3.2.88 \(\int \frac {\coth ^3(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\) [188]

Optimal. Leaf size=124 \[ -\frac {\coth ^2(c+d x)}{2 a^2 d}+\frac {\log (\cosh (c+d x))}{(a+b)^2 d}+\frac {(a-2 b) \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (3 a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 (a+b)^2 d}-\frac {b^2}{2 a^2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 457, 90} \begin {gather*} \frac {b^2 (3 a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 d (a+b)^2}+\frac {(a-2 b) \log (\tanh (c+d x))}{a^3 d}-\frac {b^2}{2 a^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\coth ^2(c+d x)}{2 a^2 d}+\frac {\log (\cosh (c+d x))}{d (a+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Coth[c + d*x]^2/(a^2*d) + Log[Cosh[c + d*x]]/((a + b)^2*d) + ((a - 2*b)*Log[Tanh[c + d*x]])/(a^3*d) + (b^
2*(3*a + 2*b)*Log[a + b*Tanh[c + d*x]^2])/(2*a^3*(a + b)^2*d) - b^2/(2*a^2*(a + b)*d*(a + b*Tanh[c + d*x]^2))

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^3 \left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{(1-x) x^2 (a+b x)^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{(a+b)^2 (-1+x)}+\frac {1}{a^2 x^2}+\frac {a-2 b}{a^3 x}+\frac {b^3}{a^2 (a+b) (a+b x)^2}+\frac {b^3 (3 a+2 b)}{a^3 (a+b)^2 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\coth ^2(c+d x)}{2 a^2 d}+\frac {\log (\cosh (c+d x))}{(a+b)^2 d}+\frac {(a-2 b) \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (3 a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 (a+b)^2 d}-\frac {b^2}{2 a^2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 93, normalized size = 0.75 \begin {gather*} \frac {-\frac {\coth ^2(c+d x)}{a^2}+\frac {b^3}{a^3 (a+b) \left (b+a \coth ^2(c+d x)\right )}+\frac {b^2 (3 a+2 b) \log \left (b+a \coth ^2(c+d x)\right )}{a^3 (a+b)^2}+\frac {2 \log (\sinh (c+d x))}{(a+b)^2}}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 3.16, size = 226, normalized size = 1.82

method	result	size
derivativedivides	\(\frac {-\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2}}+\frac {b^{2} \left (\frac {2 b \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (3 a +2 b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )}{2}\right )}{\left (a +b \right )^{2} a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{\left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{\left (a +b \right )^{2}}-\frac {1}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 b +4 a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}}{d}\)	\(226\)
default	\(\frac {-\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2}}+\frac {b^{2} \left (\frac {2 b \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (3 a +2 b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )}{2}\right )}{\left (a +b \right )^{2} a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{\left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{\left (a +b \right )^{2}}-\frac {1}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 b +4 a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}}{d}\)	\(226\)
risch	\(\frac {x}{a^{2}+2 a b +b^{2}}+\frac {4 b x}{a^{3}}+\frac {4 b c}{a^{3} d}-\frac {2 x}{a^{2}}-\frac {2 c}{a^{2} d}-\frac {6 b^{2} x}{a^{2} \left (a^{2}+2 a b +b^{2}\right )}-\frac {6 b^{2} c}{a^{2} d \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 b^{3} x}{a^{3} \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 b^{3} c}{a^{3} d \left (a^{2}+2 a b +b^{2}\right )}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{3} {\mathrm e}^{4 d x +4 c}+3 a^{2} b \,{\mathrm e}^{4 d x +4 c}+3 a \,b^{2} {\mathrm e}^{4 d x +4 c}+2 b^{3} {\mathrm e}^{4 d x +4 c}+2 a^{3} {\mathrm e}^{2 d x +2 c}+2 a^{2} b \,{\mathrm e}^{2 d x +2 c}-2 a \,b^{2} {\mathrm e}^{2 d x +2 c}-4 b^{3} {\mathrm e}^{2 d x +2 c}+a^{3}+3 a^{2} b +3 a \,b^{2}+2 b^{3}\right )}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}-\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 a^{2} d \left (a^{2}+2 a b +b^{2}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{a^{3} d \left (a^{2}+2 a b +b^{2}\right )}\)	\(518\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (118) = 236\).
