3.2.41 \(\int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx\) [141]

Optimal. Leaf size=76 \[ -\frac {b x}{a^2-b^2}+\frac {b \coth (x)}{a^2}-\frac {\coth ^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}+\frac {b^4 \log (a \cosh (x)+b \sinh (x))}{a^3 \left (a^2-b^2\right )} \]

[Out]

-b*x/(a^2-b^2)+b*coth(x)/a^2-1/2*coth(x)^2/a+(a^2+b^2)*ln(sinh(x))/a^3+b^4*ln(a*cosh(x)+b*sinh(x))/a^3/(a^2-b^
2)

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Rubi [A]
time = 0.21, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3650, 3730, 3733, 3611, 3556} \begin {gather*} -\frac {b x}{a^2-b^2}+\frac {b \coth (x)}{a^2}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}+\frac {b^4 \log (a \cosh (x)+b \sinh (x))}{a^3 \left (a^2-b^2\right )}-\frac {\coth ^2(x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(a + b*Tanh[x]),x]

[Out]

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

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3733

Int[((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C) - b*(A*d - C*d))*(x/((a^2 + b^2)*(c^2 + d^2))), x] + (Dist[
(A*b^2 + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C
+ A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c,
d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

Rubi steps

\begin {align*} \int \frac {\coth ^3(x)}{a+b \tanh (x)} \, dx &=-\frac {\coth ^2(x)}{2 a}-\frac {i \int \frac {\coth ^2(x) \left (-2 i b+2 i a \tanh (x)+2 i b \tanh ^2(x)\right )}{a+b \tanh (x)} \, dx}{2 a}\\ &=\frac {b \coth (x)}{a^2}-\frac {\coth ^2(x)}{2 a}-\frac {\int \frac {\coth (x) \left (-2 \left (a^2+b^2\right )+2 b^2 \tanh ^2(x)\right )}{a+b \tanh (x)} \, dx}{2 a^2}\\ &=-\frac {b x}{a^2-b^2}+\frac {b \coth (x)}{a^2}-\frac {\coth ^2(x)}{2 a}+\frac {\left (i b^4\right ) \int \frac {-i b-i a \tanh (x)}{a+b \tanh (x)} \, dx}{a^3 \left (a^2-b^2\right )}+\frac {\left (a^2+b^2\right ) \int \coth (x) \, dx}{a^3}\\ &=-\frac {b x}{a^2-b^2}+\frac {b \coth (x)}{a^2}-\frac {\coth ^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sinh (x))}{a^3}+\frac {b^4 \log (a \cosh (x)+b \sinh (x))}{a^3 \left (a^2-b^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 91, normalized size = 1.20 \begin {gather*} \frac {-2 a^3 b x+2 a b \left (a^2-b^2\right ) \coth (x)+\left (-a^4+a^2 b^2\right ) \text {csch}^2(x)+2 a^4 \log (\sinh (x))-2 b^4 \log (\sinh (x))+2 b^4 \log (a \cosh (x)+b \sinh (x))}{2 a^3 (a-b) (a+b)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^3/(a + b*Tanh[x]),x]

[Out]

(-2*a^3*b*x + 2*a*b*(a^2 - b^2)*Coth[x] + (-a^4 + a^2*b^2)*Csch[x]^2 + 2*a^4*Log[Sinh[x]] - 2*b^4*Log[Sinh[x]]
+ 2*b^4*Log[a*Cosh[x] + b*Sinh[x]])/(2*a^3*(a - b)*(a + b))

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Maple [A]
time = 0.64, size = 135, normalized size = 1.78

method	result	size
risch	\(\frac {x}{a +b}-\frac {2 x}{a}-\frac {2 x \,b^{2}}{a^{3}}-\frac {2 x \,b^{4}}{a^{3} \left (a^{2}-b^{2}\right )}-\frac {2 \left ({\mathrm e}^{2 x} a -b \,{\mathrm e}^{2 x}+b \right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} a^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) b^{2}}{a^{3}}+\frac {b^{4} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{3} \left (a^{2}-b^{2}\right )}\)	\(132\)
default	\(-\frac {\frac {a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}-2 b \tanh \left (\frac {x}{2}\right )}{4 a^{2}}+\frac {b^{4} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 b \tanh \left (\frac {x}{2}\right )+a \right )}{a^{3} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a +b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a -b}-\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {x}{2}\right )}\)	\(135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(a+b*tanh(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 121, normalized size = 1.59 \begin {gather*} \frac {b^{4} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{5} - a^{3} b^{2}} + \frac {2 \, {\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} - b\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} - a^{2} e^{\left (-4 \, x\right )} - a^{2}} + \frac {x}{a + b} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*tanh(x)),x, algorithm="maxima")

[Out]

