3.2.55 \(\int \frac {\coth ^2(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [155]

Optimal. Leaf size=121 \[ \frac {x}{a^2}-\frac {b^{3/2} (5 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{5/2} d}-\frac {(2 a-b) \coth (c+d x)}{2 a (a+b)^2 d}-\frac {b \coth (c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \]

[Out]

x/a^2-1/2*b^(3/2)*(5*a+2*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^2/(a+b)^(5/2)/d-1/2*(2*a-b)*coth(d*x+c)
/a/(a+b)^2/d-1/2*b*coth(d*x+c)/a/(a+b)/d/(a+b-b*tanh(d*x+c)^2)

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Rubi [A]
time = 0.20, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4226, 2000, 483, 597, 536, 212, 214} \begin {gather*} -\frac {b^{3/2} (5 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^2 d (a+b)^{5/2}}+\frac {x}{a^2}-\frac {(2 a-b) \coth (c+d x)}{2 a d (a+b)^2}-\frac {b \coth (c+d x)}{2 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/a^2 - (b^(3/2)*(5*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(5/2)*d) - ((2*a - b
)*Coth[c + d*x])/(2*a*(a + b)^2*d) - (b*Coth[c + d*x])/(2*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 483

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*(e*
x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a*d)
*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n
*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ
[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 2000

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 4226

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2
*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])

Rubi steps

\begin {align*} \int \frac {\coth ^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b \coth (c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-2 a+b-3 b x^2}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 a (a+b) d}\\ &=-\frac {(2 a-b) \coth (c+d x)}{2 a (a+b)^2 d}-\frac {b \coth (c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {2 a^2+6 a b+b^2-(2 a-b) b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 a (a+b)^2 d}\\ &=-\frac {(2 a-b) \coth (c+d x)}{2 a (a+b)^2 d}-\frac {b \coth (c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{a^2 d}-\frac {\left (b^2 (5 a+2 b)\right ) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^2 (a+b)^2 d}\\ &=\frac {x}{a^2}-\frac {b^{3/2} (5 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{5/2} d}-\frac {(2 a-b) \coth (c+d x)}{2 a (a+b)^2 d}-\frac {b \coth (c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(268\) vs. \(2(121)=242\).
time = 1.91, size = 268, normalized size = 2.21 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (\frac {2 x (a+2 b+a \cosh (2 (c+d x)))}{a^2}-\frac {b^2 (5 a+2 b) \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (a+2 b+a \cosh (2 (c+d x))) (\cosh (2 c)-\sinh (2 c))}{a^2 (a+b)^{5/2} d \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {2 (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}(c+d x) \sinh (d x)}{(a+b)^2 d}+\frac {b^2 \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2 (a+b)^2 d}\right )}{8 \left (a+b \text {sech}^2(c+d x)\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((2*x*(a + 2*b + a*Cosh[2*(c + d*x)]))/a^2 - (b^2*(5*a + 2*b)
*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(
Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d*x)])*(Cosh[2*c] - Sinh[2*c]))/(a^2*(a + b)^(5/2)*d*Sqrt[b*(
Cosh[c] - Sinh[c])^4]) + (2*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x])/((a + b)^2*d) + (
b^2*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sinh[2*d*x]))/(a^2*(a + b)^2*d)))/(8*(a + b*Sech[c + d*x]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(107)=214\).
time = 2.71, size = 288, normalized size = 2.38

