3.2.57 \(\int \frac {\coth ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [157]

Optimal. Leaf size=161 \[ \frac {x}{a^2}-\frac {b^{5/2} (7 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{7/2} d}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{2 a (a+b)^3 d}-\frac {(2 a-3 b) \coth ^3(c+d x)}{6 a (a+b)^2 d}-\frac {b \coth ^3(c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )} \]

[Out]

x/a^2-1/2*b^(5/2)*(7*a+2*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^2/(a+b)^(7/2)/d-1/2*(2*a^2+6*a*b-b^2)*c
oth(d*x+c)/a/(a+b)^3/d-1/6*(2*a-3*b)*coth(d*x+c)^3/a/(a+b)^2/d-1/2*b*coth(d*x+c)^3/a/(a+b)/d/(a+b-b*tanh(d*x+c
)^2)

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Rubi [A]
time = 0.29, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, 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^{5/2} (7 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^2 d (a+b)^{7/2}}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{2 a d (a+b)^3}+\frac {x}{a^2}-\frac {(2 a-3 b) \coth ^3(c+d x)}{6 a d (a+b)^2}-\frac {b \coth ^3(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]^4/(a + b*Sech[c + d*x]^2)^2,x]

[Out]

x/a^2 - (b^(5/2)*(7*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(7/2)*d) - ((2*a^2 +
6*a*b - b^2)*Coth[c + d*x])/(2*a*(a + b)^3*d) - ((2*a - 3*b)*Coth[c + d*x]^3)/(6*a*(a + b)^2*d) - (b*Coth[c +
d*x]^3)/(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 ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \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^4 \left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b \coth ^3(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+3 b-5 b x^2}{x^4 \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-3 b) \coth ^3(c+d x)}{6 a (a+b)^2 d}-\frac {b \coth ^3(c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {3 \left (2 a^2+6 a b-b^2\right )-3 (2 a-3 b) b x^2}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{6 a (a+b)^2 d}\\ &=-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{2 a (a+b)^3 d}-\frac {(2 a-3 b) \coth ^3(c+d x)}{6 a (a+b)^2 d}-\frac {b \coth ^3(c+d x)}{2 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-3 \left (2 a^3+8 a^2 b+12 a b^2+b^3\right )+3 b \left (2 a^2+6 a b-b^2\right ) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{6 a (a+b)^3 d}\\ &=-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{2 a (a+b)^3 d}-\frac {(2 a-3 b) \coth ^3(c+d x)}{6 a (a+b)^2 d}-\frac {b \coth ^3(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^3 (7 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)^3 d}\\ &=\frac {x}{a^2}-\frac {b^{5/2} (7 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{7/2} d}-\frac {\left (2 a^2+6 a b-b^2\right ) \coth (c+d x)}{2 a (a+b)^3 d}-\frac {(2 a-3 b) \coth ^3(c+d x)}{6 a (a+b)^2 d}-\frac {b \coth ^3(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. \(350\) vs. \(2(161)=322\).
time = 3.37, size = 350, normalized size = 2.17 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^4(c+d x) \left (\frac {6 x (a+2 b+a \cosh (2 (c+d x)))}{a^2}-\frac {2 (a+2 b+a \cosh (2 (c+d x))) \coth (c) \text {csch}^2(c+d x)}{(a+b)^2 d}-\frac {3 b^3 (7 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)^{7/2} d \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {4 (2 a+5 b) (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}(c+d x) \sinh (d x)}{(a+b)^3 d}+\frac {2 (a+2 b+a \cosh (2 (c+d x))) \text {csch}(c) \text {csch}^3(c+d x) \sinh (d x)}{(a+b)^2 d}+\frac {3 b^3 \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a^2 (a+b)^3 d}\right )}{24 \left (a+b \text {sech}^2(c+d x)\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^4*((6*x*(a + 2*b + a*Cosh[2*(c + d*x)]))/a^2 - (2*(a + 2*b + a*
Cosh[2*(c + d*x)])*Coth[c]*Csch[c + d*x]^2)/((a + b)^2*d) - (3*b^3*(7*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)^(7/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + (4*(2*a
+ 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c]*Csch[c + d*x]*Sinh[d*x])/((a + b)^3*d) + (2*(a + 2*b + a*Cosh[
2*(c + d*x)])*Csch[c]*Csch[c + d*x]^3*Sinh[d*x])/((a + b)^2*d) + (3*b^3*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a*Sin
h[2*d*x]))/(a^2*(a + b)^3*d)))/(24*(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. \(364\) vs. \(2(145)=290\).
time = 3.15, size = 365, normalized size = 2.27

