3.2.68 \(\int \frac {\coth ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [168]
Optimal. Leaf size=232 \[ \frac {x}{a^3}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{9/2} d}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
[Out]
x/a^3-1/8*b^(5/2)*(63*a^2+36*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^3/(a+b)^(9/2)/d-1/8*(8*a^3+
32*a^2*b-15*a*b^2-4*b^3)*coth(d*x+c)/a^2/(a+b)^4/d-1/24*(8*a^2-39*a*b-12*b^2)*coth(d*x+c)^3/a^2/(a+b)^3/d-1/4*
b*coth(d*x+c)^3/a/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^2-1/8*b*(11*a+4*b)*coth(d*x+c)^3/a^2/(a+b)^2/d/(a+b-b*tanh(d*x
+c)^2)
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Rubi [A]
time = 0.35, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4226, 2000,
483, 593, 597, 536, 212, 214} \begin {gather*} \frac {x}{a^3}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 d (a+b)^3}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 d (a+b)^{9/2}}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 d (a+b)^4}-\frac {b \coth ^3(c+d x)}{4 a d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
[Out]
x/a^3 - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(9/2)*
d) - ((8*a^3 + 32*a^2*b - 15*a*b^2 - 4*b^3)*Coth[c + d*x])/(8*a^2*(a + b)^4*d) - ((8*a^2 - 39*a*b - 12*b^2)*Co
th[c + d*x]^3)/(24*a^2*(a + b)^3*d) - (b*Coth[c + d*x]^3)/(4*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(
11*a + 4*b)*Coth[c + d*x]^3)/(8*a^2*(a + b)^2*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 593
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
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 )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^4 \left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )^3} \, 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 )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-4 a+3 b-7 b x^2}{x^4 \left (1-x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a+b) d}\\ &=-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {8 a^2-39 a b-12 b^2+5 b (11 a+4 b) x^2}{x^4 \left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-3 \left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right )+3 b \left (8 a^2-39 a b-12 b^2\right ) x^2}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{24 a^2 (a+b)^3 d}\\ &=-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {3 \left (8 a^4+40 a^3 b+80 a^2 b^2+17 a b^3+4 b^4\right )-3 b \left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{24 a^2 (a+b)^4 d}\\ &=-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 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^3 d}-\frac {\left (b^3 \left (63 a^2+36 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a+b)^4 d}\\ &=\frac {x}{a^3}-\frac {b^{5/2} \left (63 a^2+36 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 (a+b)^{9/2} d}-\frac {\left (8 a^3+32 a^2 b-15 a b^2-4 b^3\right ) \coth (c+d x)}{8 a^2 (a+b)^4 d}-\frac {\left (8 a^2-39 a b-12 b^2\right ) \coth ^3(c+d x)}{24 a^2 (a+b)^3 d}-\frac {b \coth ^3(c+d x)}{4 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (11 a+4 b) \coth ^3(c+d x)}{8 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 7.16, size = 3334, normalized size = 14.37 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
