3.2.47 \(\int \frac {\coth ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) [147]

Optimal. Leaf size=87 \[ \frac {x}{a}-\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a (a+b)^{5/2} d}-\frac {(a+2 b) \coth (c+d x)}{(a+b)^2 d}-\frac {\coth ^3(c+d x)}{3 (a+b) d} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 87, 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, 491, 597, 536, 212, 214} \begin {gather*} -\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a d (a+b)^{5/2}}-\frac {\coth ^3(c+d x)}{3 d (a+b)}-\frac {(a+2 b) \coth (c+d x)}{d (a+b)^2}+\frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[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)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )} \, 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 )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\coth ^3(c+d x)}{3 (a+b) d}+\frac {\text {Subst}\left (\int \frac {3 (a+2 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 )}{3 (a+b) d}\\ &=-\frac {(a+2 b) \coth (c+d x)}{(a+b)^2 d}-\frac {\coth ^3(c+d x)}{3 (a+b) d}-\frac {\text {Subst}\left (\int \frac {-3 \left (a^2+3 a b+3 b^2\right )+3 b (a+2 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{3 (a+b)^2 d}\\ &=-\frac {(a+2 b) \coth (c+d x)}{(a+b)^2 d}-\frac {\coth ^3(c+d x)}{3 (a+b) d}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{a d}-\frac {b^3 \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{a (a+b)^2 d}\\ &=\frac {x}{a}-\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a (a+b)^{5/2} d}-\frac {(a+2 b) \coth (c+d x)}{(a+b)^2 d}-\frac {\coth ^3(c+d x)}{3 (a+b) d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(380\) vs. \(2(87)=174\).
time = 2.22, size = 380, normalized size = 4.37 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left (3 b^3 \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 ) (-\cosh (2 c)+\sinh (2 c))+\frac {1}{8} \sqrt {a+b} \text {csch}(c) \text {csch}^3(c+d x) \sqrt {b (\cosh (c)-\sinh (c))^4} \left (9 (a+b)^2 d x \cosh (d x)-9 (a+b)^2 d x \cosh (2 c+d x)-3 a^2 d x \cosh (2 c+3 d x)-6 a b d x \cosh (2 c+3 d x)-3 b^2 d x \cosh (2 c+3 d x)+3 a^2 d x \cosh (4 c+3 d x)+6 a b d x \cosh (4 c+3 d x)+3 b^2 d x \cosh (4 c+3 d x)-12 a^2 \sinh (d x)-24 a b \sinh (d x)-12 a^2 \sinh (2 c+d x)-18 a b \sinh (2 c+d x)+8 a^2 \sinh (2 c+3 d x)+14 a b \sinh (2 c+3 d x)\right )\right )}{6 a (a+b)^{5/2} d \left (a+b \text {sech}^2(c+d x)\right ) \sqrt {b (\cosh (c)-\sinh (c))^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(3*b^3*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])]*(-Cosh[2*c] + Sinh[2*c]) + (Sqr
t[a + b]*Csch[c]*Csch[c + d*x]^3*Sqrt[b*(Cosh[c] - Sinh[c])^4]*(9*(a + b)^2*d*x*Cosh[d*x] - 9*(a + b)^2*d*x*Co
sh[2*c + d*x] - 3*a^2*d*x*Cosh[2*c + 3*d*x] - 6*a*b*d*x*Cosh[2*c + 3*d*x] - 3*b^2*d*x*Cosh[2*c + 3*d*x] + 3*a^
2*d*x*Cosh[4*c + 3*d*x] + 6*a*b*d*x*Cosh[4*c + 3*d*x] + 3*b^2*d*x*Cosh[4*c + 3*d*x] - 12*a^2*Sinh[d*x] - 24*a*
b*Sinh[d*x] - 12*a^2*Sinh[2*c + d*x] - 18*a*b*Sinh[2*c + d*x] + 8*a^2*Sinh[2*c + 3*d*x] + 14*a*b*Sinh[2*c + 3*
d*x]))/8))/(6*a*(a + b)^(5/2)*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(251\) vs. \(2(77)=154\).
time = 3.10, size = 252, normalized size = 2.90

