3.11.24 \(\int \frac {(a+b \coth (x))^2 \text {csch}^2(x)}{c+d \coth (x)} \, dx\) [1024]

Optimal. Leaf size=53 \[ \frac {b (b c-a d) \coth (x)}{d^2}-\frac {(a+b \coth (x))^2}{2 d}-\frac {(b c-a d)^2 \log (c+d \coth (x))}{d^3} \]

[Out]

b*(-a*d+b*c)*coth(x)/d^2-1/2*(a+b*coth(x))^2/d-(-a*d+b*c)^2*ln(c+d*coth(x))/d^3

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Rubi [A]
time = 0.10, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 = {4429, 45} \begin {gather*} -\frac {(b c-a d)^2 \log (c+d \coth (x))}{d^3}+\frac {b \coth (x) (b c-a d)}{d^2}-\frac {(a+b \coth (x))^2}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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])

Rule 4429

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])

Rubi steps

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

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Mathematica [A]
time = 0.37, size = 62, normalized size = 1.17 \begin {gather*} \frac {2 b d (b c-2 a d) \coth (x)-b^2 d^2 \text {csch}^2(x)+2 (b c-a d)^2 (\log (\sinh (x))-\log (d \cosh (x)+c \sinh (x)))}{2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(148\) vs. \(2(51)=102\).
time = 0.95, size = 149, normalized size = 2.81

