3.101 \(\int \frac{A+B \cosh (x)}{\sqrt{a+a \cosh (x)}} \, dx\)

Optimal. Leaf size=56 \[ \frac{\sqrt{2} (A-B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a \cosh (x)+a}}\right )}{\sqrt{a}}+\frac{2 B \sinh (x)}{\sqrt{a \cosh (x)+a}} \]

[Out]

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

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Rubi [A]  time = 0.0655941, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2751, 2649, 206} \[ \frac{\sqrt{2} (A-B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a \cosh (x)+a}}\right )}{\sqrt{a}}+\frac{2 B \sinh (x)}{\sqrt{a \cosh (x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/Sqrt[a + a*Cosh[x]],x]

[Out]

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

Rule 2751

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*Sin
[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,
-2^(-1)]

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{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)) \operatorname{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*}

Mathematica [A]  time = 0.0341246, size = 41, normalized size = 0.73 \[ \frac{2 \cosh \left (\frac{x}{2}\right ) \left ((A-B) \tan ^{-1}\left (\sinh \left (\frac{x}{2}\right )\right )+2 B \sinh \left (\frac{x}{2}\right )\right )}{\sqrt{a (\cosh (x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x])/Sqrt[a + a*Cosh[x]],x]

[Out]

(2*Cosh[x/2]*((A - B)*ArcTan[Sinh[x/2]] + 2*B*Sinh[x/2]))/Sqrt[a*(1 + Cosh[x])]

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Maple [B]  time = 0.053, size = 128, normalized size = 2.3 \begin{align*} -{\frac{\sqrt{2}}{a}\cosh \left ({\frac{x}{2}} \right ) \sqrt{ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}a} \left ( \ln \left ( 2\,{\frac{\sqrt{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}\sqrt{-a}-a}{\cosh \left ( x/2 \right ) }} \right ) aA-2\,B\sqrt{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}\sqrt{-a}-\ln \left ( 2\,{\frac{\sqrt{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}\sqrt{-a}-a}{\cosh \left ( x/2 \right ) }} \right ) aB \right ){\frac{1}{\sqrt{-a}}} \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{x}{2}} \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cosh(x))/(a+a*cosh(x))^(1/2),x)

[Out]

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

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Maxima [B]  time = 1.94619, size = 235, normalized size = 4.2 \begin{align*} 2 \,{\left (\sqrt{2}{\left (\frac{\arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{\sqrt{a}} + \frac{e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{a} e^{x} + \sqrt{a}}\right )} - \frac{\sqrt{2} e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{a} e^{x} + \sqrt{a}}\right )} A - \frac{1}{3} \,{\left (3 \, \sqrt{2}{\left (\frac{\arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{\sqrt{a}} - \frac{e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{a} e^{x} + \sqrt{a}}\right )} - \sqrt{2}{\left (\frac{3 \, \arctan \left (e^{\left (-\frac{1}{2} \, x\right )}\right )}{\sqrt{a}} - \frac{2 \, e^{\left (-\frac{1}{2} \, x\right )}}{\sqrt{a}} - \frac{e^{\left (-\frac{1}{2} \, x\right )}}{\sqrt{a} e^{\left (-x\right )} + \sqrt{a}}\right )} - \frac{3 \, \sqrt{2} \sqrt{a} e^{\left (\frac{3}{2} \, x\right )} - \sqrt{2} \sqrt{a} e^{\left (-\frac{1}{2} \, x\right )}}{a e^{x} + a}\right )} B \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+a*cosh(x))^(1/2),x, algorithm="maxima")

[Out]

2*(sqrt(2)*(arctan(e^(1/2*x))/sqrt(a) + e^(1/2*x)/(sqrt(a)*e^x + sqrt(a))) - sqrt(2)*e^(1/2*x)/(sqrt(a)*e^x +
sqrt(a)))*A - 1/3*(3*sqrt(2)*(arctan(e^(1/2*x))/sqrt(a) - e^(1/2*x)/(sqrt(a)*e^x + sqrt(a))) - sqrt(2)*(3*arct
an(e^(-1/2*x))/sqrt(a) - 2*e^(-1/2*x)/sqrt(a) - e^(-1/2*x)/(sqrt(a)*e^(-x) + sqrt(a))) - (3*sqrt(2)*sqrt(a)*e^
(3/2*x) - sqrt(2)*sqrt(a)*e^(-1/2*x))/(a*e^x + a))*B

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Fricas [A]  time = 2.16759, size = 244, normalized size = 4.36 \begin{align*} \frac{2 \,{\left (\sqrt{2}{\left (A - B\right )} \sqrt{a} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{1}{2}} \sqrt{\frac{a}{\cosh \left (x\right ) + \sinh \left (x\right )}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\sqrt{a}}\right ) + \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 )}}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+a*cosh(x))^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+a*cosh(x))**(1/2),x)

[Out]

Integral((A + B*cosh(x))/sqrt(a*(cosh(x) + 1)), x)

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Giac [C]  time = 1.23595, size = 100, normalized size = 1.79 \begin{align*} \frac{1}{4} \, \sqrt{2}{\left (\frac{8 \,{\left (A - B\right )} \arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{\sqrt{a}} + \frac{4 \, B e^{\left (\frac{1}{2} \, x\right )}}{\sqrt{a}} - \frac{4 \, B e^{\left (-\frac{1}{2} \, x\right )}}{\sqrt{a}} + \frac{8 i \, A \sqrt{-a} \arctan \left (-i\right ) - 8 i \, B \sqrt{-a} \arctan \left (-i\right ) + 8 \, B \sqrt{-a}}{a}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+a*cosh(x))^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(2)*(8*(A - B)*arctan(e^(1/2*x))/sqrt(a) + 4*B*e^(1/2*x)/sqrt(a) - 4*B*e^(-1/2*x)/sqrt(a) + (8*I*A*sqr
t(-a)*arctan(-I) - 8*I*B*sqrt(-a)*arctan(-I) + 8*B*sqrt(-a))/a)