### 3.1025 $$\int \frac{(a+b \coth (x))^3 \text{csch}^2(x)}{c+d \coth (x)} \, dx$$

Optimal. Leaf size=78 $-\frac{b \coth (x) (b c-a d)^2}{d^3}+\frac{(b c-a d) (a+b \coth (x))^2}{2 d^2}+\frac{(b c-a d)^3 \log (c+d \coth (x))}{d^4}-\frac{(a+b \coth (x))^3}{3 d}$

[Out]

-((b*(b*c - a*d)^2*Coth[x])/d^3) + ((b*c - a*d)*(a + b*Coth[x])^2)/(2*d^2) - (a + b*Coth[x])^3/(3*d) + ((b*c -
a*d)^3*Log[c + d*Coth[x]])/d^4

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Rubi [A]  time = 0.153484, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.095, Rules used = {4344, 43} $-\frac{b \coth (x) (b c-a d)^2}{d^3}+\frac{(b c-a d) (a+b \coth (x))^2}{2 d^2}+\frac{(b c-a d)^3 \log (c+d \coth (x))}{d^4}-\frac{(a+b \coth (x))^3}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[((a + b*Coth[x])^3*Csch[x]^2)/(c + d*Coth[x]),x]

[Out]

-((b*(b*c - a*d)^2*Coth[x])/d^3) + ((b*c - a*d)*(a + b*Coth[x])^2)/(2*d^2) - (a + b*Coth[x])^3/(3*d) + ((b*c -
a*d)^3*Log[c + d*Coth[x]])/d^4

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))^3 \text{csch}^2(x)}{c+d \coth (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^3}{c+d x} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx,x,\coth (x)\right )\\ &=-\frac{b (b c-a d)^2 \coth (x)}{d^3}+\frac{(b c-a d) (a+b \coth (x))^2}{2 d^2}-\frac{(a+b \coth (x))^3}{3 d}+\frac{(b c-a d)^3 \log (c+d \coth (x))}{d^4}\\ \end{align*}

Mathematica [A]  time = 1.14586, size = 136, normalized size = 1.74 $\frac{(a+b \coth (x))^3 (c \sinh (x)+d \cosh (x)) \left (-b d \left (\sinh (2 x) \left (9 a^2 d^2-9 a b c d+b^2 \left (3 c^2+d^2\right )\right )-3 b d (b c-3 a d)\right )-6 \sinh ^2(x) (b c-a d)^3 (\log (\sinh (x))-\log (c \sinh (x)+d \cosh (x)))-2 b^3 d^3 \coth (x)\right )}{6 d^4 (c+d \coth (x)) (a \sinh (x)+b \cosh (x))^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((a + b*Coth[x])^3*Csch[x]^2)/(c + d*Coth[x]),x]

[Out]

((a + b*Coth[x])^3*(d*Cosh[x] + c*Sinh[x])*(-2*b^3*d^3*Coth[x] - 6*(b*c - a*d)^3*(Log[Sinh[x]] - Log[d*Cosh[x]
+ c*Sinh[x]])*Sinh[x]^2 - b*d*(-3*b*d*(b*c - 3*a*d) + (-9*a*b*c*d + 9*a^2*d^2 + b^2*(3*c^2 + d^2))*Sinh[2*x])
))/(6*d^4*(c + d*Coth[x])*(b*Cosh[x] + a*Sinh[x])^3)

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Maple [B]  time = 0.073, size = 378, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*coth(x))^3*csch(x)^2/(c+d*coth(x)),x)

[Out]

