3.102 \(\int \frac{A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx\)

Optimal. Leaf size=65 \[ \frac{(A+3 B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a \cosh (x)+a}}\right )}{2 \sqrt{2} a^{3/2}}+\frac{(A-B) \sinh (x)}{2 (a \cosh (x)+a)^{3/2}} \]

[Out]

((A + 3*B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a + a*Cosh[x]])])/(2*Sqrt[2]*a^(3/2)) + ((A - B)*Sinh[x])/(2
*(a + a*Cosh[x])^(3/2))

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Rubi [A]  time = 0.0681411, antiderivative size = 65, 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 = {2750, 2649, 206} \[ \frac{(A+3 B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a \cosh (x)+a}}\right )}{2 \sqrt{2} a^{3/2}}+\frac{(A-B) \sinh (x)}{2 (a \cosh (x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[x])/(a + a*Cosh[x])^(3/2),x]

[Out]

((A + 3*B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a + a*Cosh[x]])])/(2*Sqrt[2]*a^(3/2)) + ((A - B)*Sinh[x])/(2
*(a + a*Cosh[x])^(3/2))

Rule 2750

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((b
*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)
), Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, 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)}{(a+a \cosh (x))^{3/2}} \, dx &=\frac{(A-B) \sinh (x)}{2 (a+a \cosh (x))^{3/2}}+\frac{(A+3 B) \int \frac{1}{\sqrt{a+a \cosh (x)}} \, dx}{4 a}\\ &=\frac{(A-B) \sinh (x)}{2 (a+a \cosh (x))^{3/2}}+\frac{(i (A+3 B)) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{i a \sinh (x)}{\sqrt{a+a \cosh (x)}}\right )}{2 a}\\ &=\frac{(A+3 B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a+a \cosh (x)}}\right )}{2 \sqrt{2} a^{3/2}}+\frac{(A-B) \sinh (x)}{2 (a+a \cosh (x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0772987, size = 44, normalized size = 0.68 \[ \frac{\frac{1}{2} (A-B) \sinh (x)+(A+3 B) \cosh ^3\left (\frac{x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac{x}{2}\right )\right )}{(a (\cosh (x)+1))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x])/(a + a*Cosh[x])^(3/2),x]

[Out]

((A + 3*B)*ArcTan[Sinh[x/2]]*Cosh[x/2]^3 + ((A - B)*Sinh[x])/2)/(a*(1 + Cosh[x]))^(3/2)

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Maple [B]  time = 0.062, size = 159, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{2}}{4\,{a}^{2}}\sqrt{ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}a} \left ( A\ln \left ( 2\,{\frac{\sqrt{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}\sqrt{-a}-a}{\cosh \left ( x/2 \right ) }} \right ) \left ( \cosh \left ({\frac{x}{2}} \right ) \right ) ^{2}a+3\,B\ln \left ( 2\,{\frac{\sqrt{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}\sqrt{-a}-a}{\cosh \left ( x/2 \right ) }} \right ) a \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-A\sqrt{-a}\sqrt{ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}a}+B\sqrt{ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}a}\sqrt{-a} \right ) \left ( \cosh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\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))^(3/2),x)

[Out]

-1/4*(sinh(1/2*x)^2*a)^(1/2)*(A*ln(2/cosh(1/2*x)*((sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2)-a))*cosh(1/2*x)^2*a+3*B*l
n(2/cosh(1/2*x)*((sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2)-a))*a*cosh(1/2*x)^2-A*(-a)^(1/2)*(sinh(1/2*x)^2*a)^(1/2)+B
*(sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2))/cosh(1/2*x)/a^2/(-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 = 2.03622, size = 405, normalized size = 6.23 \begin{align*} \frac{1}{6} \,{\left (\sqrt{2}{\left (\frac{3 \, e^{\left (\frac{5}{2} \, x\right )} + 8 \, e^{\left (\frac{3}{2} \, x\right )} - 3 \, e^{\left (\frac{1}{2} \, x\right )}}{a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}} + \frac{3 \, \arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{a^{\frac{3}{2}}}\right )} - \frac{8 \, \sqrt{2} e^{\left (\frac{3}{2} \, x\right )}}{a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}}\right )} A + \frac{1}{20} \,{\left (\sqrt{2}{\left (\frac{15 \, e^{\left (\frac{5}{2} \, x\right )} + 40 \, e^{\left (\frac{3}{2} \, x\right )} + 33 \, e^{\left (\frac{1}{2} \, x\right )}}{a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}} + \frac{15 \, \arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{a^{\frac{3}{2}}}\right )} + 5 \, \sqrt{2}{\left (\frac{3 \, e^{\left (\frac{5}{2} \, x\right )} - 8 \, e^{\left (\frac{3}{2} \, x\right )} - 3 \, e^{\left (\frac{1}{2} \, x\right )}}{a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}} + \frac{3 \, \arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{a^{\frac{3}{2}}}\right )} - \frac{8 \,{\left (5 \, \sqrt{2} \sqrt{a} e^{\left (\frac{5}{2} \, x\right )} + \sqrt{2} \sqrt{a} e^{\left (\frac{1}{2} \, x\right )}\right )}}{a^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} e^{x} + a^{2}}\right )} B \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+a*cosh(x))^(3/2),x, algorithm="maxima")