method	result	size
derivativedivides	\(\frac {-\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+13 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {2 b^{3} \left (\frac {-\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (7 a +2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{a^{2} \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {1}{24 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +13 b}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(365\)
default	\(\frac {-\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+13 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {2 b^{3} \left (\frac {-\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (7 a +2 b \right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{a^{2} \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {1}{24 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +13 b}{8 \left (a +b \right )^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(365\)
risch	\(\frac {x}{a^{2}}-\frac {12 a^{4} {\mathrm e}^{8 d x +8 c}+24 a^{3} b \,{\mathrm e}^{8 d x +8 c}-3 a \,b^{3} {\mathrm e}^{8 d x +8 c}-6 b^{4} {\mathrm e}^{8 d x +8 c}+12 a^{4} {\mathrm e}^{6 d x +6 c}+60 a^{3} b \,{\mathrm e}^{6 d x +6 c}+96 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}+6 a \,b^{3} {\mathrm e}^{6 d x +6 c}+18 b^{4} {\mathrm e}^{6 d x +6 c}-4 a^{4} {\mathrm e}^{4 d x +4 c}-76 a^{3} b \,{\mathrm e}^{4 d x +4 c}-144 a^{2} b^{2} {\mathrm e}^{4 d x +4 c}-18 b^{4} {\mathrm e}^{4 d x +4 c}+4 a^{4} {\mathrm e}^{2 d x +2 c}+36 a^{3} b \,{\mathrm e}^{2 d x +2 c}+80 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}-6 a \,b^{3} {\mathrm e}^{2 d x +2 c}+6 b^{4} {\mathrm e}^{2 d x +2 c}+8 a^{4}+20 a^{3} b +3 a \,b^{3}}{3 d \left (a +b \right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3} a^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {7 \sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right )}{4 \left (a +b \right )^{4} d a}+\frac {\sqrt {b \left (a +b \right )}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right )}{2 \left (a +b \right )^{4} d \,a^{2}}-\frac {7 \sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right )}{4 \left (a +b \right )^{4} d a}-\frac {\sqrt {b \left (a +b \right )}\, b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right )}{2 \left (a +b \right )^{4} d \,a^{2}}\)	\(572\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/8/(a^2+2*a*b+b^2)/(a+b)*(1/3*a*tanh(1/2*d*x+1/2*c)^3+1/3*b*tanh(1/2*d*x+1/2*c)^3+5*a*tanh(1/2*d*x+1/2*
c)+13*b*tanh(1/2*d*x+1/2*c))+1/a^2*ln(tanh(1/2*d*x+1/2*c)+1)+2*b^3/a^2/(a+b)^3*((-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*(7*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)-1/24/(a+b)^2/tanh(1/2*d*x+1/2*c)^3-1/8*(5*a+13*b
)/(a+b)^3/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. 2961 vs. \(2 (148) = 296\).
time = 0.77, size = 2961, normalized size = 18.39 \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)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/4*(a^2*b + 3*a*b^2 + b^3)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^5 + 3*a^4*b + 3*a^3*b
^2 + a^2*b^3)*d) - 1/2*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*d) - 1/4*(a^2*b + 3*a*b^2 + b^3)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^5 + 3*a^4
*b + 3*a^3*b^2 + a^2*b^3)*d) + 1/2*b*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^3 + 3*a^2*
