\(\int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx\) [434]

Optimal result

Integrand size = 32, antiderivative size = 88 \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)\right )}{\sqrt {2} \sqrt {\sqrt {b^2+c^2}+\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}}\right )}{\sqrt [4]{b^2+c^2} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3194, 2728, 212} \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt [4]{b^2+c^2} \sin \left (-\tan ^{-1}(b,c)+d+e x\right )}{\sqrt {2} \sqrt {\sqrt {b^2+c^2} \cos \left (-\tan ^{-1}(b,c)+d+e x\right )+\sqrt {b^2+c^2}}}\right )}{e \sqrt [4]{b^2+c^2}} \]

[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 212

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

Rule 2728

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 3194

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]

Rubi steps \begin{align*} \text {integral}& = \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 \text {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} \text {arctanh}\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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 54.77 (sec) , antiderivative size = 63264, normalized size of antiderivative = 718.91 \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\text {Result too large to show} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(171\) vs. \(2(75)=150\).

Time = 5.57 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.95

method	result	size
default	\(-\frac {\left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )+1\right ) \sqrt {-\sqrt {b^{2}+c^{2}}\, \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\sqrt {b^{2}+c^{2}}\, \left (\sin \left (e x +d -\arctan \left (-b , c\right )\right )-1\right )}\, \sqrt {2}}{2 \left (b^{2}+c^{2}\right )^{\frac {1}{4}}}\right )}{\left (b^{2}+c^{2}\right )^{\frac {1}{4}} \cos \left (e x +d -\arctan \left (-b , c\right )\right ) \sqrt {\frac {b^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+c^{2} \sin \left (e x +d -\arctan \left (-b , c\right )\right )+b^{2}+c^{2}}{\sqrt {b^{2}+c^{2}}}}\, e}\)	\(172\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (81) = 162\).

Time = 0.42 (sec) , antiderivative size = 349, normalized size of antiderivative = 3.97 \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\frac {\sqrt {2} \log \left (\frac {{\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} + {\left (b^{2} c + 4 \, c^{3}\right )} \cos \left (e x + d\right ) - {\left (3 \, b^{3} + 4 \, b c^{2} + {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right ) + \frac {2 \, \sqrt {2} {\left (2 \, {\left (b^{3} + b c^{2}\right )} \cos \left (e x + d\right ) + 2 \, {\left (b^{2} c + c^{3}\right )} \sin \left (e x + d\right ) - {\left (2 \, b c \cos \left (e x + d\right ) \sin \left (e x + d\right ) + {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} + b^{2} + 2 \, c^{2}\right )} \sqrt {b^{2} + c^{2}}\right )} \sqrt {b \cos \left (e x + d\right ) + c \sin \left (e x + d\right ) + \sqrt {b^{2} + c^{2}}}}{{\left (b^{2} + c^{2}\right )}^{\frac {1}{4}}} - 4 \, {\left (2 \, b c \cos \left (e x + d\right )^{2} - {\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) - b c\right )} \sqrt {b^{2} + c^{2}}}{3 \, b^{2} c \cos \left (e x + d\right ) - {\left (3 \, b^{2} c - c^{3}\right )} \cos \left (e x + d\right )^{3} - {\left (b^{3} - {\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (e x + d\right )^{2}\right )} \sin \left (e x + d\right )}\right )}{2 \, {\left (b^{2} + c^{2}\right )}^{\frac {1}{4}} e} \]

[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]

1/2*sqrt(2)*log(((3*b^2*c - c^3)*cos(e*x + d)^3 + (b^2*c + 4*c^3)*cos(e*x + d) - (3*b^3 + 4*b*c^2 + (b^3 - 3*b
*c^2)*cos(e*x + d)^2)*sin(e*x + d) + 2*sqrt(2)*(2*(b^3 + b*c^2)*cos(e*x + d) + 2*(b^2*c + c^3)*sin(e*x + d) -
(2*b*c*cos(e*x + d)*sin(e*x + d) + (b^2 - c^2)*cos(e*x + d)^2 + b^2 + 2*c^2)*sqrt(b^2 + c^2))*sqrt(b*cos(e*x +
d) + c*sin(e*x + d) + sqrt(b^2 + c^2))/(b^2 + c^2)^(1/4) - 4*(2*b*c*cos(e*x + d)^2 - (b^2 - c^2)*cos(e*x + d)
*sin(e*x + d) - b*c)*sqrt(b^2 + c^2))/(3*b^2*c*cos(e*x + d) - (3*b^2*c - c^3)*cos(e*x + d)^3 - (b^3 - (b^3 - 3
*b*c^2)*cos(e*x + d)^2)*sin(e*x + d)))/((b^2 + c^2)^(1/4)*e)

Sympy [F]

\[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {b \cos {\left (d + e x \right )} + c \sin {\left (d + e x \right )} + \sqrt {b^{2} + c^{2}}}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\text {Exception raised: RuntimeError} \]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\text {Exception raised: TypeError} \]

[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 >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\sqrt {b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {b\,\cos \left (d+e\,x\right )+c\,\sin \left (d+e\,x\right )+\sqrt {b^2+c^2}}} \,d x \]

[In]

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

[Out]

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