method	result	size
risch	\(\frac {x}{a}-\frac {2 \left (6 a \,{\mathrm e}^{4 d x +4 c}+9 b \,{\mathrm e}^{4 d x +4 c}-6 a \,{\mathrm e}^{2 d x +2 c}-12 b \,{\mathrm e}^{2 d x +2 c}+4 a +7 b \right )}{3 d \left (a +b \right )^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {\sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right )}{2 \left (a +b \right )^{3} d a}-\frac {\sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right )}{2 \left (a +b \right )^{3} d a}\)	\(192\)
derivativedivides	\(\frac {-\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+9 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right )^{2}}+\frac {2 b^{3} \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a \left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}-\frac {1}{24 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +9 b}{8 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(252\)
default	\(\frac {-\frac {\frac {a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+5 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+9 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right )^{2}}+\frac {2 b^{3} \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a \left (a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a}-\frac {1}{24 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a +9 b}{8 \left (a +b \right )^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/8/(a+b)^2*(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)+9*b*tanh(1/
2*d*x+1/2*c))+2*b^3/a/(a+b)^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))+1/4/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*ln(tanh(1/2*d*x+1/2*c)+1)-1/a*ln(tanh(1/2*d*x+1/2*c)-1)-1/24/(a+b)/tanh(1/2*d*x+1/2*c
)^3-1/8*(5*a+9*b)/(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. 1435 vs. \(2 (77) = 154\).
time = 0.59, size = 1435, normalized size = 16.49 \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),x, algorithm="maxima")

[Out]

3/16*a*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) + 1/8*(a*b + 2*b^2)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x
+ 2*c) + a)/((a^3 + 2*a^2*b + a*b^2)*d) - 1/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^
2 + 2*a*b + b^2)*d) - 1/8*(a*b + 2*b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^3 + 2*a
^2*b + a*b^2)*d) + 1/4*b*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^2 + 2*a*b + b^2)*d) +
1/4*(2*a + 3*b)*log(e^(2*d*x + 2*c) - 1)/((a^2 + 2*a*b + b^2)*d) + 1/2*b*log(e^(2*d*x + 2*c) - 1)/((a^2 + 2*a*
b + b^2)*d) - 1/4*(2*a + 3*b)*log(e^(-2*d*x - 2*c) - 1)/((a^2 + 2*a*b + b^2)*d) - 1/2*b*log(e^(-2*d*x - 2*c) -
1)/((a^2 + 2*a*b + b^2)*d) - 1/32*(a^2*b + 8*a*b^2 + 8*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^3 + 2*a^2*b + a*b^2)*sqrt((a + b)*b)*d) + 1/8*(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^2 + 2*a*b + b^2)*sqrt((a + b)*b)*d) + 1/32*(a^2*b + 8*a*b^2 + 8*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^3 + 2*a^2*b + a*b^2)*sqrt((a +
b)*b)*d) - 1/8*(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^2 + 2*a*b + b^2)*sqrt((a + b)*b)*d) + 1/24*(3*(12*a + 13*b)*e^(4*d*x + 4*c) - 6*
(9*a + 10*b)*e^(2*d*x + 2*c) + 22*a + 25*b)/((a^2 + 2*a*b + b^2 - (a^2 + 2*a*b + b^2)*e^(6*d*x + 6*c) + 3*(a^2
+ 2*a*b + b^2)*e^(4*d*x + 4*c) - 3*(a^2 + 2*a*b + b^2)*e^(2*d*x + 2*c))*d) + 1/6*(3*(4*a + 5*b)*e^(4*d*x + 4*
c) - 6*(2*a + 3*b)*e^(2*d*x + 2*c) + 4*a + 7*b)/((a^2 + 2*a*b + b^2 - (a^2 + 2*a*b + b^2)*e^(6*d*x + 6*c) + 3*
(a^2 + 2*a*b + b^2)*e^(4*d*x + 4*c) - 3*(a^2 + 2*a*b + b^2)*e^(2*d*x + 2*c))*d) + 1/24*(6*(9*a + 10*b)*e^(-2*d
*x - 2*c) - 3*(12*a + 13*b)*e^(-4*d*x - 4*c) - 22*a - 25*b)/((a^2 + 2*a*b + b^2 - 3*(a^2 + 2*a*b + b^2)*e^(-2*
d*x - 2*c) + 3*(a^2 + 2*a*b + b^2)*e^(-4*d*x - 4*c) - (a^2 + 2*a*b + b^2)*e^(-6*d*x - 6*c))*d) + 1/6*(6*(2*a +
3*b)*e^(-2*d*x - 2*c) - 3*(4*a + 5*b)*e^(-4*d*x - 4*c) - 4*a - 7*b)/((a^2 + 2*a*b + b^2 - 3*(a^2 + 2*a*b + b^
2)*e^(-2*d*x - 2*c) + 3*(a^2 + 2*a*b + b^2)*e^(-4*d*x - 4*c) - (a^2 + 2*a*b + b^2)*e^(-6*d*x - 6*c))*d) - 1/4*
(6*a*e^(-2*d*x - 2*c) + 3*b*e^(-4*d*x - 4*c) - 2*a + b)/((a^2 + 2*a*b + b^2 - 3*(a^2 + 2*a*b + b^2)*e^(-2*d*x
- 2*c) + 3*(a^2 + 2*a*b + b^2)*e^(-4*d*x - 4*c) - (a^2 + 2*a*b + b^2)*e^(-6*d*x - 6*c))*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1214 vs. \(2 (77) = 154\).
time = 0.45, size = 2705, normalized size = 31.09 \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),x, algorithm="fricas")