b + 3*a*b^2 + b^3)*d) + 1/2*(a + 2*b)*log(e^(2*d*x + 2*c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + b*log(e^(
2*d*x + 2*c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/2*(a + 2*b)*log(e^(-2*d*x - 2*c) - 1)/((a^3 + 3*a^2*
b + 3*a*b^2 + b^3)*d) - b*log(e^(-2*d*x - 2*c) - 1)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 1/64*(3*a^3*b + 38*a
^2*b^2 + 56*a*b^3 + 16*b^4)*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^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt((a + b)*b)*d) + 1/16*(3*a*b + 8*b^2)*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^3 + 3*a
^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*b)*d) + 1/64*(3*a^3*b + 38*a^2*b^2 + 56*a*b^3 + 16*b^4)*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^5 + 3*a^4*b + 3*a^
3*b^2 + a^2*b^3)*sqrt((a + b)*b)*d) - 1/16*(3*a*b + 8*b^2)*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^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*b)*d) +
3/32*(3*a*b - 2*b^2)*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^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*b)*d) + 1/48*(44*a^4 + 59*a^3*b + 24*a^2*b^2 +
24*a*b^3 + 3*(24*a^4 + 27*a^3*b - 18*a^2*b^2 - 48*a*b^3 - 32*b^4)*e^(8*d*x + 8*c) + 6*(6*a^4 + 55*a^3*b + 79*
a^2*b^2 + 68*a*b^3 + 48*b^4)*e^(6*d*x + 6*c) - 2*(50*a^4 + 278*a^3*b + 309*a^2*b^2 + 180*a*b^3 + 144*b^4)*e^(4
*d*x + 4*c) - 2*(10*a^4 - 75*a^3*b - 103*a^2*b^2 - 36*a*b^3 - 48*b^4)*e^(2*d*x + 2*c))/((a^6 + 3*a^5*b + 3*a^4
*b^2 + a^3*b^3 - (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*e^(10*d*x + 10*c) + (a^6 - a^5*b - 9*a^4*b^2 - 11*a^3*b
^3 - 4*a^2*b^4)*e^(8*d*x + 8*c) + 2*(a^6 + 9*a^5*b + 21*a^4*b^2 + 19*a^3*b^3 + 6*a^2*b^4)*e^(6*d*x + 6*c) - 2*
(a^6 + 9*a^5*b + 21*a^4*b^2 + 19*a^3*b^3 + 6*a^2*b^4)*e^(4*d*x + 4*c) - (a^6 - a^5*b - 9*a^4*b^2 - 11*a^3*b^3
- 4*a^2*b^4)*e^(2*d*x + 2*c))*d) - 1/48*(44*a^4 + 59*a^3*b + 24*a^2*b^2 + 24*a*b^3 - 2*(10*a^4 - 75*a^3*b - 10
3*a^2*b^2 - 36*a*b^3 - 48*b^4)*e^(-2*d*x - 2*c) - 2*(50*a^4 + 278*a^3*b + 309*a^2*b^2 + 180*a*b^3 + 144*b^4)*e
^(-4*d*x - 4*c) + 6*(6*a^4 + 55*a^3*b + 79*a^2*b^2 + 68*a*b^3 + 48*b^4)*e^(-6*d*x - 6*c) + 3*(24*a^4 + 27*a^3*
b - 18*a^2*b^2 - 48*a*b^3 - 32*b^4)*e^(-8*d*x - 8*c))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 - (a^6 - a^5*b - 9
*a^4*b^2 - 11*a^3*b^3 - 4*a^2*b^4)*e^(-2*d*x - 2*c) - 2*(a^6 + 9*a^5*b + 21*a^4*b^2 + 19*a^3*b^3 + 6*a^2*b^4)*
e^(-4*d*x - 4*c) + 2*(a^6 + 9*a^5*b + 21*a^4*b^2 + 19*a^3*b^3 + 6*a^2*b^4)*e^(-6*d*x - 6*c) + (a^6 - a^5*b - 9
*a^4*b^2 - 11*a^3*b^3 - 4*a^2*b^4)*e^(-8*d*x - 8*c) - (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*e^(-10*d*x - 10*c)
)*d) + 1/12*(8*a^3 + 17*a^2*b - 6*a*b^2 + 3*(8*a^3 + 13*a^2*b + 8*a*b^2 + 8*b^3)*e^(8*d*x + 8*c) + 6*(4*a^3 +
19*a^2*b + 13*a*b^2 - 12*b^3)*e^(6*d*x + 6*c) - 2*(8*a^3 + 68*a^2*b + 69*a*b^2 - 36*b^3)*e^(4*d*x + 4*c) - 2*(
4*a^3 - 15*a^2*b - 37*a*b^2 + 12*b^3)*e^(2*d*x + 2*c))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 - (a^5 + 3*a^4*b
+ 3*a^3*b^2 + a^2*b^3)*e^(10*d*x + 10*c) + (a^5 - a^4*b - 9*a^3*b^2 - 11*a^2*b^3 - 4*a*b^4)*e^(8*d*x + 8*c) +
2*(a^5 + 9*a^4*b + 21*a^3*b^2 + 19*a^2*b^3 + 6*a*b^4)*e^(6*d*x + 6*c) - 2*(a^5 + 9*a^4*b + 21*a^3*b^2 + 19*a^2
*b^3 + 6*a*b^4)*e^(4*d*x + 4*c) - (a^5 - a^4*b - 9*a^3*b^2 - 11*a^2*b^3 - 4*a*b^4)*e^(2*d*x + 2*c))*d) - 1/12*
(8*a^3 + 17*a^2*b - 6*a*b^2 - 2*(4*a^3 - 15*a^2*b - 37*a*b^2 + 12*b^3)*e^(-2*d*x - 2*c) - 2*(8*a^3 + 68*a^2*b
