3.5.23 \(\int \frac {\coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) [423]

Optimal. Leaf size=34 \[ \frac {\log (\sinh (c+d x))}{a d}-\frac {\log (a+b \sinh (c+d x))}{a d} \]

[Out]

ln(sinh(d*x+c))/a/d-ln(a+b*sinh(d*x+c))/a/d

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Rubi [A]
time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2800, 36, 29, 31} \begin {gather*} \frac {\log (\sinh (c+d x))}{a d}-\frac {\log (a+b \sinh (c+d x))}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]/(a + b*Sinh[c + d*x]),x]

[Out]

Log[Sinh[c + d*x]]/(a*d) - Log[a + b*Sinh[c + d*x]]/(a*d)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2800

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

\begin {align*} \int \frac {\coth (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,b \sinh (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (c+d x)\right )}{a d}\\ &=\frac {\log (\sinh (c+d x))}{a d}-\frac {\log (a+b \sinh (c+d x))}{a d}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 28, normalized size = 0.82 \begin {gather*} \frac {\log (\sinh (c+d x))-\log (a+b \sinh (c+d x))}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]/(a + b*Sinh[c + d*x]),x]

[Out]

(Log[Sinh[c + d*x]] - Log[a + b*Sinh[c + d*x]])/(a*d)

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Maple [A]
time = 0.74, size = 33, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (a +b \sinh \left (d x +c \right )\right )}{a}+\frac {\ln \left (\sinh \left (d x +c \right )\right )}{a}}{d}\)	\(33\)
default	\(\frac {-\frac {\ln \left (a +b \sinh \left (d x +c \right )\right )}{a}+\frac {\ln \left (\sinh \left (d x +c \right )\right )}{a}}{d}\)	\(33\)
risch	\(\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a d}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/a*ln(a+b*sinh(d*x+c))+1/a*ln(sinh(d*x+c)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (34) = 68\).
time = 0.28, size = 75, normalized size = 2.21 \begin {gather*} -\frac {\log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a*d) + log(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1)/(
a*d)

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Fricas [A]
time = 0.40, size = 67, normalized size = 1.97 \begin {gather*} -\frac {\log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) - log(2*sinh(d*x + c)/(cosh(d*x + c) - 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 )}}{a + b \sinh {\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+b*sinh(d*x+c)),x)

[Out]

Integral(coth(c + d*x)/(a + b*sinh(c + d*x)), x)

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Giac [A]
time = 0.48, size = 61, normalized size = 1.79 \begin {gather*} -\frac {\frac {\log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a} - \frac {\log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

-(log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/a - log(abs(e^(d*x + c) - e^(-d*x - c)))/a)/d

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Mupad [B]
time = 0.41, size = 254, normalized size = 7.47 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-a^2\,d^2}+b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^2\,d^2}-2\,a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^2\,d^2}-b\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {-a^2\,d^2}}{a^2\,d}\right )}{\sqrt {-a^2\,d^2}}-\frac {2\,\mathrm {atan}\left (\left (4\,a^4\,b\,d\,\sqrt {-a^2\,d^2}+4\,a^2\,b^3\,d\,\sqrt {-a^2\,d^2}\right )\,\left (\frac {1}{8\,a\,b\,d^2\,{\left (a^2+b^2\right )}^2}-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {1}{16\,b^2\,d^2\,{\left (a^2+b^2\right )}^2}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^4\,b^2\,d^2\,{\left (a^2+b^2\right )}^2}\right )+\frac {a^2+2\,b^2}{8\,a^3\,b\,d^2\,{\left (a^2+b^2\right )}^2}\right )\right )}{\sqrt {-a^2\,d^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)/(a + b*sinh(c + d*x)),x)

[Out]

(2*atan((a*(-a^2*d^2)^(1/2) + b*exp(d*x)*exp(c)*(-a^2*d^2)^(1/2) - 2*a*exp(2*c)*exp(2*d*x)*(-a^2*d^2)^(1/2) -
b*exp(3*c)*exp(3*d*x)*(-a^2*d^2)^(1/2))/(a^2*d)))/(-a^2*d^2)^(1/2) - (2*atan((4*a^4*b*d*(-a^2*d^2)^(1/2) + 4*a
^2*b^3*d*(-a^2*d^2)^(1/2))*(1/(8*a*b*d^2*(a^2 + b^2)^2) - exp(d*x)*exp(c)*(1/(16*b^2*d^2*(a^2 + b^2)^2) - (a^2
+ 2*b^2)^2/(16*a^4*b^2*d^2*(a^2 + b^2)^2)) + (a^2 + 2*b^2)/(8*a^3*b*d^2*(a^2 + b^2)^2))))/(-a^2*d^2)^(1/2)

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