method	result	size
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 b^{2} \left (\frac {-\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (5 a +2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{\left (a +b \right )^{2} a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {1}{2 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(288\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 b^{2} \left (\frac {-\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (5 a +2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{\left (a +b \right )^{2} a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}-\frac {1}{2 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(288\)
risch	\(\frac {x}{a^{2}}-\frac {2 a^{3} {\mathrm e}^{4 d x +4 c}-a \,b^{2} {\mathrm e}^{4 d x +4 c}-2 b^{3} {\mathrm e}^{4 d x +4 c}+4 a^{3} {\mathrm e}^{2 d x +2 c}+8 a^{2} b \,{\mathrm e}^{2 d x +2 c}+2 b^{3} {\mathrm e}^{2 d x +2 c}+2 a^{3}+a \,b^{2}}{d \left (a +b \right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {5 \sqrt {b \left (a +b \right )}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right )}{4 \left (a +b \right )^{3} d a}+\frac {\sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right )}{2 \left (a +b \right )^{3} d \,a^{2}}-\frac {5 \sqrt {b \left (a +b \right )}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right )}{4 \left (a +b \right )^{3} d a}-\frac {\sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right )}{2 \left (a +b \right )^{3} d \,a^{2}}\)	\(379\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/2/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)+2*b^2/(a+b)^2/a^2*((-1/2*a*tanh(1/2*d*x+1/2*c)^3-1/2*a*tanh(1/2*
d*x+1/2*c))/(a*tanh(1/2*d*x+1/2*c)^4+b*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)
^2+a+b)+1/2*(5*a+2*b)*(-1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(
1/2)+(a+b)^(1/2))+1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(
a+b)^(1/2))))-1/a^2*ln(tanh(1/2*d*x+1/2*c)-1)+1/a^2*ln(tanh(1/2*d*x+1/2*c)+1)-1/2/(a+b)^2/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. 1070 vs. \(2 (110) = 220\).
time = 0.55, size = 1070, normalized size = 8.84 \begin {gather*} \frac {{\left (2 \, a b + b^{2}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d} - \frac {{\left (2 \, a b + b^{2}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} d} - \frac {{\left (3 \, a^{2} b + 10 \, a b^{2} + 4 \, b^{3}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {{\left (3 \, a^{2} b + 10 \, a b^{2} + 4 \, b^{3}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {3 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {2 \, a^{3} + a^{2} b + 2 \, a b^{2} + {\left (2 \, a^{3} - a^{2} b - 8 \, a b^{2} - 8 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (2 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} + 4 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{4 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2} - {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )} - {\left (a^{5} + 6 \, a^{4} b + 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )} + {\left (a^{5} + 6 \, a^{4} b + 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} d} - \frac {2 \, a^{3} + a^{2} b + 2 \, a b^{2} + 2 \, {\left (2 \, a^{3} + 4 \, a^{2} b + 3 \, a b^{2} + 4 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (2 \, a^{3} - a^{2} b - 8 \, a b^{2} - 8 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{4 \, {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2} + {\left (a^{5} + 6 \, a^{4} b + 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{5} + 6 \, a^{4} b + 9 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} - \frac {2 \, a^{2} - a b + 2 \, {\left (2 \, a^{2} + 4 \, a b - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (2 \, a^{2} + a b + 2 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{2 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} + 6 \, a^{3} b + 9 \, a^{2} b^{2} + 4 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{4} + 6 \, a^{3} b + 9 \, a^{2} b^{2} + 4 \, a b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*a*b + b^2)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^4 + 2*a^3*b + a^2*b^2)*d) - 1/4