time = 0.28, size = 402, normalized size = 3.24 \begin {gather*} \frac {{\left (3 \, a b^{2} + 2 \, b^{3}\right )} \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d} + \frac {d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {2 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{3} + a^{2} b - a b^{2} - 2 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} - 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (a^{5} - a^{4} b - 5 \, a^{3} b^{2} - 3 \, a^{2} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {{\left (a - 2 \, b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {{\left (a - 2 \, b\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*a*b^2 + 2*b^3)*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^5 + 2*a^4*b + a^3
*b^2)*d) + (d*x + c)/((a^2 + 2*a*b + b^2)*d) - 2*((a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*e^(-2*d*x - 2*c) + 2*(a^3
+ a^2*b - a*b^2 - 2*b^3)*e^(-4*d*x - 4*c) + (a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*e^(-6*d*x - 6*c))/((a^5 + 3*a^4*
b + 3*a^3*b^2 + a^2*b^3 - 4*(a^4*b + 2*a^3*b^2 + a^2*b^3)*e^(-2*d*x - 2*c) - 2*(a^5 - a^4*b - 5*a^3*b^2 - 3*a^
2*b^3)*e^(-4*d*x - 4*c) - 4*(a^4*b + 2*a^3*b^2 + a^2*b^3)*e^(-6*d*x - 6*c) + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*
b^3)*e^(-8*d*x - 8*c))*d) + (a - 2*b)*log(e^(-d*x - c) + 1)/(a^3*d) + (a - 2*b)*log(e^(-d*x - c) - 1)/(a^3*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3468 vs. \(2 (118) = 236\).
time = 0.52, size = 3468, normalized size = 27.97 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(a^4 + a^3*b)*d*x*cosh(d*x + c)^8 + 16*(a^4 + a^3*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 2*(a^4 + a^3*
b)*d*x*sinh(d*x + c)^8 - 4*(2*a^3*b*d*x - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^6 - 4*(2*a^3*b*d*
x - 14*(a^4 + a^3*b)*d*x*cosh(d*x + c)^2 - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*sinh(d*x + c)^6 + 8*(14*(a^4 +
a^3*b)*d*x*cosh(d*x + c)^3 - 3*(2*a^3*b*d*x - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x +
c)^5 + 4*(2*a^4 + 2*a^3*b - 2*a^2*b^2 - 4*a*b^3 - (a^4 - 3*a^3*b)*d*x)*cosh(d*x + c)^4 + 4*(35*(a^4 + a^3*b)*d
*x*cosh(d*x + c)^4 + 2*a^4 + 2*a^3*b - 2*a^2*b^2 - 4*a*b^3 - (a^4 - 3*a^3*b)*d*x - 15*(2*a^3*b*d*x - a^4 - 3*a
^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(7*(a^4 + a^3*b)*d*x*cosh(d*x + c)^5 - 5*(2*
a^3*b*d*x - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^3 + (2*a^4 + 2*a^3*b - 2*a^2*b^2 - 4*a*b^3 - (a
^4 - 3*a^3*b)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^4 + a^3*b)*d*x - 4*(2*a^3*b*d*x - a^4 - 3*a^3*b - 3*a
^2*b^2 - 2*a*b^3)*cosh(d*x + c)^2 + 4*(14*(a^4 + a^3*b)*d*x*cosh(d*x + c)^6 - 2*a^3*b*d*x - 15*(2*a^3*b*d*x -
a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^4 + a^4 + 3*a^3*b + 3*a^2*b^2 + 2*a*b^3 + 6*(2*a^4 + 2*a^3*
b - 2*a^2*b^2 - 4*a*b^3 - (a^4 - 3*a^3*b)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^2*b^2 + 5*a*b^3 + 2*b^
4)*cosh(d*x + c)^8 + 8*(3*a^2*b^2 + 5*a*b^3 + 2*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2*b^2 + 5*a*b^3 + 2*
b^4)*sinh(d*x + c)^8 - 4*(3*a*b^3 + 2*b^4)*cosh(d*x + c)^6 - 4*(3*a*b^3 + 2*b^4 - 7*(3*a^2*b^2 + 5*a*b^3 + 2*b
^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2*b^2 + 5*a*b^3 + 2*b^4)*cosh(d*x + c)^3 - 3*(3*a*b^3 + 2*b^4
)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(3*a^2*b^2 - 7*a*b^3 - 6*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^2*b^2 + 5*a*b^
3 + 2*b^4)*cosh(d*x + c)^4 - 3*a^2*b^2 + 7*a*b^3 + 6*b^4 - 30*(3*a*b^3 + 2*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)
^4 + 3*a^2*b^2 + 5*a*b^3 + 2*b^4 + 8*(7*(3*a^2*b^2 + 5*a*b^3 + 2*b^4)*cosh(d*x + c)^5 - 10*(3*a*b^3 + 2*b^4)*c
osh(d*x + c)^3 - (3*a^2*b^2 - 7*a*b^3 - 6*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(3*a*b^3 + 2*b^4)*cosh(d*x +
c)^2 + 4*(7*(3*a^2*b^2 + 5*a*b^3 + 2*b^4)*cosh(d*x + c)^6 - 15*(3*a*b^3 + 2*b^4)*cosh(d*x + c)^4 - 3*a*b^3 -
2*b^4 - 3*(3*a^2*b^2 - 7*a*b^3 - 6*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^2*b^2 + 5*a*b^3 + 2*b^4)*co
sh(d*x + c)^7 - 3*(3*a*b^3 + 2*b^4)*cosh(d*x + c)^5 - (3*a^2*b^2 - 7*a*b^3 - 6*b^4)*cosh(d*x + c)^3 - (3*a*b^3
