∫ A+B cosh(x)___∘ a − a cosh(x) dx [104]

Optimal. Leaf size=57 \[ -\frac {\sqrt {2} (A+B) \text {ArcTan}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}}+\frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}} \]

-(A+B)*arctan(1/2*sinh(x)*a^(1/2)*2^(1/2)/(a-a*cosh(x))^(1/2))*2^(1/2)/a^(1/2)+2*B*sinh(x)/(a-a*cosh(x))^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2830, 2728, 212} \begin {gather*} \frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}}-\frac {\sqrt {2} (A+B) \text {ArcTan}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}} \end {gather*}

-((Sqrt[2]*(A + B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a - a*Cosh[x]])])/Sqrt[a]) + (2*B*Sinh[x])/Sqrt[a -

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]

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]

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d

)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S

in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m

\begin {align*} \int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx &=\frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}}+(A+B) \int \frac {1}{\sqrt {a-a \cosh (x)}} \, dx\\ &=\frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}}+(2 i (A+B)) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {i a \sinh (x)}{\sqrt {a-a \cosh (x)}}\right )\\ &=-\frac {\sqrt {2} (A+B) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a-a \cosh (x)}}\right )}{\sqrt {a}}+\frac {2 B \sinh (x)}{\sqrt {a-a \cosh (x)}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 40, normalized size = 0.70 \begin {gather*} \frac {2 \left (2 B \cosh \left (\frac {x}{2}\right )+(A+B) \log \left (\tanh \left (\frac {x}{4}\right )\right )\right ) \sinh \left (\frac {x}{2}\right )}{\sqrt {a-a \cosh (x)}} \end {gather*}

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method	result	size

default	\(\frac {\sinh \left (\frac {x}{2}\right ) \left (\ln \left (\cosh \left (\frac {x}{2}\right )-1\right ) A -\ln \left (\cosh \left (\frac {x}{2}\right )+1\right ) A +\ln \left (\cosh \left (\frac {x}{2}\right )-1\right ) B -\ln \left (\cosh \left (\frac {x}{2}\right )+1\right ) B +4 B \cosh \left (\frac {x}{2}\right )\right )}{\sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) a}}\)	\(63\)

sinh(1/2*x)*(ln(cosh(1/2*x)-1)*A-ln(cosh(1/2*x)+1)*A+ln(cosh(1/2*x)-1)*B-ln(cosh(1/2*x)+1)*B+4*B*cosh(1/2*x))/

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (46) = 92\).
time = 0.36, size = 99, normalized size = 1.74 \begin {gather*} \frac {\sqrt {2} {\left (A + B\right )} a \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}} \sqrt {-\frac {1}{a}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - \cosh \left (x\right ) - \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - 2 \, \sqrt {\frac {1}{2}} {\left (B \cosh \left (x\right ) + B \sinh \left (x\right ) + B\right )} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{a} \end {gather*}

(sqrt(2)*(A + B)*a*sqrt(-1/a)*log((2*sqrt(2)*sqrt(1/2)*sqrt(-a/(cosh(x) + sinh(x)))*sqrt(-1/a)*(cosh(x) + sinh

(x)) - cosh(x) - sinh(x) - 1)/(cosh(x) + sinh(x) - 1)) - 2*sqrt(1/2)*(B*cosh(x) + B*sinh(x) + B)*sqrt(-a/(cosh

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B \cosh {\left (x \right )}}{\sqrt {- a \left (\cosh {\left (x \right )} - 1\right )}}\, dx \end {gather*}

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Giac [A]
time = 0.43, size = 84, normalized size = 1.47 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} A + \sqrt {2} B\right )} \arctan \left (\frac {\sqrt {-a e^{x}}}{\sqrt {a}}\right )}{\sqrt {a} \mathrm {sgn}\left (-e^{x} + 1\right )} - \frac {\sqrt {2} B}{\sqrt {-a e^{x}} \mathrm {sgn}\left (-e^{x} + 1\right )} + \frac {\sqrt {2} \sqrt {-a e^{x}} B}{a \mathrm {sgn}\left (-e^{x} + 1\right )} \end {gather*}

-2*(sqrt(2)*A + sqrt(2)*B)*arctan(sqrt(-a*e^x)/sqrt(a))/(sqrt(a)*sgn(-e^x + 1)) - sqrt(2)*B/(sqrt(-a*e^x)*sgn(

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {A+B\,\mathrm {cosh}\left (x\right )}{\sqrt {a-a\,\mathrm {cosh}\left (x\right )}} \,d x \end {gather*}

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3.2.4 \(\int \frac {A+B \cosh (x)}{\sqrt {a-a \cosh (x)}} \, dx\) [104]