[Out]

1/6*(sqrt(2)*((3*e^(5/2*x) + 8*e^(3/2*x) - 3*e^(1/2*x))/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^x +
a^(3/2)) + 3*arctan(e^(1/2*x))/a^(3/2)) - 8*sqrt(2)*e^(3/2*x)/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2
)*e^x + a^(3/2)))*A + 1/20*(sqrt(2)*((15*e^(5/2*x) + 40*e^(3/2*x) + 33*e^(1/2*x))/(a^(3/2)*e^(3*x) + 3*a^(3/2)
*e^(2*x) + 3*a^(3/2)*e^x + a^(3/2)) + 15*arctan(e^(1/2*x))/a^(3/2)) + 5*sqrt(2)*((3*e^(5/2*x) - 8*e^(3/2*x) -
3*e^(1/2*x))/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^x + a^(3/2)) + 3*arctan(e^(1/2*x))/a^(3/2)) -
8*(5*sqrt(2)*sqrt(a)*e^(5/2*x) + sqrt(2)*sqrt(a)*e^(1/2*x))/(a^2*e^(3*x) + 3*a^2*e^(2*x) + 3*a^2*e^x + a^2))*B

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Fricas [B]  time = 2.18368, size = 581, normalized size = 8.94 \begin{align*} -\frac{\sqrt{2}{\left ({\left (A + 3 \, B\right )} \cosh \left (x\right )^{2} +{\left (A + 3 \, B\right )} \sinh \left (x\right )^{2} + 2 \,{\left (A + 3 \, B\right )} \cosh \left (x\right ) + 2 \,{\left ({\left (A + 3 \, B\right )} \cosh \left (x\right ) + A + 3 \, B\right )} \sinh \left (x\right ) + A + 3 \, B\right )} \sqrt{a} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{1}{2}} \sqrt{a} \sqrt{\frac{a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{a}\right ) - 2 \, \sqrt{\frac{1}{2}}{\left ({\left (A - B\right )} \cosh \left (x\right )^{2} +{\left (A - B\right )} \sinh \left (x\right )^{2} -{\left (A - B\right )} \cosh \left (x\right ) +{\left (2 \,{\left (A - B\right )} \cosh \left (x\right ) - A + B\right )} \sinh \left (x\right )\right )} \sqrt{\frac{a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{2 \,{\left (a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a^{2} + 2 \,{\left (a^{2} \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+a*cosh(x))^(3/2),x, algorithm="fricas")

[Out]

-1/2*(sqrt(2)*((A + 3*B)*cosh(x)^2 + (A + 3*B)*sinh(x)^2 + 2*(A + 3*B)*cosh(x) + 2*((A + 3*B)*cosh(x) + A + 3*
B)*sinh(x) + A + 3*B)*sqrt(a)*arctan(sqrt(2)*sqrt(1/2)*sqrt(a)*sqrt(a/(cosh(x) + sinh(x)))/a) - 2*sqrt(1/2)*((
A - B)*cosh(x)^2 + (A - B)*sinh(x)^2 - (A - B)*cosh(x) + (2*(A - B)*cosh(x) - A + B)*sinh(x))*sqrt(a/(cosh(x)
+ sinh(x))))/(a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a^2*cosh(x) + a^2 + 2*(a^2*cosh(x) + a^2)*sinh(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+a*cosh(x))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.33583, size = 105, normalized size = 1.62 \begin{align*} \frac{{\left (\sqrt{2} A + 3 \, \sqrt{2} B\right )} \arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{2 \, a^{\frac{3}{2}}} + \frac{\sqrt{2}{\left (A a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} - B a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} - A a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} + B a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )}\right )}}{2 \,{\left (a e^{x} + a\right )}^{2} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(x))/(a+a*cosh(x))^(3/2),x, algorithm="giac")

[Out]

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