+ 2*b^4)*cosh(d*x + c))*sinh(d*x + c))*log(2*((a + b)*cosh(d*x + c)^2 + (a + b)*sinh(d*x + c)^2 + a - b)/(cos
h(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*((a^4 + a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b
^4)*cosh(d*x + c)^8 + 8*(a^4 + a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + a^3
*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*sinh(d*x + c)^8 - 4*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^6 - 4*(a^3*b - 3
*a*b^3 - 2*b^4 - 7*(a^4 + a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 +
a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)^3 - 3*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c))*sinh(d*x + c
)^5 - 2*(a^4 - 3*a^3*b - 3*a^2*b^2 + 7*a*b^3 + 6*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 + a^3*b - 3*a^2*b^2 - 5*a*b
^3 - 2*b^4)*cosh(d*x + c)^4 - a^4 + 3*a^3*b + 3*a^2*b^2 - 7*a*b^3 - 6*b^4 - 30*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(
d*x + c)^2)*sinh(d*x + c)^4 + a^4 + a^3*b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4 + 8*(7*(a^4 + a^3*b - 3*a^2*b^2 - 5*a*
b^3 - 2*b^4)*cosh(d*x + c)^5 - 10*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^3 - (a^4 - 3*a^3*b - 3*a^2*b^2 + 7*a
*b^3 + 6*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^2 + 4*(7*(a^4 + a^3*b
- 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)^6 - 15*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^4 - a^3*b + 3*a*b
^3 + 2*b^4 - 3*(a^4 - 3*a^3*b - 3*a^2*b^2 + 7*a*b^3 + 6*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 + a^3*
b - 3*a^2*b^2 - 5*a*b^3 - 2*b^4)*cosh(d*x + c)^7 - 3*(a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c)^5 - (a^4 - 3*a^3*
b - 3*a^2*b^2 + 7*a*b^3 + 6*b^4)*cosh(d*x + c)^3 - (a^3*b - 3*a*b^3 - 2*b^4)*cosh(d*x + c))*sinh(d*x + c))*log
(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(2*(a^4 + a^3*b)*d*x*cosh(d*x + c)^7 - 3*(2*a^3*b*d*x -
a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c)^5 + 2*(2*a^4 + 2*a^3*b - 2*a^2*b^2 - 4*a*b^3 - (a^4 - 3*a^3
*b)*d*x)*cosh(d*x + c)^3 - (2*a^3*b*d*x - a^4 - 3*a^3*b - 3*a^2*b^2 - 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/(
(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^8 + 8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x +
c)*sinh(d*x + c)^7 + (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*sinh(d*x + c)^8 - 4*(a^5*b + 2*a^4*b^2 + a^3*b^3
)*d*cosh(d*x + c)^6 + 4*(7*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^2 - (a^5*b + 2*a^4*b^2 + a^3*
b^3)*d)*sinh(d*x + c)^6 - 2*(a^6 - a^5*b - 5*a^4*b^2 - 3*a^3*b^3)*d*cosh(d*x + c)^4 + 8*(7*(a^6 + 3*a^5*b + 3*
a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^3 - 3*(a^5*b + 2*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3
5*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**3/(a+b*tanh(d*x+c)**2)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (118) = 236\).
time = 0.52, size = 323, normalized size = 2.60 \begin {gather*} \frac {\frac {{\left (3 \, a b^{2} + 2 \, b^{3}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}} - \frac {2 \, {\left (d x + c\right )}}{a^{2} + 2 \, a b + b^{2}} + \frac {2 \, {\left (a - 2 \, b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{3}} - \frac {4 \, {\left (\frac {{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{a + b} + \frac {2 \, {\left (a^{4} + a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{a + b} + \frac {{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{a + b}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} {\left (a + b\right )} a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((3*a*b^2 + 2*b^3)*log(a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) +
a + b)/(a^5 + 2*a^4*b + a^3*b^2) - 2*(d*x + c)/(a^2 + 2*a*b + b^2) + 2*(a - 2*b)*log(abs(e^(2*d*x + 2*c) - 1)
)/a^3 - 4*((a^4 + 3*a^3*b + 3*a^2*b^2 + 2*a*b^3)*e^(6*d*x + 6*c)/(a + b) + 2*(a^4 + a^3*b - a^2*b^2 - 2*a*b^3)
*e^(4*d*x + 4*c)/(a + b) + (a^4 + 3*a^3*b + 3*a^2*b^2 + 2*a*b^3)*e^(2*d*x + 2*c)/(a + b))/((a*e^(4*d*x + 4*c)
+ b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)*(a + b)*a^3*(e^(2*d*x + 2*c) - 1)^2))
/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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