+ 69*a*b^2 - 36*b^3)*e^(-4*d*x - 4*c) + 6*(4*a^3 + 19*a^2*b + 13*a*b^2 - 12*b^3)*e^(-6*d*x - 6*c) + 3*(8*a^3 +
13*a^2*b + 8*a*b^2 + 8*b^3)*e^(-8*d*x - 8*c))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 - (a^5 - a^4*b - 9*a^3*b^
2 - 11*a^2*b^3 - 4*a*b^4)*e^(-2*d*x - 2*c) - 2*(a^5 + 9*a^4*b + 21*a^3*b^2 + 19*a^2*b^3 + 6*a*b^4)*e^(-4*d*x -
4*c) + 2*(a^5 + 9*a^4*b + 21*a^3*b^2 + 19*a^2*b^3 + 6*a*b^4)*e^(-6*d*x - 6*c) + (a^5 - a^4*b - 9*a^3*b^2 - 11
*a^2*b^3 - 4*a*b^4)*e^(-8*d*x - 8*c) - (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*e^(-10*d*x - 10*c))*d) + 1/8*(4*a
^2 - 11*a*b - 2*(2*a^2 - 9*a*b + 19*b^2)*e^(-2*d*x - 2*c) - 2*(10*a^2 + 22*a*b - 33*b^2)*e^(-4*d*x - 4*c) - 6*
(2*a^2 + 3*a*b + 11*b^2)*e^(-6*d*x - 6*c) - 3*(3*a*b - 2*b^2)*e^(-8*d*x - 8*c))/((a^4 + 3*a^3*b + 3*a^2*b^2 +
a*b^3 - (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*e^(-2*d*x - 2*c) - 2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b
^3 + 6*b^4)*e^(-4*d*x - 4*c) + 2*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*e^(-6*d*x - 6*c) + (a^4 - a^3
*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*e^(-8*d*x - ...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4786 vs. \(2 (148) = 296\).
time = 0.49, size = 9849, normalized size = 61.17 \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)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/12*(12*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^10 + 120*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d
*x*cosh(d*x + c)*sinh(d*x + c)^9 + 12*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*sinh(d*x + c)^10 - 12*(4*a^4 + 8
*a^3*b - a*b^3 - 2*b^4 + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*x)*cosh(d*x + c)^8 + 12*(45*(a^4 + 3*a
^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^2 - 4*a^4 - 8*a^3*b + a*b^3 + 2*b^4 - (a^4 - a^3*b - 9*a^2*b^2 - 1
1*a*b^3 - 4*b^4)*d*x)*sinh(d*x + c)^8 + 96*(15*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^3 - (4*a^
4 + 8*a^3*b - a*b^3 - 2*b^4 + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7
- 24*(2*a^4 + 10*a^3*b + 16*a^2*b^2 + a*b^3 + 3*b^4 + (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*x)*co
sh(d*x + c)^6 + 24*(105*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^4 - 2*a^4 - 10*a^3*b - 16*a^2*b^
2 - a*b^3 - 3*b^4 - (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*x - 14*(4*a^4 + 8*a^3*b - a*b^3 - 2*b^4
+ (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 48*(63*(a^4 + 3*a^3*b +
3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^5 - 14*(4*a^4 + 8*a^3*b - a*b^3 - 2*b^4 + (a^4 - a^3*b - 9*a^2*b^2 - 11*
a*b^3 - 4*b^4)*d*x)*cosh(d*x + c)^3 - 3*(2*a^4 + 10*a^3*b + 16*a^2*b^2 + a*b^3 + 3*b^4 + (a^4 + 9*a^3*b + 21*a
^2*b^2 + 19*a*b^3 + 6*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 8*(2*a^4 + 38*a^3*b + 72*a^2*b^2 + 9*b^4 + 3*
(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*x)*cosh(d*x + c)^4 + 8*(315*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b
^3)*d*x*cosh(d*x + c)^6 - 105*(4*a^4 + 8*a^3*b - a*b^3 - 2*b^4 + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*
d*x)*cosh(d*x + c)^4 + 2*a^4 + 38*a^3*b + 72*a^2*b^2 + 9*b^4 + 3*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^
4)*d*x - 45*(2*a^4 + 10*a^3*b + 16*a^2*b^2 + a*b^3 + 3*b^4 + (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d
*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 32*a^4 - 80*a^3*b - 12*a*b^3 + 32*(45*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^