-1/24*b^3/d*tanh(1/2*x)^3-3/8*b^2/d*tanh(1/2*x)^2*a+1/8*b^3/d^2*tanh(1/2*x)^2*c-3/2*b/d*a^2*tanh(1/2*x)+3/2*b^
2/d^2*c*a*tanh(1/2*x)-1/2*b^3/d^3*c^2*tanh(1/2*x)-1/8*b^3/d*tanh(1/2*x)-1/d*ln(tanh(1/2*x)^2*d+2*c*tanh(1/2*x)
+d)*a^3+3/d^2*ln(tanh(1/2*x)^2*d+2*c*tanh(1/2*x)+d)*a^2*b*c-3/d^3*ln(tanh(1/2*x)^2*d+2*c*tanh(1/2*x)+d)*c^2*b^
2*a+1/d^4*ln(tanh(1/2*x)^2*d+2*c*tanh(1/2*x)+d)*c^3*b^3-1/24*b^3/d/tanh(1/2*x)^3+1/d*ln(tanh(1/2*x))*a^3-3/d^2
*ln(tanh(1/2*x))*a^2*b*c+3/d^3*ln(tanh(1/2*x))*c^2*b^2*a-1/d^4*ln(tanh(1/2*x))*c^3*b^3-3/2*b/d/tanh(1/2*x)*a^2
+3/2*b^2/d^2/tanh(1/2*x)*c*a-1/2*b^3/d^3/tanh(1/2*x)*c^2-1/8*b^3/d/tanh(1/2*x)-3/8*b^2/d/tanh(1/2*x)^2*a+1/8*b
^3/d^2/tanh(1/2*x)^2*c

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Maxima [B]  time = 1.23211, size = 427, normalized size = 5.47 \begin{align*} \frac{1}{3} \, b^{3}{\left (\frac{2 \,{\left (3 \, c^{2} + d^{2} - 3 \,{\left (2 \, c^{2} + c d\right )} e^{\left (-2 \, x\right )} + 3 \,{\left (c^{2} + c d + d^{2}\right )} e^{\left (-4 \, x\right )}\right )}}{3 \, d^{3} e^{\left (-2 \, x\right )} - 3 \, d^{3} e^{\left (-4 \, x\right )} + d^{3} e^{\left (-6 \, x\right )} - d^{3}} + \frac{3 \, c^{3} \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{4}} - \frac{3 \, c^{3} \log \left (e^{\left (-x\right )} + 1\right )}{d^{4}} - \frac{3 \, c^{3} \log \left (e^{\left (-x\right )} - 1\right )}{d^{4}}\right )} + 3 \, a b^{2}{\left (\frac{2 \,{\left ({\left (c + d\right )} e^{\left (-2 \, x\right )} - c\right )}}{2 \, d^{2} e^{\left (-2 \, x\right )} - d^{2} e^{\left (-4 \, x\right )} - d^{2}} - \frac{c^{2} \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{3}} + \frac{c^{2} \log \left (e^{\left (-x\right )} + 1\right )}{d^{3}} + \frac{c^{2} \log \left (e^{\left (-x\right )} - 1\right )}{d^{3}}\right )} + 3 \, a^{2} 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^{3} \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))^3*csch(x)^2/(c+d*coth(x)),x, algorithm="maxima")

[Out]

1/3*b^3*(2*(3*c^2 + d^2 - 3*(2*c^2 + c*d)*e^(-2*x) + 3*(c^2 + c*d + d^2)*e^(-4*x))/(3*d^3*e^(-2*x) - 3*d^3*e^(
-4*x) + d^3*e^(-6*x) - d^3) + 3*c^3*log(-(c - d)*e^(-2*x) + c + d)/d^4 - 3*c^3*log(e^(-x) + 1)/d^4 - 3*c^3*log
(e^(-x) - 1)/d^4) + 3*a*b^2*(2*((c + d)*e^(-2*x) - c)/(2*d^2*e^(-2*x) - d^2*e^(-4*x) - d^2) - c^2*log(-(c - d)
*e^(-2*x) + c + d)/d^3 + c^2*log(e^(-x) + 1)/d^3 + c^2*log(e^(-x) - 1)/d^3) + 3*a^2*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^3*log(d*coth(x) + c)/d

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Fricas [B]  time = 2.78804, size = 4383, normalized size = 56.19 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(x))^3*csch(x)^2/(c+d*coth(x)),x, algorithm="fricas")

