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

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

[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 = 0.0728487, antiderivative size = 65, normalized size of antiderivative = 1., 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 = {2750, 2649, 206} \[ -\frac{(A-3 B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a-a \cosh (x)}}\right )}{2 \sqrt{2} a^{3/2}}-\frac{(A+B) \sinh (x)}{2 (a-a \cosh (x))^{3/2}} \]

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 = 0.149487, size = 71, normalized size = 1.09 \[ \frac{\sinh ^3\left (\frac{x}{2}\right ) \left ((A+B) \text{csch}^2\left (\frac{x}{4}\right )+(A+B) \text{sech}^2\left (\frac{x}{4}\right )+4 (A-3 B) \log \left (\tanh \left (\frac{x}{4}\right )\right )\right )}{4 a (\cosh (x)-1) \sqrt{a-a \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((A + B)*Csch[x/4]^2 + 4*(A - 3*B)*Log[Tanh[x/4]] + (A + B)*Sech[x/4]^2)*Sinh[x/2]^3)/(4*a*(-1 + Cosh[x])*Sqr
t[a - a*Cosh[x]])

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Maple [A]  time = 0.07, size = 83, normalized size = 1.3 \begin{align*} -{\frac{1}{4\,a} \left ( \cosh \left ({\frac{x}{2}} \right ) \left ( -2\,A-2\,B \right ) + \left ( \ln \left ( 1+\cosh \left ({\frac{x}{2}} \right ) \right ) A-\ln \left ( -1+\cosh \left ({\frac{x}{2}} \right ) \right ) A-3\,B\ln \left ( 1+\cosh \left ( x/2 \right ) \right ) +3\,B\ln \left ( -1+\cosh \left ( x/2 \right ) \right ) \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}}}} \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/a*(cosh(1/2*x)*(-2*A-2*B)+(ln(1+cosh(1/2*x))*A-ln(-1+cosh(1/2*x))*A-3*B*ln(1+cosh(1/2*x))+3*B*ln(-1+cosh(
1/2*x)))*sinh(1/2*x)^2)/sinh(1/2*x)/(-2*sinh(1/2*x)^2*a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cosh \left (x\right ) + A}{{\left (-a \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \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]

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

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Fricas [B]  time = 2.19713, size = 680, normalized size = 10.46 \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} \log \left (\frac{2 \, \sqrt{2} \sqrt{\frac{1}{2}} \sqrt{-a} \sqrt{-\frac{a}{\cosh \left (x\right ) + \sinh \left (x\right )}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a \cosh \left (x\right ) - a \sinh \left (x\right ) - a}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - 4 \, \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 )}}}{4 \,{\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]

1/4*(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)*log((2*sqrt(2)*sqrt(1/2)*sqrt(-a)*sqrt(-a/(cosh(x) + sinh(x)))*(cosh(x) + sinh(x
)) - a*cosh(x) - a*sinh(x) - a)/(cosh(x) + sinh(x) - 1)) - 4*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*si
nh(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 [B]  time = 1.22786, size = 150, normalized size = 2.31 \begin{align*} -\frac{{\left (\sqrt{2} A - 3 \, \sqrt{2} B\right )} \arctan \left (\frac{\sqrt{-a e^{x}}}{\sqrt{a}}\right )}{2 \, a^{\frac{3}{2}} \mathrm{sgn}\left (-e^{x} + 1\right )} + \frac{\sqrt{2}{\left (\sqrt{-a e^{x}} A a e^{x} + \sqrt{-a e^{x}} B a e^{x} + \sqrt{-a e^{x}} A a + \sqrt{-a e^{x}} B a\right )}}{2 \,{\left (a e^{x} - a\right )}^{2} a \mathrm{sgn}\left (-e^{x} + 1\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="giac")

[Out]

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