### 3.1023 $$\int \frac{(a+b \coth (x)) \text{csch}^2(x)}{c+d \coth (x)} \, dx$$

Optimal. Leaf size=28 $\frac{(b c-a d) \log (c+d \coth (x))}{d^2}-\frac{b \coth (x)}{d}$

[Out]

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

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Rubi [A]  time = 0.0967657, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.105, Rules used = {4344, 43} $\frac{(b c-a d) \log (c+d \coth (x))}{d^2}-\frac{b \coth (x)}{d}$

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*Coth[x])*Csch[x]^2)/(c + d*Coth[x]),x]

[Out]

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

Rule 4344

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -Dist[d
/(b*c), Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d], x] /; FunctionOfQ[Cot[c*(
a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.300637, size = 56, normalized size = 2. $\frac{\sinh (x) (a+b \coth (x)) (-(b c-a d) (\log (\sinh (x))-\log (c \sinh (x)+d \cosh (x)))-b d \coth (x))}{d^2 (a \sinh (x)+b \cosh (x))}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*Coth[x])*Csch[x]^2)/(c + d*Coth[x]),x]

[Out]

((a + b*Coth[x])*(-(b*d*Coth[x]) - (b*c - a*d)*(Log[Sinh[x]] - Log[d*Cosh[x] + c*Sinh[x]]))*Sinh[x])/(d^2*(b*C
osh[x] + a*Sinh[x]))

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Maple [B]  time = 0.036, size = 94, normalized size = 3.4 \begin{align*} -{\frac{b}{2\,d}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{a}{d}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}d+2\,c\tanh \left ( x/2 \right ) +d \right ) }+{\frac{cb}{{d}^{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}d+2\,c\tanh \left ( x/2 \right ) +d \right ) }-{\frac{b}{2\,d} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{a}{d}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{cb}{{d}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*coth(x))*csch(x)^2/(c+d*coth(x)),x)

[Out]

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

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Maxima [B]  time = 1.19942, size = 104, normalized size = 3.71 \begin{align*} b{\left (\frac{c \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{2}} - \frac{c \log \left (e^{\left (-x\right )} + 1\right )}{d^{2}} - \frac{c \log \left (e^{\left (-x\right )} - 1\right )}{d^{2}} + \frac{2}{d e^{\left (-2 \, x\right )} - d}\right )} - \frac{a \log \left (d \coth \left (x\right ) + c\right )}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.20974, size = 467, normalized size = 16.68 \begin{align*} -\frac{2 \, b d -{\left ({\left (b c - a d\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b c - a d\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b c - a d\right )} \sinh \left (x\right )^{2} - b c + a d\right )} \log \left (\frac{2 \,{\left (d \cosh \left (x\right ) + c \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) +{\left ({\left (b c - a d\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b c - a d\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b c - a d\right )} \sinh \left (x\right )^{2} - b c + a d\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{d^{2} \cosh \left (x\right )^{2} + 2 \, d^{2} \cosh \left (x\right ) \sinh \left (x\right ) + d^{2} \sinh \left (x\right )^{2} - d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*d - ((b*c - a*d)*cosh(x)^2 + 2*(b*c - a*d)*cosh(x)*sinh(x) + (b*c - a*d)*sinh(x)^2 - b*c + a*d)*log(2*(d
*cosh(x) + c*sinh(x))/(cosh(x) - sinh(x))) + ((b*c - a*d)*cosh(x)^2 + 2*(b*c - a*d)*cosh(x)*sinh(x) + (b*c - a
*d)*sinh(x)^2 - b*c + a*d)*log(2*sinh(x)/(cosh(x) - sinh(x))))/(d^2*cosh(x)^2 + 2*d^2*cosh(x)*sinh(x) + d^2*si
nh(x)^2 - d^2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(x))*csch(x)**2/(c+d*coth(x)),x)

[Out]

Integral((a + b*coth(x))*csch(x)**2/(c + d*coth(x)), x)

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Giac [B]  time = 1.12092, size = 153, normalized size = 5.46 \begin{align*} \frac{{\left (b c^{2} - a c d + b c d - a d^{2}\right )} \log \left ({\left | c e^{\left (2 \, x\right )} + d e^{\left (2 \, x\right )} - c + d \right |}\right )}{c d^{2} + d^{3}} - \frac{{\left (b c - a d\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{d^{2}} + \frac{b c e^{\left (2 \, x\right )} - a d e^{\left (2 \, x\right )} - b c + a d - 2 \, b d}{d^{2}{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c^2 - a*c*d + b*c*d - a*d^2)*log(abs(c*e^(2*x) + d*e^(2*x) - c + d))/(c*d^2 + d^3) - (b*c - a*d)*log(abs(e^
(2*x) - 1))/d^2 + (b*c*e^(2*x) - a*d*e^(2*x) - b*c + a*d - 2*b*d)/(d^2*(e^(2*x) - 1))