[Out]
((63*a^2 + 36*a*b + 8*b^2)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((I/64)*b^3*ArcTan[Sech[d*x]*(((
-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[
4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt
[b*Cosh[4*c] - b*Sinh[4*c]]) - ((I/64)*b^3*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c
] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sin
h[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(a^3*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^4*(a +
b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]^3*Sech[2*c]*Sech[c + d*x]^6*(-3
6*a^6*d*x*Cosh[d*x] - 336*a^5*b*d*x*Cosh[d*x] - 1560*a^4*b^2*d*x*Cosh[d*x] - 3600*a^3*b^3*d*x*Cosh[d*x] - 4260
*a^2*b^4*d*x*Cosh[d*x] - 2496*a*b^5*d*x*Cosh[d*x] - 576*b^6*d*x*Cosh[d*x] + 36*a^6*d*x*Cosh[3*d*x] + 240*a^5*b
*d*x*Cosh[3*d*x] + 408*a^4*b^2*d*x*Cosh[3*d*x] - 48*a^3*b^3*d*x*Cosh[3*d*x] - 732*a^2*b^4*d*x*Cosh[3*d*x] - 67
2*a*b^5*d*x*Cosh[3*d*x] - 192*b^6*d*x*Cosh[3*d*x] + 36*a^6*d*x*Cosh[2*c - d*x] + 336*a^5*b*d*x*Cosh[2*c - d*x]
+ 1560*a^4*b^2*d*x*Cosh[2*c - d*x] + 3600*a^3*b^3*d*x*Cosh[2*c - d*x] + 4260*a^2*b^4*d*x*Cosh[2*c - d*x] + 24
96*a*b^5*d*x*Cosh[2*c - d*x] + 576*b^6*d*x*Cosh[2*c - d*x] + 36*a^6*d*x*Cosh[2*c + d*x] + 336*a^5*b*d*x*Cosh[2
*c + d*x] + 1560*a^4*b^2*d*x*Cosh[2*c + d*x] + 3600*a^3*b^3*d*x*Cosh[2*c + d*x] + 4260*a^2*b^4*d*x*Cosh[2*c +
d*x] + 2496*a*b^5*d*x*Cosh[2*c + d*x] + 576*b^6*d*x*Cosh[2*c + d*x] - 36*a^6*d*x*Cosh[4*c + d*x] - 336*a^5*b*d
*x*Cosh[4*c + d*x] - 1560*a^4*b^2*d*x*Cosh[4*c + d*x] - 3600*a^3*b^3*d*x*Cosh[4*c + d*x] - 4260*a^2*b^4*d*x*Co
sh[4*c + d*x] - 2496*a*b^5*d*x*Cosh[4*c + d*x] - 576*b^6*d*x*Cosh[4*c + d*x] - 36*a^6*d*x*Cosh[2*c + 3*d*x] -
240*a^5*b*d*x*Cosh[2*c + 3*d*x] - 408*a^4*b^2*d*x*Cosh[2*c + 3*d*x] + 48*a^3*b^3*d*x*Cosh[2*c + 3*d*x] + 732*a
^2*b^4*d*x*Cosh[2*c + 3*d*x] + 672*a*b^5*d*x*Cosh[2*c + 3*d*x] + 192*b^6*d*x*Cosh[2*c + 3*d*x] + 36*a^6*d*x*Co
sh[4*c + 3*d*x] + 240*a^5*b*d*x*Cosh[4*c + 3*d*x] + 408*a^4*b^2*d*x*Cosh[4*c + 3*d*x] - 48*a^3*b^3*d*x*Cosh[4*
c + 3*d*x] - 732*a^2*b^4*d*x*Cosh[4*c + 3*d*x] - 672*a*b^5*d*x*Cosh[4*c + 3*d*x] - 192*b^6*d*x*Cosh[4*c + 3*d*
x] - 36*a^6*d*x*Cosh[6*c + 3*d*x] - 240*a^5*b*d*x*Cosh[6*c + 3*d*x] - 408*a^4*b^2*d*x*Cosh[6*c + 3*d*x] + 48*a
^3*b^3*d*x*Cosh[6*c + 3*d*x] + 732*a^2*b^4*d*x*Cosh[6*c + 3*d*x] + 672*a*b^5*d*x*Cosh[6*c + 3*d*x] + 192*b^6*d
*x*Cosh[6*c + 3*d*x] - 12*a^6*d*x*Cosh[2*c + 5*d*x] - 144*a^5*b*d*x*Cosh[2*c + 5*d*x] - 456*a^4*b^2*d*x*Cosh[2
*c + 5*d*x] - 624*a^3*b^3*d*x*Cosh[2*c + 5*d*x] - 396*a^2*b^4*d*x*Cosh[2*c + 5*d*x] - 96*a*b^5*d*x*Cosh[2*c +
5*d*x] + 12*a^6*d*x*Cosh[4*c + 5*d*x] + 144*a^5*b*d*x*Cosh[4*c + 5*d*x] + 456*a^4*b^2*d*x*Cosh[4*c + 5*d*x] +