*(2*a*b + b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^4 + 2*a^3*b + a^2*b^2)*d) - 1/16
*(3*a^2*b + 10*a*b^2 + 4*b^3)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2
*b + 2*sqrt((a + b)*b)))/((a^4 + 2*a^3*b + a^2*b^2)*sqrt((a + b)*b)*d) + 1/16*(3*a^2*b + 10*a*b^2 + 4*b^3)*log
((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^4
+ 2*a^3*b + a^2*b^2)*sqrt((a + b)*b)*d) - 3/8*b*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-
2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2 + 2*a*b + b^2)*sqrt((a + b)*b)*d) + 1/4*(2*a^3 + a^2*b + 2*
a*b^2 + (2*a^3 - a^2*b - 8*a*b^2 - 8*b^3)*e^(4*d*x + 4*c) + 2*(2*a^3 + 4*a^2*b + 3*a*b^2 + 4*b^3)*e^(2*d*x + 2
*c))/((a^5 + 2*a^4*b + a^3*b^2 - (a^5 + 2*a^4*b + a^3*b^2)*e^(6*d*x + 6*c) - (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^
2*b^3)*e^(4*d*x + 4*c) + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*e^(2*d*x + 2*c))*d) - 1/4*(2*a^3 + a^2*b + 2*
a*b^2 + 2*(2*a^3 + 4*a^2*b + 3*a*b^2 + 4*b^3)*e^(-2*d*x - 2*c) + (2*a^3 - a^2*b - 8*a*b^2 - 8*b^3)*e^(-4*d*x -
4*c))/((a^5 + 2*a^4*b + a^3*b^2 + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*e^(-2*d*x - 2*c) - (a^5 + 6*a^4*b +
9*a^3*b^2 + 4*a^2*b^3)*e^(-4*d*x - 4*c) - (a^5 + 2*a^4*b + a^3*b^2)*e^(-6*d*x - 6*c))*d) - 1/2*(2*a^2 - a*b +
2*(2*a^2 + 4*a*b - b^2)*e^(-2*d*x - 2*c) + (2*a^2 + a*b + 2*b^2)*e^(-4*d*x - 4*c))/((a^4 + 2*a^3*b + a^2*b^2
+ (a^4 + 6*a^3*b + 9*a^2*b^2 + 4*a*b^3)*e^(-2*d*x - 2*c) - (a^4 + 6*a^3*b + 9*a^2*b^2 + 4*a*b^3)*e^(-4*d*x - 4
*c) - (a^4 + 2*a^3*b + a^2*b^2)*e^(-6*d*x - 6*c))*d) + 1/2*log(e^(2*d*x + 2*c) - 1)/((a^2 + 2*a*b + b^2)*d) -
1/2*log(e^(-2*d*x - 2*c) - 1)/((a^2 + 2*a*b + b^2)*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1674 vs. \(2 (110) = 220\).
time = 0.45, size = 3624, normalized size = 29.95 \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)^2/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/4*(4*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)^6 + 24*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)*sinh(d*x +
c)^5 + 4*(a^3 + 2*a^2*b + a*b^2)*d*x*sinh(d*x + c)^6 - 4*(2*a^3 - a*b^2 - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4
*b^3)*d*x)*cosh(d*x + c)^4 + 4*(15*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)^2 - 2*a^3 + a*b^2 + 2*b^3 + (a^3
+ 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*sinh(d*x + c)^4 + 16*(5*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)^3 - (2*a^3
- a*b^2 - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*a^3 - 4*a*b^2 - 4
*(a^3 + 2*a^2*b + a*b^2)*d*x - 4*(4*a^3 + 8*a^2*b + 2*b^3 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*cosh(d*x +
c)^2 + 4*(15*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)^4 - 4*a^3 - 8*a^2*b - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2
+ 4*b^3)*d*x - 6*(2*a^3 - a*b^2 - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c
)^2 + ((5*a^2*b + 2*a*b^2)*cosh(d*x + c)^6 + 6*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (5*a^2*b +
2*a*b^2)*sinh(d*x + c)^6 + (5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + (5*a^2*b + 22*a*b^2 + 8*b^3 + 15*(5*
a^2*b + 2*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + (5*a^2*b + 22*a
*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 5*a^2*b - 2*a*b^2 - (5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(d*x + c)^
2 + (15*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^4 - 5*a^2*b - 22*a*b^2 - 8*b^3 + 6*(5*a^2*b + 22*a*b^2 + 8*b^3)*cosh
(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + 2*(5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(
d*x + c)^3 - (5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a + b))*log((a^2*cosh(d*x + c)
^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*co
sh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*co
sh(d*x + c))*sinh(d*x + c) + 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2
