3.5.52 \(\int \frac {\coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [452]

Optimal. Leaf size=50 \[ -\frac {\text {csch}(c+d x)}{a d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b \log (a+b \sinh (c+d x))}{a^2 d} \]

[Out]

-csch(d*x+c)/a/d-b*ln(sinh(d*x+c))/a^2/d+b*ln(a+b*sinh(d*x+c))/a^2/d

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Rubi [A]
time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 46} \begin {gather*} -\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b \log (a+b \sinh (c+d x))}{a^2 d}-\frac {\text {csch}(c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 50, normalized size = 1.00 \begin {gather*} -\frac {\text {csch}(c+d x)}{a d}-\frac {b \log (\sinh (c+d x))}{a^2 d}+\frac {b \log (a+b \sinh (c+d x))}{a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.76, size = 34, normalized size = 0.68

method	result	size
derivativedivides	\(-\frac {\frac {\mathrm {csch}\left (d x +c \right )}{a}-\frac {b \ln \left (a \,\mathrm {csch}\left (d x +c \right )+b \right )}{a^{2}}}{d}\)	\(34\)
default	\(-\frac {\frac {\mathrm {csch}\left (d x +c \right )}{a}-\frac {b \ln \left (a \,\mathrm {csch}\left (d x +c \right )+b \right )}{a^{2}}}{d}\)	\(34\)
risch	\(-\frac {2 \,{\mathrm e}^{d x +c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a^{2} d}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*(1/a*csch(d*x+c)-1/a^2*b*ln(a*csch(d*x+c)+b))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (50) = 100\).
time = 0.27, size = 110, normalized size = 2.20 \begin {gather*} \frac {2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} + \frac {b \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (50) = 100\).
time = 0.42, size = 211, normalized size = 4.22 \begin {gather*} -\frac {2 \, a \cosh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \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 ) + {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, a \sinh \left (d x + c\right )}{a^{2} d \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d \sinh \left (d x + c\right )^{2} - a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*cosh(d*x + c) - (b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*log(2*(b*s
inh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + (b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*
sinh(d*x + c)^2 - b)*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 2*a*sinh(d*x + c))/(a^2*d*cosh(d*x
+ c)^2 + 2*a^2*d*cosh(d*x + c)*sinh(d*x + c) + a^2*d*sinh(d*x + c)^2 - a^2*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 )} \operatorname {csch}{\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)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (50) = 100\).
time = 0.43, size = 110, normalized size = 2.20 \begin {gather*} \frac {\frac {b \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{2}} - \frac {b \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{2}} + \frac {b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, a}{a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.87, size = 409, normalized size = 8.18 \begin {gather*} \frac {\left (2\,\mathrm {atan}\left (\left (4\,a^3\,b\,d\,{\left (b^2\right )}^{5/2}\,\sqrt {-a^4\,d^2}+4\,a^5\,b\,d\,{\left (b^2\right )}^{3/2}\,\sqrt {-a^4\,d^2}\right )\,\left (\frac {1}{8\,a^3\,b^4\,d^2\,{\left (a^2+b^2\right )}^2}-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {1}{16\,a^2\,b^5\,d^2\,{\left (a^2+b^2\right )}^2}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^6\,b^5\,d^2\,{\left (a^2+b^2\right )}^2}\right )+\frac {a^2+2\,b^2}{8\,a^5\,b^4\,d^2\,{\left (a^2+b^2\right )}^2}\right )\right )+2\,\mathrm {atan}\left (-\frac {4\,a^3\,b^5\,\sqrt {-a^4\,d^2}+4\,a\,b^7\,\sqrt {-a^4\,d^2}-4\,b^8\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {-a^4\,d^2}+4\,b^8\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^4\,d^2}-8\,a\,b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^4\,d^2}+4\,a^2\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^4\,d^2}-8\,a^3\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^4\,d^2}-4\,a^2\,b^6\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {-a^4\,d^2}}{b^4\,\left (4\,a^3\,d\,{\left (b^2\right )}^{3/2}+4\,a^5\,d\,\sqrt {b^2}\right )}\right )\right )\,\sqrt {b^2}}{\sqrt {-a^4\,d^2}}-\frac {1}{a\,d\,\mathrm {sinh}\left (c+d\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*atan((4*a^3*b*d*(b^2)^(5/2)*(-a^4*d^2)^(1/2) + 4*a^5*b*d*(b^2)^(3/2)*(-a^4*d^2)^(1/2))*(1/(8*a^3*b^4*d^2*(
a^2 + b^2)^2) - exp(d*x)*exp(c)*(1/(16*a^2*b^5*d^2*(a^2 + b^2)^2) - (a^2 + 2*b^2)^2/(16*a^6*b^5*d^2*(a^2 + b^2
)^2)) + (a^2 + 2*b^2)/(8*a^5*b^4*d^2*(a^2 + b^2)^2))) + 2*atan(-(4*a^3*b^5*(-a^4*d^2)^(1/2) + 4*a*b^7*(-a^4*d^
2)^(1/2) - 4*b^8*exp(3*c)*exp(3*d*x)*(-a^4*d^2)^(1/2) + 4*b^8*exp(d*x)*exp(c)*(-a^4*d^2)^(1/2) - 8*a*b^7*exp(2
*c)*exp(2*d*x)*(-a^4*d^2)^(1/2) + 4*a^2*b^6*exp(d*x)*exp(c)*(-a^4*d^2)^(1/2) - 8*a^3*b^5*exp(2*c)*exp(2*d*x)*(
-a^4*d^2)^(1/2) - 4*a^2*b^6*exp(3*c)*exp(3*d*x)*(-a^4*d^2)^(1/2))/(b^4*(4*a^3*d*(b^2)^(3/2) + 4*a^5*d*(b^2)^(1
/2)))))*(b^2)^(1/2))/(-a^4*d^2)^(1/2) - 1/(a*d*sinh(c + d*x))

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