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]

-1/2*sinh(d*x+c)/d/(a-a*cosh(d*x+c))^(3/2)-1/4*arctan(1/2*sinh(d*x+c)*a^(1/2)*2^(1/2)/(a-a*cosh(d*x+c))^(1/2))
/a^(3/2)/d*2^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.17, 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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fricas [B]  time = 0.61, size = 274, normalized size = 3.47 \[ -\frac {\sqrt {2} {\left (\cosh \left (d x + c\right )^{2} + 2 \, {\left (\cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) + 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-a} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt {\frac {1}{2}} {\left (\cosh \left (d x + c\right )^{2} + {\left (2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + \cosh \left (d x + c\right )\right )} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{4 \, {\left (a^{2} d \cosh \left (d x + c\right )^{2} + a^{2} d \sinh \left (d x + c\right )^{2} - 2 \, a^{2} d \cosh \left (d x + c\right ) + a^{2} d + 2 \, {\left (a^{2} d \cosh \left (d x + c\right ) - a^{2} d\right )} \sinh \left (d x + c\right )\right )}} \]

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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giac [A]  time = 0.25, size = 107, normalized size = 1.35 \[ -\frac {\sqrt {2} {\left (\frac {\arctan \left (\frac {\sqrt {-a e^{\left (d x + c\right )}}}{\sqrt {a}}\right )}{\sqrt {a} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} - \frac {\sqrt {-a e^{\left (d x + c\right )}} a e^{\left (d x + c\right )} + \sqrt {-a e^{\left (d x + c\right )}} a}{{\left (a e^{\left (d x + c\right )} - a\right )}^{2} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )}\right )}}{2 \, a d} \]

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

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maple [A]  time = 0.37, size = 87, normalized size = 1.10 \[ \frac {2 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\ln \left (-1+\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\ln \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right ) \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]

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(cosh(1/2*d*x+1/2*c)+1))*sinh(1/2*d*x+1/2*c)^2)/sin
h(1/2*d*x+1/2*c)/(-2*a*sinh(1/2*d*x+1/2*c)^2)^(1/2)/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a \cosh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a-a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - a*cosh(c + d*x))^(3/2),x)

[Out]

int(1/(a - a*cosh(c + d*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- a \cosh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \]

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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