624*a^3*b^3*d*x*Cosh[4*c + 5*d*x] + 396*a^2*b^4*d*x*Cosh[4*c + 5*d*x] + 96*a*b^5*d*x*Cosh[4*c + 5*d*x] - 12*a^
6*d*x*Cosh[6*c + 5*d*x] - 144*a^5*b*d*x*Cosh[6*c + 5*d*x] - 456*a^4*b^2*d*x*Cosh[6*c + 5*d*x] - 624*a^3*b^3*d*
x*Cosh[6*c + 5*d*x] - 396*a^2*b^4*d*x*Cosh[6*c + 5*d*x] - 96*a*b^5*d*x*Cosh[6*c + 5*d*x] + 12*a^6*d*x*Cosh[8*c
+ 5*d*x] + 144*a^5*b*d*x*Cosh[8*c + 5*d*x] + 456*a^4*b^2*d*x*Cosh[8*c + 5*d*x] + 624*a^3*b^3*d*x*Cosh[8*c + 5
*d*x] + 396*a^2*b^4*d*x*Cosh[8*c + 5*d*x] + 96*a*b^5*d*x*Cosh[8*c + 5*d*x] - 12*a^6*d*x*Cosh[4*c + 7*d*x] - 48
*a^5*b*d*x*Cosh[4*c + 7*d*x] - 72*a^4*b^2*d*x*Cosh[4*c + 7*d*x] - 48*a^3*b^3*d*x*Cosh[4*c + 7*d*x] - 12*a^2*b^
4*d*x*Cosh[4*c + 7*d*x] + 12*a^6*d*x*Cosh[6*c + 7*d*x] + 48*a^5*b*d*x*Cosh[6*c + 7*d*x] + 72*a^4*b^2*d*x*Cosh[
6*c + 7*d*x] + 48*a^3*b^3*d*x*Cosh[6*c + 7*d*x] + 12*a^2*b^4*d*x*Cosh[6*c + 7*d*x] - 12*a^6*d*x*Cosh[8*c + 7*d
*x] - 48*a^5*b*d*x*Cosh[8*c + 7*d*x] - 72*a^4*b^2*d*x*Cosh[8*c + 7*d*x] - 48*a^3*b^3*d*x*Cosh[8*c + 7*d*x] - 1
2*a^2*b^4*d*x*Cosh[8*c + 7*d*x] + 12*a^6*d*x*Cosh[10*c + 7*d*x] + 48*a^5*b*d*x*Cosh[10*c + 7*d*x] + 72*a^4*b^2
*d*x*Cosh[10*c + 7*d*x] + 48*a^3*b^3*d*x*Cosh[10*c + 7*d*x] + 12*a^2*b^4*d*x*Cosh[10*c + 7*d*x] - 128*a^6*Sinh
[d*x] - 440*a^5*b*Sinh[d*x] - 1152*a^4*b^2*Sinh[d*x] - 1920*a^3*b^3*Sinh[d*x] + 228*a^2*b^4*Sinh[d*x] + 1320*a
*b^5*Sinh[d*x] + 432*b^6*Sinh[d*x] + 48*a^6*Sinh[3*d*x] + 104*a^5*b*Sinh[3*d*x] + 640*a^4*b^2*Sinh[3*d*x] + 15
11*a^3*b^3*Sinh[3*d*x] - 528*a^2*b^4*Sinh[3*d*x] + 264*a*b^5*Sinh[3*d*x] + 144*b^6*Sinh[3*d*x] - 32*a^6*Sinh[2
*c - d*x] + 384*a^5*b*Sinh[2*c - d*x] + 2048*a^4*b^2*Sinh[2*c - d*x] + 3072*a^3*b^3*Sinh[2*c - d*x] + 228*a^2*
b^4*Sinh[2*c - d*x] + 1320*a*b^5*Sinh[2*c - d*x] + 432*b^6*Sinh[2*c - d*x] + 32*a^6*Sinh[2*c + d*x] - 384*a^5*
b*Sinh[2*c + d*x] - 2048*a^4*b^2*Sinh[2*c + d*x] - 2919*a^3*b^3*Sinh[2*c + d*x] + 642*a^2*b^4*Sinh[2*c + d*x]
+ 1416*a*b^5*Sinh[2*c + d*x] + 432*b^6*Sinh[2*c + d*x] - 128*a^6*Sinh[4*c + d*x] - 440*a^5*b*Sinh[4*c + d*x] -
1152*a^4*b^2*Sinh[4*c + d*x] - 2073*a^3*b^3*Sinh[4*c + d*x] - 642*a^2*b^4*Sinh[4*c + d*x] - 1416*a*b^5*Sinh[4
*c + d*x] - 432*b^6*Sinh[4*c + d*x] - 144*a^6*S...
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(467\) vs.
\(2(214)=428\).
time = 3.47, size = 468, normalized size = 2.02 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/8/(a^3+3*a^2*b+3*a*b^2+b^3)/(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)+17*b*tanh(1/2*d*x+1/2*c))+2*b^3/(a+b)^4/a^3*(((-17/8*a^3-21/8*a^2*b-1/2*a*b^2)*tanh(1/2*d*x+1/2*c
)^7-1/8*(51*a^2+3*a*b-4*b^2)*a*tanh(1/2*d*x+1/2*c)^5-1/8*(51*a^2+3*a*b-4*b^2)*a*tanh(1/2*d*x+1/2*c)^3+(-17/8*a
^3-21/8*a^2*b-1/2*a*b^2)*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)^2+1/8*(63*a^2+36*a*b+8*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))))-1/24/(a+b)^3/tanh(1/2*d*x+1/2*c)^3-1/8*(5*a+17*b
)/(a+b)^4/tanh(1/2*d*x+1/2*c)+1/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1))
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 4920 vs.