[Out]

-1/3*(6*b^3*c^2*d - 18*a*b^2*c*d^2 + 6*(b^3*c^2*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*cos
h(x)^4 + 24*(b^3*c^2*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*cosh(x)*sinh(x)^3 + 6*(b^3*c^2
*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*sinh(x)^4 + 2*(9*a^2*b + b^3)*d^3 - 6*(2*b^3*c^2*d
- (6*a*b^2 + b^3)*c*d^2 + 3*(2*a^2*b + a*b^2)*d^3)*cosh(x)^2 - 6*(2*b^3*c^2*d - (6*a*b^2 + b^3)*c*d^2 + 3*(2*
a^2*b + a*b^2)*d^3 - 6*(b^3*c^2*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*cosh(x)^2)*sinh(x)^
2 - 3*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^6 + 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*
d^2 - a^3*d^3)*cosh(x)*sinh(x)^5 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sinh(x)^6 - b^3*c^3 + 3
*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^4 - 3*(
b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 - 5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh
(x)^2)*sinh(x)^4 + 4*(5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^3 - 3*(b^3*c^3 - 3*a*b^2*c
^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x))*sinh(x)^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cos
h(x)^2 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 + 5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a
^3*d^3)*cosh(x)^4 - 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^2)*sinh(x)^2 + 6*((b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^5 - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*co
sh(x)^3 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x))*sinh(x))*log(2*(d*cosh(x) + c*sinh(x))/
(cosh(x) - sinh(x))) + 3*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^6 + 6*(b^3*c^3 - 3*a*b^2
*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)*sinh(x)^5 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sinh
(x)^6 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d
^3)*cosh(x)^4 - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 - 5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*
d^2 - a^3*d^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^3 - 3*(
b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x))*sinh(x)^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c
*d^2 - a^3*d^3)*cosh(x)^2 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 + 5*(b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^4 - 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^2)*sinh(x
)^2 + 6*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^5 - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*
c*d^2 - a^3*d^3)*cosh(x)^3 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x))*sinh(x))*log(2*sinh(
x)/(cosh(x) - sinh(x))) + 12*(2*(b^3*c^2*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*cosh(x)^3
- (2*b^3*c^2*d - (6*a*b^2 + b^3)*c*d^2 + 3*(2*a^2*b + a*b^2)*d^3)*cosh(x))*sinh(x))/(d^4*cosh(x)^6 + 6*d^4*cos
h(x)*sinh(x)^5 + d^4*sinh(x)^6 - 3*d^4*cosh(x)^4 + 3*d^4*cosh(x)^2 + 3*(5*d^4*cosh(x)^2 - d^4)*sinh(x)^4 - d^4
+ 4*(5*d^4*cosh(x)^3 - 3*d^4*cosh(x))*sinh(x)^3 + 3*(5*d^4*cosh(x)^4 - 6*d^4*cosh(x)^2 + d^4)*sinh(x)^2 + 6*(
d^4*cosh(x)^5 - 2*d^4*cosh(x)^3 + d^4*cosh(x))*sinh(x))

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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 )^{3} \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))**3*csch(x)**2/(c+d*coth(x)),x)

[Out]

Integral((a + b*coth(x))**3*csch(x)**2/(c + d*coth(x)), x)