+ a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*
x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2
+ 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 8*(3*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh
(d*x + c)^5 - 2*(2*a^3 - a*b^2 - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*cosh(d*x + c)^3 - (4*a^3 + 8*a
^2*b + 2*b^3 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + 2*a^4*b + a^3*b^2)
*d*cosh(d*x + c)^6 + 6*(a^5 + 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^5 + 2*a^4*b + a^3*b^2)*d
*sinh(d*x + c)^6 + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*d*cosh(d*x + c)^4 + (15*(a^5 + 2*a^4*b + a^3*b^2)*d
*cosh(d*x + c)^2 + (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*d)*sinh(d*x + c)^4 - (a^5 + 6*a^4*b + 9*a^3*b^2 + 4
*a^2*b^3)*d*cosh(d*x + c)^2 + 4*(5*(a^5 + 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^3 + (a^5 + 6*a^4*b + 9*a^3*b^2 +
4*a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^5 + 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^4 + 6*(a^5 + 6*a^4
*b + 9*a^3*b^2 + 4*a^2*b^3)*d*cosh(d*x + c)^2 - (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*d)*sinh(d*x + c)^2 - (
a^5 + 2*a^4*b + a^3*b^2)*d + 2*(3*(a^5 + 2*a^4*b + a^3*b^2)*d*cosh(d*x + c)^5 + 2*(a^5 + 6*a^4*b + 9*a^3*b^2 +
4*a^2*b^3)*d*cosh(d*x + c)^3 - (a^5 + 6*a^4*b + 9*a^3*b^2 + 4*a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(
2*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)^6 + 12*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^5 +
2*(a^3 + 2*a^2*b + a*b^2)*d*x*sinh(d*x + c)^6 - 2*(2*a^3 - a*b^2 - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*
d*x)*cosh(d*x + c)^4 + 2*(15*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)^2 - 2*a^3 + a*b^2 + 2*b^3 + (a^3 + 6*a^
2*b + 9*a*b^2 + 4*b^3)*d*x)*sinh(d*x + c)^4 + 8*(5*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)^3 - (2*a^3 - a*b^
2 - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*a^3 - 2*a*b^2 - 2*(a^3 +
2*a^2*b + a*b^2)*d*x - 2*(4*a^3 + 8*a^2*b + 2*b^3 + (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*cosh(d*x + c)^2 +
2*(15*(a^3 + 2*a^2*b + a*b^2)*d*x*cosh(d*x + c)^4 - 4*a^3 - 8*a^2*b - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3
)*d*x - 6*(2*a^3 - a*b^2 - 2*b^3 - (a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - (
(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^6 + 6*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (5*a^2*b + 2*a*b^2
)*sinh(d*x + c)^6 + (5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + (5*a^2*b + 22*a*b^2 + 8*b^3 + 15*(5*a^2*b +
2*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^3 + (5*a^2*b + 22*a*b^2 +
8*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 5*a^2*b - 2*a*b^2 - (5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(d*x + c)^2 + (15
*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^4 - 5*a^2*b - 22*a*b^2 - 8*b^3 + 6*(5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 2*(3*(5*a^2*b + 2*a*b^2)*cosh(d*x + c)^5 + 2*(5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(d*x + c
)^3 - (5*a^2*b + 22*a*b^2 + 8*b^3)*cosh(d*x + c...

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

[Out]

Integral(coth(c + d*x)**2/(a + b*sech(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. 273 vs. \(2 (110) = 220\).
time = 1.19, size = 273, normalized size = 2.26 \begin {gather*} -\frac {\frac {{\left (5 \, a b^{2} + 2 \, b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (2 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} + a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} - a e^{\left (2 \, d x + 2 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} - a\right )}} - \frac {2 \, {\left (d x + c\right )}}{a^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\mathrm {coth}\left (c+d\,x\right )}^2}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((cosh(c + d*x)^4*coth(c + d*x)^2)/(b + a*cosh(c + d*x)^2)^2, x)

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