3.52 \(\int \frac{1}{(a-a \cosh (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{2} \sqrt{a-a \cosh (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0414287, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2650, 2649, 206} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{2} \sqrt{a-a \cosh (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Cosh[c + d*x])^(-3/2),x]

[Out]

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

Rule 2650

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

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{1}{(a-a \cosh (c+d x))^{3/2}} \, dx &=-\frac{\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}+\frac{\int \frac{1}{\sqrt{a-a \cosh (c+d x)}} \, dx}{4 a}\\ &=-\frac{\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}+\frac{i \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{i a \sinh (c+d x)}{\sqrt{a-a \cosh (c+d x)}}\right )}{2 a d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{2} \sqrt{a-a \cosh (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.164053, size = 85, normalized size = 1.08 \[ \frac{\sinh ^3\left (\frac{1}{2} (c+d x)\right ) \left (\text{csch}^2\left (\frac{1}{4} (c+d x)\right )+\text{sech}^2\left (\frac{1}{4} (c+d x)\right )+4 \log \left (\tanh \left (\frac{1}{4} (c+d x)\right )\right )\right )}{4 a d (\cosh (c+d x)-1) \sqrt{a-a \cosh (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Csch[(c + d*x)/4]^2 + 4*Log[Tanh[(c + d*x)/4]] + Sech[(c + d*x)/4]^2)*Sinh[(c + d*x)/2]^3)/(4*a*d*(-1 + Cosh
[c + d*x])*Sqrt[a - a*Cosh[c + d*x]])

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Maple [A]  time = 0.053, size = 87, normalized size = 1.1 \begin{align*} -{\frac{1}{4\,ad} \left ( -2\,\cosh \left ( 1/2\,dx+c/2 \right ) + \left ( -\ln \left ( -1+\cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) +\ln \left ( 1+\cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \right ) \left ( \sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) \left ( \sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-a*cosh(d*x+c))^(3/2),x)

[Out]

-1/4/a*(-2*cosh(1/2*d*x+1/2*c)+(-ln(-1+cosh(1/2*d*x+1/2*c))+ln(1+cosh(1/2*d*x+1/2*c)))*sinh(1/2*d*x+1/2*c)^2)/
sinh(1/2*d*x+1/2*c)/(-2*sinh(1/2*d*x+1/2*c)^2*a)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a \cosh \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*cosh(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((-a*cosh(d*x + c) + a)^(-3/2), x)

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Fricas [B]  time = 1.92949, size = 783, normalized size = 9.91 \begin{align*} -\frac{\sqrt{2}{\left (\cosh \left (d x + c\right )^{2} + 2 \,{\left (\cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) + 1\right )} \sqrt{-a} \log \left (-\frac{2 \, \sqrt{2} \sqrt{\frac{1}{2}} \sqrt{-a} \sqrt{-\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}{\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt{\frac{1}{2}}{\left (\cosh \left (d x + c\right )^{2} +{\left (2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + \cosh \left (d x + c\right )\right )} \sqrt{-\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{4 \,{\left (a^{2} d \cosh \left (d x + c\right )^{2} + a^{2} d \sinh \left (d x + c\right )^{2} - 2 \, a^{2} d \cosh \left (d x + c\right ) + a^{2} d + 2 \,{\left (a^{2} d \cosh \left (d x + c\right ) - a^{2} d\right )} \sinh \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*cosh(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/4*(sqrt(2)*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) - 1)*sinh(d*x + c) + sinh(d*x + c)^2 - 2*cosh(d*x + c) + 1)*
sqrt(-a)*log(-(2*sqrt(2)*sqrt(1/2)*sqrt(-a)*sqrt(-a/(cosh(d*x + c) + sinh(d*x + c)))*(cosh(d*x + c) + sinh(d*x
+ c)) + a*cosh(d*x + c) + a*sinh(d*x + c) + a)/(cosh(d*x + c) + sinh(d*x + c) - 1)) + 4*sqrt(1/2)*(cosh(d*x +
c)^2 + (2*cosh(d*x + c) + 1)*sinh(d*x + c) + sinh(d*x + c)^2 + cosh(d*x + c))*sqrt(-a/(cosh(d*x + c) + sinh(d
*x + c))))/(a^2*d*cosh(d*x + c)^2 + a^2*d*sinh(d*x + c)^2 - 2*a^2*d*cosh(d*x + c) + a^2*d + 2*(a^2*d*cosh(d*x
+ c) - a^2*d)*sinh(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- a \cosh{\left (c + d x \right )} + a\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*cosh(d*x+c))**(3/2),x)

[Out]

Integral((-a*cosh(c + d*x) + a)**(-3/2), x)

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Giac [A]  time = 1.27652, size = 153, normalized size = 1.94 \begin{align*} -\frac{\frac{\sqrt{2} \arctan \left (\frac{\sqrt{-a e^{\left (d x + c\right )}}}{\sqrt{a}}\right )}{\sqrt{a} d \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right )} - \frac{\sqrt{2}{\left (\sqrt{-a e^{\left (d x + c\right )}} a e^{\left (d x + c\right )} + \sqrt{-a e^{\left (d x + c\right )}} a\right )}}{{\left (a e^{\left (d x + c\right )} - a\right )}^{2} d \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right )}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*cosh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

-1/2*(sqrt(2)*arctan(sqrt(-a*e^(d*x + c))/sqrt(a))/(sqrt(a)*d*sgn(-e^(d*x + c) + 1)) - sqrt(2)*(sqrt(-a*e^(d*x
+ c))*a*e^(d*x + c) + sqrt(-a*e^(d*x + c))*a)/((a*e^(d*x + c) - a)^2*d*sgn(-e^(d*x + c) + 1)))/a