[Out]

[1/6*(6*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^6 + 36*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^5 + 6
*(a^2 + 2*a*b + b^2)*d*x*sinh(d*x + c)^6 - 6*(3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 6*a*b)*cosh(d*x + c)^4 + 6*(
15*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^2 - 3*(a^2 + 2*a*b + b^2)*d*x - 4*a^2 - 6*a*b)*sinh(d*x + c)^4 + 24*(
5*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^3 - (3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 6*a*b)*cosh(d*x + c))*sinh(d*
x + c)^3 - 6*(a^2 + 2*a*b + b^2)*d*x + 6*(3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 8*a*b)*cosh(d*x + c)^2 + 6*(15*(
a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^4 + 3*(a^2 + 2*a*b + b^2)*d*x - 6*(3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 6*
a*b)*cosh(d*x + c)^2 + 4*a^2 + 8*a*b)*sinh(d*x + c)^2 + 3*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x
+ c)^5 + b^2*sinh(d*x + c)^6 - 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 - b^2)*sinh(d*x + c)^4 + 3*b^2
*cosh(d*x + c)^2 + 4*(5*b^2*cosh(d*x + c)^3 - 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^2*cosh(d*x + c)^4
- 6*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^2 - b^2 + 6*(b^2*cosh(d*x + c)^5 - 2*b^2*cosh(d*x + c)^3 + b^2*co
sh(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*cosh(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)*cosh(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)*c
osh(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)) - 16*a^2 - 28*a*b + 12*(3*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^5 - 2*(3*(a^2 + 2*a
*b + b^2)*d*x + 4*a^2 + 6*a*b)*cosh(d*x + c)^3 + (3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 8*a*b)*cosh(d*x + c))*si
nh(d*x + c))/((a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)*sinh(d*x +
c)^5 + (a^3 + 2*a^2*b + a*b^2)*d*sinh(d*x + c)^6 - 3*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^4 + 3*(5*(a^3 +
2*a^2*b + a*b^2)*d*cosh(d*x + c)^2 - (a^3 + 2*a^2*b + a*b^2)*d)*sinh(d*x + c)^4 + 3*(a^3 + 2*a^2*b + a*b^2)*d*
cosh(d*x + c)^2 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^3 - 3*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c))*
sinh(d*x + c)^3 + 3*(5*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^4 - 6*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^2
+ (a^3 + 2*a^2*b + a*b^2)*d)*sinh(d*x + c)^2 - (a^3 + 2*a^2*b + a*b^2)*d + 6*((a^3 + 2*a^2*b + a*b^2)*d*cosh(
d*x + c)^5 - 2*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c))*sinh(d*x +
c)), 1/3*(3*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^6 + 18*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^
5 + 3*(a^2 + 2*a*b + b^2)*d*x*sinh(d*x + c)^6 - 3*(3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 6*a*b)*cosh(d*x + c)^4
+ 3*(15*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^2 - 3*(a^2 + 2*a*b + b^2)*d*x - 4*a^2 - 6*a*b)*sinh(d*x + c)^4 +
12*(5*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^3 - (3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 6*a*b)*cosh(d*x + c))*si
nh(d*x + c)^3 - 3*(a^2 + 2*a*b + b^2)*d*x + 3*(3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 8*a*b)*cosh(d*x + c)^2 + 3*
(15*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^4 + 3*(a^2 + 2*a*b + b^2)*d*x - 6*(3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2
+ 6*a*b)*cosh(d*x + c)^2 + 4*a^2 + 8*a*b)*sinh(d*x + c)^2 - 3*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh
(d*x + c)^5 + b^2*sinh(d*x + c)^6 - 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 - b^2)*sinh(d*x + c)^4 +
3*b^2*cosh(d*x + c)^2 + 4*(5*b^2*cosh(d*x + c)^3 - 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^2*cosh(d*x +
c)^4 - 6*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^2 - b^2 + 6*(b^2*cosh(d*x + c)^5 - 2*b^2*cosh(d*x + c)^3 + b
^2*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/(a + b))*arctan(1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x +
c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(-b/(a + b))/b) - 8*a^2 - 14*a*b + 6*(3*(a^2 + 2*a*b + b^2)*d*x*cosh(d*
x + c)^5 - 2*(3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2 + 6*a*b)*cosh(d*x + c)^3 + (3*(a^2 + 2*a*b + b^2)*d*x + 4*a^2
+ 8*a*b)*cosh(d*x + c))*sinh(d*x + c))/((a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*
d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^3 + 2*a^2*b + a*b^2)*d*sinh(d*x + c)^6 - 3*(a^3 + 2*a^2*b + a*b^2)*d*cosh
(d*x + c)^4 + 3*(5*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^2 - (a^3 + 2*a^2*b + a*b^2)*d)*sinh(d*x + c)^4 + 3*
(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^2 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^3 - 3*(a^3 + 2*a^2*b
+ a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^4 - 6*(a^3 + 2*a^2*b
+ a*b^2)*d*cosh(d*x + c)^2 + (a^3 + 2*a^2*b + a*b^2)*d)*sinh(d*x + c)^2 - (a^3 + 2*a^2*b + a*b^2)*d + 6*((a^3
+ 2*a^2*b + a*b^2)*d*cosh(d*x + c)^5 - 2*(a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*d
*cosh(d*x + c))*sinh(d*x + c))]