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Giac [B]  time = 1.20912, size = 734, normalized size = 9.41 \begin{align*} \frac{{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + b^{3} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - 3 \, a b^{2} c^{2} d^{2} - a^{3} c d^{3} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \log \left ({\left | c e^{\left (2 \, x\right )} + d e^{\left (2 \, x\right )} - c + d \right |}\right )}{c d^{4} + d^{5}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{d^{4}} + \frac{11 \, b^{3} c^{3} e^{\left (6 \, x\right )} - 33 \, a b^{2} c^{2} d e^{\left (6 \, x\right )} + 33 \, a^{2} b c d^{2} e^{\left (6 \, x\right )} - 11 \, a^{3} d^{3} e^{\left (6 \, x\right )} - 33 \, b^{3} c^{3} e^{\left (4 \, x\right )} + 99 \, a b^{2} c^{2} d e^{\left (4 \, x\right )} - 12 \, b^{3} c^{2} d e^{\left (4 \, x\right )} - 99 \, a^{2} b c d^{2} e^{\left (4 \, x\right )} + 36 \, a b^{2} c d^{2} e^{\left (4 \, x\right )} + 12 \, b^{3} c d^{2} e^{\left (4 \, x\right )} + 33 \, a^{3} d^{3} e^{\left (4 \, x\right )} - 36 \, a^{2} b d^{3} e^{\left (4 \, x\right )} - 36 \, a b^{2} d^{3} e^{\left (4 \, x\right )} - 12 \, b^{3} d^{3} e^{\left (4 \, x\right )} + 33 \, b^{3} c^{3} e^{\left (2 \, x\right )} - 99 \, a b^{2} c^{2} d e^{\left (2 \, x\right )} + 24 \, b^{3} c^{2} d e^{\left (2 \, x\right )} + 99 \, a^{2} b c d^{2} e^{\left (2 \, x\right )} - 72 \, a b^{2} c d^{2} e^{\left (2 \, x\right )} - 12 \, b^{3} c d^{2} e^{\left (2 \, x\right )} - 33 \, a^{3} d^{3} e^{\left (2 \, x\right )} + 72 \, a^{2} b d^{3} e^{\left (2 \, x\right )} + 36 \, a b^{2} d^{3} e^{\left (2 \, x\right )} - 11 \, b^{3} c^{3} + 33 \, a b^{2} c^{2} d - 12 \, b^{3} c^{2} d - 33 \, a^{2} b c d^{2} + 36 \, a b^{2} c d^{2} + 11 \, a^{3} d^{3} - 36 \, a^{2} b d^{3} - 4 \, b^{3} d^{3}}{6 \, d^{4}{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(x))^3*csch(x)^2/(c+d*coth(x)),x, algorithm="giac")

[Out]

(b^3*c^4 - 3*a*b^2*c^3*d + b^3*c^3*d + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^2*d^2 - a^3*c*d^3 + 3*a^2*b*c*d^3 - a^3*d^4
)*log(abs(c*e^(2*x) + d*e^(2*x) - c + d))/(c*d^4 + d^5) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*
log(abs(e^(2*x) - 1))/d^4 + 1/6*(11*b^3*c^3*e^(6*x) - 33*a*b^2*c^2*d*e^(6*x) + 33*a^2*b*c*d^2*e^(6*x) - 11*a^3
*d^3*e^(6*x) - 33*b^3*c^3*e^(4*x) + 99*a*b^2*c^2*d*e^(4*x) - 12*b^3*c^2*d*e^(4*x) - 99*a^2*b*c*d^2*e^(4*x) + 3
6*a*b^2*c*d^2*e^(4*x) + 12*b^3*c*d^2*e^(4*x) + 33*a^3*d^3*e^(4*x) - 36*a^2*b*d^3*e^(4*x) - 36*a*b^2*d^3*e^(4*x
) - 12*b^3*d^3*e^(4*x) + 33*b^3*c^3*e^(2*x) - 99*a*b^2*c^2*d*e^(2*x) + 24*b^3*c^2*d*e^(2*x) + 99*a^2*b*c*d^2*e
^(2*x) - 72*a*b^2*c*d^2*e^(2*x) - 12*b^3*c*d^2*e^(2*x) - 33*a^3*d^3*e^(2*x) + 72*a^2*b*d^3*e^(2*x) + 36*a*b^2*
d^3*e^(2*x) - 11*b^3*c^3 + 33*a*b^2*c^2*d - 12*b^3*c^2*d - 33*a^2*b*c*d^2 + 36*a*b^2*c*d^2 + 11*a^3*d^3 - 36*a
^2*b*d^3 - 4*b^3*d^3)/(d^4*(e^(2*x) - 1)^3)