3)*d*x*cosh(d*x + c)^7 - 21*(4*a^4 + 8*a^3*b - a*b^3 - 2*b^4 + (a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*
x)*cosh(d*x + c)^5 - 15*(2*a^4 + 10*a^3*b + 16*a^2*b^2 + a*b^3 + 3*b^4 + (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^
3 + 6*b^4)*d*x)*cosh(d*x + c)^3 + (2*a^4 + 38*a^3*b + 72*a^2*b^2 + 9*b^4 + 3*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*
a*b^3 + 6*b^4)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 12*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x - 4*(4*a^4 + 3
6*a^3*b + 80*a^2*b^2 - 6*a*b^3 + 6*b^4 - 3*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*x)*cosh(d*x + c)^2 +
4*(135*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cosh(d*x + c)^8 - 84*(4*a^4 + 8*a^3*b - a*b^3 - 2*b^4 + (a^4 -
a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*x)*cosh(d*x + c)^6 - 90*(2*a^4 + 10*a^3*b + 16*a^2*b^2 + a*b^3 + 3*b^
4 + (a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*x)*cosh(d*x + c)^4 - 4*a^4 - 36*a^3*b - 80*a^2*b^2 + 6*a
*b^3 - 6*b^4 + 3*(a^4 - a^3*b - 9*a^2*b^2 - 11*a*b^3 - 4*b^4)*d*x + 12*(2*a^4 + 38*a^3*b + 72*a^2*b^2 + 9*b^4
+ 3*(a^4 + 9*a^3*b + 21*a^2*b^2 + 19*a*b^3 + 6*b^4)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((7*a^2*b^2 + 2*
a*b^3)*cosh(d*x + c)^10 + 10*(7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^9 + (7*a^2*b^2 + 2*a*b^3)*sinh(
d*x + c)^10 - (7*a^2*b^2 - 26*a*b^3 - 8*b^4)*cosh(d*x + c)^8 - (7*a^2*b^2 - 26*a*b^3 - 8*b^4 - 45*(7*a^2*b^2 +
2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3 - (7*a^2*b^2 - 26*a*b
^3 - 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(7*a^2*b^2 + 44*a*b^3 + 12*b^4)*cosh(d*x + c)^6 + 2*(105*(7*a^2
*b^2 + 2*a*b^3)*cosh(d*x + c)^4 - 7*a^2*b^2 - 44*a*b^3 - 12*b^4 - 14*(7*a^2*b^2 - 26*a*b^3 - 8*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^6 + 4*(63*(7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^5 - 14*(7*a^2*b^2 - 26*a*b^3 - 8*b^4)*cosh(
d*x + c)^3 - 3*(7*a^2*b^2 + 44*a*b^3 + 12*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(7*a^2*b^2 + 44*a*b^3 + 12*b
^4)*cosh(d*x + c)^4 + 2*(105*(7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^6 - 35*(7*a^2*b^2 - 26*a*b^3 - 8*b^4)*cosh(d*
x + c)^4 + 7*a^2*b^2 + 44*a*b^3 + 12*b^4 - 15*(7*a^2*b^2 + 44*a*b^3 + 12*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4
- 7*a^2*b^2 - 2*a*b^3 + 8*(15*(7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^7 - 7*(7*a^2*b^2 - 26*a*b^3 - 8*b^4)*cosh(d
*x + c)^5 - 5*(7*a^2*b^2 + 44*a*b^3 + 12*b^4)*cosh(d*x + c)^3 + (7*a^2*b^2 + 44*a*b^3 + 12*b^4)*cosh(d*x + c))
*sinh(d*x + c)^3 + (7*a^2*b^2 - 26*a*b^3 - 8*b^4)*cosh(d*x + c)^2 + (45*(7*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^8
- 28*(7*a^2*b^2 - 26*a*b^3 - 8*b^4)*cosh(d*x + c)^6 - 30*(7*a^2*b^2 + 44*a*b^3 + 12*b^4)*cosh(d*x + c)^4 + 7*a
^2*b^2 - 26*a*b^3 - 8*b^4 + 12*(7*a^2*b^2 + 44*a*b^3 + 12*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(7*a^2*
b^2 + 2*a*b^3)*cosh(d*x + c)^9 - 4*(7*a^2*b^2 - 26*a*b^3 - 8*b^4)*cosh(d*x + c)^7 - 6*(7*a^2*b^2 + 44*a*b^3 +
12*b^4)*cosh(d*x + c)^5 + 4*(7*a^2*b^2 + 44*a*b^3 + 12*b^4)*cosh(d*x + c)^3 + (7*a^2*b^2 - 26*a*b^3 - 8*b^4)*c
osh(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 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{4}{\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)**4/(a+b*sech(d*x+c)**2)**2,x)