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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 )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, 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),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (77) = 154\).
time = 1.60, size = 164, normalized size = 1.89 \begin {gather*} -\frac {\frac {3 \, b^{3} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt {-a b - b^{2}}} - \frac {3 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 12 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a + 7 \, b\right )}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, 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),x, algorithm="giac")

[Out]

-1/3*(3*b^3*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^3 + 2*a^2*b + a*b^2)*sqrt(-a*b - b^
2)) - 3*(d*x + c)/a + 2*(6*a*e^(4*d*x + 4*c) + 9*b*e^(4*d*x + 4*c) - 6*a*e^(2*d*x + 2*c) - 12*b*e^(2*d*x + 2*c
) + 4*a + 7*b)/((a^2 + 2*a*b + b^2)*(e^(2*d*x + 2*c) - 1)^3))/d

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Mupad [B]
time = 4.15, size = 779, normalized size = 8.95 \begin {gather*} \frac {x}{a}-\frac {8}{3\,\left (a\,d+b\,d\right )\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {\mathrm {atan}\left (\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {4\,b^3}{a^3\,d\,{\left (a+b\right )}^2\,\sqrt {b^5}\,\left (a^3+2\,a^2\,b+a\,b^2\right )}+\frac {\left (a+2\,b\right )\,\left (a^4\,d\,\sqrt {b^5}+2\,a\,b^3\,d\,\sqrt {b^5}+4\,a^3\,b\,d\,\sqrt {b^5}+5\,a^2\,b^2\,d\,\sqrt {b^5}\right )}{a^2\,b^3\,\left (a^3+2\,a^2\,b+a\,b^2\right )\,\sqrt {-a^2\,d^2\,{\left (a+b\right )}^5}\,\sqrt {-a^7\,d^2-5\,a^6\,b\,d^2-10\,a^5\,b^2\,d^2-10\,a^4\,b^3\,d^2-5\,a^3\,b^4\,d^2-a^2\,b^5\,d^2}}\right )+\frac {\left (a+2\,b\right )\,\left (a^4\,d\,\sqrt {b^5}+2\,a^3\,b\,d\,\sqrt {b^5}+a^2\,b^2\,d\,\sqrt {b^5}\right )}{a^2\,b^3\,\left (a^3+2\,a^2\,b+a\,b^2\right )\,\sqrt {-a^2\,d^2\,{\left (a+b\right )}^5}\,\sqrt {-a^7\,d^2-5\,a^6\,b\,d^2-10\,a^5\,b^2\,d^2-10\,a^4\,b^3\,d^2-5\,a^3\,b^4\,d^2-a^2\,b^5\,d^2}}\right )\,\left (\frac {a^4\,\sqrt {-a^7\,d^2-5\,a^6\,b\,d^2-10\,a^5\,b^2\,d^2-10\,a^4\,b^3\,d^2-5\,a^3\,b^4\,d^2-a^2\,b^5\,d^2}}{2}+\frac {a^2\,b^2\,\sqrt {-a^7\,d^2-5\,a^6\,b\,d^2-10\,a^5\,b^2\,d^2-10\,a^4\,b^3\,d^2-5\,a^3\,b^4\,d^2-a^2\,b^5\,d^2}}{2}+a^3\,b\,\sqrt {-a^7\,d^2-5\,a^6\,b\,d^2-10\,a^5\,b^2\,d^2-10\,a^4\,b^3\,d^2-5\,a^3\,b^4\,d^2-a^2\,b^5\,d^2}\right )\right )\,\sqrt {b^5}}{\sqrt {-a^7\,d^2-5\,a^6\,b\,d^2-10\,a^5\,b^2\,d^2-10\,a^4\,b^3\,d^2-5\,a^3\,b^4\,d^2-a^2\,b^5\,d^2}}-\frac {4\,\left (a^2+b\,a\right )}{a\,\left (a+b\right )\,\left (a\,d+b\,d\right )\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,\left (2\,a^2+3\,b\,a\right )}{a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (a+b\right )\,\left (a\,d+b\,d\right )} \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),x)

[Out]

x/a - 8/(3*(a*d + b*d)*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (atan((exp(2*c)*exp
(2*d*x)*((4*b^3)/(a^3*d*(a + b)^2*(b^5)^(1/2)*(a*b^2 + 2*a^2*b + a^3)) + ((a + 2*b)*(a^4*d*(b^5)^(1/2) + 2*a*b
^3*d*(b^5)^(1/2) + 4*a^3*b*d*(b^5)^(1/2) + 5*a^2*b^2*d*(b^5)^(1/2)))/(a^2*b^3*(a*b^2 + 2*a^2*b + a^3)*(-a^2*d^
2*(a + b)^5)^(1/2)*(- a^7*d^2 - 5*a^6*b*d^2 - a^2*b^5*d^2 - 5*a^3*b^4*d^2 - 10*a^4*b^3*d^2 - 10*a^5*b^2*d^2)^(
1/2))) + ((a + 2*b)*(a^4*d*(b^5)^(1/2) + 2*a^3*b*d*(b^5)^(1/2) + a^2*b^2*d*(b^5)^(1/2)))/(a^2*b^3*(a*b^2 + 2*a
^2*b + a^3)*(-a^2*d^2*(a + b)^5)^(1/2)*(- a^7*d^2 - 5*a^6*b*d^2 - a^2*b^5*d^2 - 5*a^3*b^4*d^2 - 10*a^4*b^3*d^2
- 10*a^5*b^2*d^2)^(1/2)))*((a^4*(- a^7*d^2 - 5*a^6*b*d^2 - a^2*b^5*d^2 - 5*a^3*b^4*d^2 - 10*a^4*b^3*d^2 - 10*
a^5*b^2*d^2)^(1/2))/2 + (a^2*b^2*(- a^7*d^2 - 5*a^6*b*d^2 - a^2*b^5*d^2 - 5*a^3*b^4*d^2 - 10*a^4*b^3*d^2 - 10*
a^5*b^2*d^2)^(1/2))/2 + a^3*b*(- a^7*d^2 - 5*a^6*b*d^2 - a^2*b^5*d^2 - 5*a^3*b^4*d^2 - 10*a^4*b^3*d^2 - 10*a^5
*b^2*d^2)^(1/2)))*(b^5)^(1/2))/(- a^7*d^2 - 5*a^6*b*d^2 - a^2*b^5*d^2 - 5*a^3*b^4*d^2 - 10*a^4*b^3*d^2 - 10*a^
5*b^2*d^2)^(1/2) - (4*(a*b + a^2))/(a*(a + b)*(a*d + b*d)*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (2*(3
*a*b + 2*a^2))/(a*(exp(2*c + 2*d*x) - 1)*(a + b)*(a*d + b*d))

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