method	result	size
default	\(-\frac {b \left (\frac {b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) d}{2}+4 a d \tanh \left (\frac {x}{2}\right )-2 b c \tanh \left (\frac {x}{2}\right )\right )}{4 d^{2}}-\frac {b^{2}}{8 d \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a^{2} d^{2}-8 a b c d +4 b^{2} c^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 d^{3}}-\frac {b \left (2 a d -b c \right )}{2 d^{2} \tanh \left (\frac {x}{2}\right )}+\frac {\left (-4 a^{2} d^{2}+8 a b c d -4 b^{2} c^{2}\right ) \ln \left (d \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 c \tanh \left (\frac {x}{2}\right )+d \right )}{4 d^{3}}\)	\(149\)
risch	\(-\frac {2 b \left (2 a d \,{\mathrm e}^{2 x}-b c \,{\mathrm e}^{2 x}+b d \,{\mathrm e}^{2 x}-2 a d +b c \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} d^{2}}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {c -d}{c +d}\right ) a^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{2 x}-\frac {c -d}{c +d}\right ) a b c}{d^{2}}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {c -d}{c +d}\right ) b^{2} c^{2}}{d^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) a^{2}}{d}-\frac {2 \ln \left ({\mathrm e}^{2 x}-1\right ) a b c}{d^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) b^{2} c^{2}}{d^{3}}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (51) = 102\).
time = 0.27, size = 177, normalized size = 3.34 \begin {gather*} 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 )} + 2 \, a 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^{2} \log \left (d \coth \left (x\right ) + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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) + 2*a*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^2*log(d*coth(x) + c)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (51) = 102\).
time = 0.40, size = 694, normalized size = 13.09 \begin {gather*} -\frac {2 \, b^{2} c d - 4 \, a b d^{2} - 2 \, {\left (b^{2} c d - {\left (2 \, a b + b^{2}\right )} d^{2}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (b^{2} c d - {\left (2 \, a b + b^{2}\right )} d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, {\left (b^{2} c d - {\left (2 \, a b + b^{2}\right )} d^{2}\right )} \sinh \left (x\right )^{2} + {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sinh \left (x\right )^{4} + b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )^{3} - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\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^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sinh \left (x\right )^{4} + b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )^{3} - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{d^{3} \cosh \left (x\right )^{4} + 4 \, d^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + d^{3} \sinh \left (x\right )^{4} - 2 \, d^{3} \cosh \left (x\right )^{2} + d^{3} + 2 \, {\left (3 \, d^{3} \cosh \left (x\right )^{2} - d^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (d^{3} \cosh \left (x\right )^{3} - d^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b^2*c*d - 4*a*b*d^2 - 2*(b^2*c*d - (2*a*b + b^2)*d^2)*cosh(x)^2 - 4*(b^2*c*d - (2*a*b + b^2)*d^2)*cosh(x)*
sinh(x) - 2*(b^2*c*d - (2*a*b + b^2)*d^2)*sinh(x)^2 + ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cosh(x)^4 + 4*(b^2*c^2
- 2*a*b*c*d + a^2*d^2)*cosh(x)*sinh(x)^3 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sinh(x)^4 + b^2*c^2 - 2*a*b*c*d + a
^2*d^2 - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cosh(x)^2 - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2 - 3*(b^2*c^2 - 2*a*b*c
*d + a^2*d^2)*cosh(x)^2)*sinh(x)^2 + 4*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cosh(x)^3 - (b^2*c^2 - 2*a*b*c*d + a^2
*d^2)*cosh(x))*sinh(x))*log(2*(d*cosh(x) + c*sinh(x))/(cosh(x) - sinh(x))) - ((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*
cosh(x)^4 + 4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cosh(x)*sinh(x)^3 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sinh(x)^4 +
b^2*c^2 - 2*a*b*c*d + a^2*d^2 - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cosh(x)^2 - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2
- 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cosh(x)^2)*sinh(x)^2 + 4*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*cosh(x)^3 - (b^
2*c^2 - 2*a*b*c*d + a^2*d^2)*cosh(x))*sinh(x))*log(2*sinh(x)/(cosh(x) - sinh(x))))/(d^3*cosh(x)^4 + 4*d^3*cosh
(x)*sinh(x)^3 + d^3*sinh(x)^4 - 2*d^3*cosh(x)^2 + d^3 + 2*(3*d^3*cosh(x)^2 - d^3)*sinh(x)^2 + 4*(d^3*cosh(x)^3
- d^3*cosh(x))*sinh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (51) = 102\).
time = 0.42, size = 265, normalized size = 5.00 \begin {gather*} -\frac {{\left (b^{2} c^{3} - 2 \, a b c^{2} d + b^{2} c^{2} d + a^{2} c d^{2} - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left ({\left | c e^{\left (2 \, x\right )} + d e^{\left (2 \, x\right )} - c + d \right |}\right )}{c d^{3} + d^{4}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{d^{3}} - \frac {3 \, b^{2} c^{2} e^{\left (4 \, x\right )} - 6 \, a b c d e^{\left (4 \, x\right )} + 3 \, a^{2} d^{2} e^{\left (4 \, x\right )} - 6 \, b^{2} c^{2} e^{\left (2 \, x\right )} + 12 \, a b c d e^{\left (2 \, x\right )} - 4 \, b^{2} c d e^{\left (2 \, x\right )} - 6 \, a^{2} d^{2} e^{\left (2 \, x\right )} + 8 \, a b d^{2} e^{\left (2 \, x\right )} + 4 \, b^{2} d^{2} e^{\left (2 \, x\right )} + 3 \, b^{2} c^{2} - 6 \, a b c d + 4 \, b^{2} c d + 3 \, a^{2} d^{2} - 8 \, a b d^{2}}{2 \, d^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*c^3 - 2*a*b*c^2*d + b^2*c^2*d + a^2*c*d^2 - 2*a*b*c*d^2 + a^2*d^3)*log(abs(c*e^(2*x) + d*e^(2*x) - c + d
))/(c*d^3 + d^4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(abs(e^(2*x) - 1))/d^3 - 1/2*(3*b^2*c^2*e^(4*x) - 6*a*b*
c*d*e^(4*x) + 3*a^2*d^2*e^(4*x) - 6*b^2*c^2*e^(2*x) + 12*a*b*c*d*e^(2*x) - 4*b^2*c*d*e^(2*x) - 6*a^2*d^2*e^(2*
x) + 8*a*b*d^2*e^(2*x) + 4*b^2*d^2*e^(2*x) + 3*b^2*c^2 - 6*a*b*c*d + 4*b^2*c*d + 3*a^2*d^2 - 8*a*b*d^2)/(d^3*(
e^(2*x) - 1)^2)

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Mupad [B]
time = 2.16, size = 107, normalized size = 2.02 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,{\left (a\,d-b\,c\right )}^2}{d^3}-\frac {\ln \left (d-c+d\,{\mathrm {e}}^{2\,x}+c\,{\mathrm {e}}^{2\,x}\right )\,{\left (a\,d-b\,c\right )}^2}{d^3}-\frac {2\,\left (b^2\,d-b^2\,c+2\,a\,b\,d\right )}{d^2\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2\,b^2}{d\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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