b^4*log(-(a - b)*e^(-2*x) - a - b)/(a^5 - a^3*b^2) + 2*((a + b)*e^(-2*x) - b)/(2*a^2*e^(-2*x) - a^2*e^(-4*x) -
a^2) + x/(a + b) + (a^2 + b^2)*log(e^(-x) + 1)/a^3 + (a^2 + b^2)*log(e^(-x) - 1)/a^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (74) = 148\).
time = 0.43, size = 641, normalized size = 8.43 \begin {gather*} -\frac {{\left (a^{4} + a^{3} b\right )} x \cosh \left (x\right )^{4} + 4 \, {\left (a^{4} + a^{3} b\right )} x \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{4} + a^{3} b\right )} x \sinh \left (x\right )^{4} + 2 \, a^{3} b - 2 \, a b^{3} + 2 \, {\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3} - {\left (a^{4} + a^{3} b\right )} x\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3} + 3 \, {\left (a^{4} + a^{3} b\right )} x \cosh \left (x\right )^{2} - {\left (a^{4} + a^{3} b\right )} x\right )} \sinh \left (x\right )^{2} + {\left (a^{4} + a^{3} b\right )} x - {\left (b^{4} \cosh \left (x\right )^{4} + 4 \, b^{4} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{4} \sinh \left (x\right )^{4} - 2 \, b^{4} \cosh \left (x\right )^{2} + b^{4} + 2 \, {\left (3 \, b^{4} \cosh \left (x\right )^{2} - b^{4}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b^{4} \cosh \left (x\right )^{3} - b^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left ({\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{4} - b^{4}\right )} \sinh \left (x\right )^{4} + a^{4} - b^{4} - 2 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{4} - b^{4} - 3 \, {\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{4} - b^{4}\right )} \cosh \left (x\right )^{3} - {\left (a^{4} - b^{4}\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 ) + 4 \, {\left ({\left (a^{4} + a^{3} b\right )} x \cosh \left (x\right )^{3} + {\left (a^{4} - a^{3} b - a^{2} b^{2} + a b^{3} - {\left (a^{4} + a^{3} b\right )} x\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{a^{5} - a^{3} b^{2} + {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{5} - a^{3} b^{2}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{5} - a^{3} b^{2} - 3 \, {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )^{3} - {\left (a^{5} - a^{3} b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*tanh(x)),x, algorithm="fricas")

[Out]

-((a^4 + a^3*b)*x*cosh(x)^4 + 4*(a^4 + a^3*b)*x*cosh(x)*sinh(x)^3 + (a^4 + a^3*b)*x*sinh(x)^4 + 2*a^3*b - 2*a*
b^3 + 2*(a^4 - a^3*b - a^2*b^2 + a*b^3 - (a^4 + a^3*b)*x)*cosh(x)^2 + 2*(a^4 - a^3*b - a^2*b^2 + a*b^3 + 3*(a^
4 + a^3*b)*x*cosh(x)^2 - (a^4 + a^3*b)*x)*sinh(x)^2 + (a^4 + a^3*b)*x - (b^4*cosh(x)^4 + 4*b^4*cosh(x)*sinh(x)
^3 + b^4*sinh(x)^4 - 2*b^4*cosh(x)^2 + b^4 + 2*(3*b^4*cosh(x)^2 - b^4)*sinh(x)^2 + 4*(b^4*cosh(x)^3 - b^4*cosh
(x))*sinh(x))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) - ((a^4 - b^4)*cosh(x)^4 + 4*(a^4 - b^4)*cosh
(x)*sinh(x)^3 + (a^4 - b^4)*sinh(x)^4 + a^4 - b^4 - 2*(a^4 - b^4)*cosh(x)^2 - 2*(a^4 - b^4 - 3*(a^4 - b^4)*cos
h(x)^2)*sinh(x)^2 + 4*((a^4 - b^4)*cosh(x)^3 - (a^4 - b^4)*cosh(x))*sinh(x))*log(2*sinh(x)/(cosh(x) - sinh(x))
) + 4*((a^4 + a^3*b)*x*cosh(x)^3 + (a^4 - a^3*b - a^2*b^2 + a*b^3 - (a^4 + a^3*b)*x)*cosh(x))*sinh(x))/(a^5 -
a^3*b^2 + (a^5 - a^3*b^2)*cosh(x)^4 + 4*(a^5 - a^3*b^2)*cosh(x)*sinh(x)^3 + (a^5 - a^3*b^2)*sinh(x)^4 - 2*(a^5
- a^3*b^2)*cosh(x)^2 - 2*(a^5 - a^3*b^2 - 3*(a^5 - a^3*b^2)*cosh(x)^2)*sinh(x)^2 + 4*((a^5 - a^3*b^2)*cosh(x)
^3 - (a^5 - a^3*b^2)*cosh(x))*sinh(x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**3/(a+b*tanh(x)),x)

[Out]

Integral(coth(x)**3/(a + b*tanh(x)), x)

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Giac [A]
time = 0.41, size = 97, normalized size = 1.28 \begin {gather*} \frac {b^{4} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{5} - a^{3} b^{2}} - \frac {x}{a - b} + \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{a^{3}} - \frac {2 \, {\left (a b + {\left (a^{2} - a b\right )} e^{\left (2 \, x\right )}\right )}}{a^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^3/(a+b*tanh(x)),x, algorithm="giac")

[Out]

b^4*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^5 - a^3*b^2) - x/(a - b) + (a^2 + b^2)*log(abs(e^(2*x) - 1))/a^
3 - 2*(a*b + (a^2 - a*b)*e^(2*x))/(a^3*(e^(2*x) - 1)^2)

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Mupad [B]
time = 1.42, size = 111, normalized size = 1.46 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^2+b^2\right )}{a^3}-\frac {x}{a-b}-\frac {2}{a\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}+\frac {b^4\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^5-a^3\,b^2}-\frac {2\,\left (a^2-b^2\right )}{a^2\,\left (a+b\right )\,\left ({\mathrm {e}}^{2\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(a + b*tanh(x)),x)

[Out]

(log(exp(2*x) - 1)*(a^2 + b^2))/a^3 - x/(a - b) - 2/(a*(exp(4*x) - 2*exp(2*x) + 1)) + (b^4*log(a - b + a*exp(2
*x) + b*exp(2*x)))/(a^5 - a^3*b^2) - (2*(a^2 - b^2))/(a^2*(a + b)*(exp(2*x) - 1))

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