3.11.38 \(\int \frac {\coth (c+d x)}{\sqrt {a \cosh ^2(c+d x)}} \, dx\) [1038]

Optimal. Leaf size=31 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]

[Out]

-arctanh((a*cosh(d*x+c)^2)^(1/2)/a^(1/2))/d/a^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3284, 65, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]/Sqrt[a*Cosh[c + d*x]^2],x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3284

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFact
ors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[x^((m - 1)/2)*((b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)
^((m + 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2
]

Rubi steps

\begin {align*} \int \frac {\coth (c+d x)}{\sqrt {a \cosh ^2(c+d x)}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cosh ^2(c+d x)}\right )}{a d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cosh ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 49, normalized size = 1.58 \begin {gather*} \frac {\cosh (c+d x) \left (-\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{d \sqrt {a \cosh ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]/Sqrt[a*Cosh[c + d*x]^2],x]

[Out]

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

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Maple [A]
time = 2.22, size = 31, normalized size = 1.00

method	result	size
default	\(-\frac {\cosh \left (d x +c \right ) \arctanh \left (\cosh \left (d x +c \right )\right )}{\sqrt {a \left (\cosh ^{2}\left (d x +c \right )\right )}\, d}\)	\(31\)
risch	\(\frac {\ln \left ({\mathrm e}^{d x}-{\mathrm e}^{-c}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right ) {\mathrm e}^{-d x -c}}{d \sqrt {a \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} {\mathrm e}^{-2 d x -2 c}}}-\frac {\ln \left ({\mathrm e}^{d x}+{\mathrm e}^{-c}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right ) {\mathrm e}^{-d x -c}}{d \sqrt {a \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} {\mathrm e}^{-2 d x -2 c}}}\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)/(a*cosh(d*x+c)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/(a*cosh(d*x+c)^2)^(1/2)*cosh(d*x+c)*arctanh(cosh(d*x+c))/d

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Maxima [A]
time = 0.51, size = 40, normalized size = 1.29 \begin {gather*} -\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{\sqrt {a} d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{\sqrt {a} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (25) = 50\).
time = 0.37, size = 174, normalized size = 5.61 \begin {gather*} \left [\frac {\sqrt {a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + a} \log \left (\frac {\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1}\right )}{a d e^{\left (2 \, d x + 2 \, c\right )} + a d}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + a} \sqrt {-a}}{a \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + a \cosh \left (d x + c\right ) + {\left (a e^{\left (2 \, d x + 2 \, c\right )} + a\right )} \sinh \left (d x + c\right )}\right )}{a d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + a)*log((cosh(d*x + c) + sinh(d*x + c) - 1)/(cosh(d*x + c) + si
nh(d*x + c) + 1))/(a*d*e^(2*d*x + 2*c) + a*d), 2*sqrt(-a)*arctan(sqrt(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c)
+ a)*sqrt(-a)/(a*cosh(d*x + c)*e^(2*d*x + 2*c) + a*cosh(d*x + c) + (a*e^(2*d*x + 2*c) + a)*sinh(d*x + c)))/(a*
d)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth {\left (c + d x \right )}}{\sqrt {a \cosh ^{2}{\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(c + d*x)/sqrt(a*cosh(c + d*x)**2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {coth}\left (c+d\,x\right )}{\sqrt {a\,{\mathrm {cosh}\left (c+d\,x\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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