[Out]

Integral(coth(c + d*x)**4/(a + b*sech(c + d*x)**2)**2, x)

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Giac [A]
time = 1.88, size = 293, normalized size = 1.82 \begin {gather*} -\frac {\frac {3 \, {\left (7 \, a b^{3} + 2 \, b^{4}\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^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt {-a b - b^{2}}} - \frac {6 \, {\left (a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + a b^{3}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}} - \frac {6 \, {\left (d x + c\right )}}{a^{2}} + \frac {8 \, {\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 5 \, b\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-1/6*(3*(7*a*b^3 + 2*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^5 + 3*a^4*b + 3*a^3*b
^2 + a^2*b^3)*sqrt(-a*b - b^2)) - 6*(a*b^3*e^(2*d*x + 2*c) + 2*b^4*e^(2*d*x + 2*c) + a*b^3)/((a^5 + 3*a^4*b +
3*a^3*b^2 + a^2*b^3)*(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)) - 6*(d*x + c)/a^2 +
8*(3*a*e^(4*d*x + 4*c) + 6*b*e^(4*d*x + 4*c) - 3*a*e^(2*d*x + 2*c) - 9*b*e^(2*d*x + 2*c) + 2*a + 5*b)/((a^3 +
3*a^2*b + 3*a*b^2 + b^3)*(e^(2*d*x + 2*c) - 1)^3))/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 )}^4}{{\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)^4/(a + b/cosh(c + d*x)^2)^2,x)

[Out]

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

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