrt{{({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}){\frac{1}{\sqrt{{b}^{2}+{c}^{2}}}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1+sin(e*x+d-arctan(-b,c)))*(-(b^2+c^2)^(1/2)*(sin(e*x+d-arctan(-b,c))-1))^(1/2)*2^(1/2)/(b^2+c^2)^(1/4)*arct
anh(1/2*(-(b^2+c^2)^(1/2)*(sin(e*x+d-arctan(-b,c))-1))^(1/2)*2^(1/2)/(b^2+c^2)^(1/4))/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

________________________________________________________________________________________

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(1/(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

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(b*cos(e*x+d)+c*sin(e*x+d)+(b**2+c**2)**(1/2))**(1/2),x)

[Out]

Integral(1/sqrt(b*cos(d + e*x) + c*sin(d + e*x) + sqrt(b**2 + c**2)), x)

________________________________________________________________________________________

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(1/(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