\(2 (220) = 440\).
time = 1.11, size = 4920, normalized size = 21.21 \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)^3,x, algorithm="maxima")
[Out]
1/8*(3*a^3*b + 12*a^2*b^2 + 8*a*b^3 + 2*b^4)*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/((a^7 +
4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d) - 3/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)
/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 1/8*(3*a^3*b + 12*a^2*b^2 + 8*a*b^3 + 2*b^4)*log(2*(a + 2*b
)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d) + 3/4*b*log
(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) + 1/4*
(2*a + 5*b)*log(e^(2*d*x + 2*c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) + 3/2*b*log(e^(2*d*x + 2*
c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 1/4*(2*a + 5*b)*log(e^(-2*d*x - 2*c) - 1)/((a^4 + 4*
a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 3/2*b*log(e^(-2*d*x - 2*c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3
+ b^4)*d) - 1/256*(15*a^4*b + 260*a^3*b^2 + 504*a^2*b^3 + 288*a*b^4 + 64*b^5)*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^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b
^3 + a^3*b^4)*sqrt((a + b)*b)*d) + 5/64*(3*a*b + 10*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^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt((a + b)*
b)*d) + 1/256*(15*a^4*b + 260*a^3*b^2 + 504*a^2*b^3 + 288*a*b^4 + 64*b^5)*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^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3
+ a^3*b^4)*sqrt((a + b)*b)*d) - 5/64*(3*a*b + 10*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^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt((a + b)*
b)*d) + 15/128*(3*a*b - 4*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^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt((a + b)*b)*d) + 1/192*(176*a^6 +
275*a^5*b + 306*a^4*b^2 + 456*a^3*b^3 + 144*a^2*b^4 + 3*(96*a^6 + 111*a^5*b - 220*a^4*b^2 - 776*a^3*b^3 - 832
*a^2*b^4 - 256*a*b^5)*e^(12*d*x + 12*c) + 6*(120*a^6 + 528*a^5*b + 525*a^4*b^2 - 52*a^3*b^3 - 896*a^2*b^4 - 12
16*a*b^5 - 384*b^6)*e^(10*d*x + 10*c) + (176*a^6 + 1337*a^5*b + 7554*a^4*b^2 + 16416*a^3*b^3 + 26880*a^2*b^4 +
25344*a*b^5 + 6912*b^6)*e^(8*d*x + 8*c) - 4*(184*a^6 + 1056*a^5*b + 2993*a^4*b^2 + 4122*a^3*b^3 + 5892*a^2*b^
4 + 6144*a*b^5 + 1728*b^6)*e^(6*d*x + 6*c) - (384*a^6 + 1177*a^5*b - 736*a^4*b^2 + 112*a^3*b^3 - 624*a^2*b^4 -
6144*a*b^5 - 2304*b^6)*e^(4*d*x + 4*c) + 2*(136*a^6 + 912*a^5*b + 1211*a^4*b^2 + 1440*a^3*b^3 + 1896*a^2*b^4
+ 576*a*b^5)*e^(2*d*x + 2*c))/((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4 - (a^9 + 4*a^8*b + 6*a^7*b^2 +
4*a^6*b^3 + a^5*b^4)*e^(14*d*x + 14*c) - (a^9 + 12*a^8*b + 38*a^7*b^2 + 52*a^6*b^3 + 33*a^5*b^4 + 8*a^4*b^5)*
e^(12*d*x + 12*c) + (3*a^9 + 20*a^8*b + 34*a^7*b^2 - 4*a^6*b^3 - 61*a^5*b^4 - 56*a^4*b^5 - 16*a^3*b^6)*e^(10*d
*x + 10*c) + (3*a^9 + 28*a^8*b + 130*a^7*b^2 + 300*a^6*b^3 + 355*a^5*b^4 + 208*a^4*b^5 + 48*a^3*b^6)*e^(8*d*x
+ 8*c) - (3*a^9 + 28*a^8*b + 130*a^7*b^2 + 300*a^6*b^3 + 355*a^5*b^4 + 208*a^4*b^5 + 48*a^3*b^6)*e^(6*d*x + 6*
c) - (3*a^9 + 20*a^8*b + 34*a^7*b^2 - 4*a^6*b^3 - 61*a^5*b^4 - 56*a^4*b^5 - 16*a^3*b^6)*e^(4*d*x + 4*c) + (a^9
+ 12*a^8*b + 38*a^7*b^2 + 52*a^6*b^3 + 33*a^5*b^4 + 8*a^4*b^5)*e^(2*d*x + 2*c))*d) - 1/192*(176*a^6 + 275*a^5
*b + 306*a^4*b^2 + 456*a^3*b^3 + 144*a^2*b^4 + 2*(136*a^6 + 912*a^5*b + 1211*a^4*b^2 + 1440*a^3*b^3 + 1896*a^2
*b^4 + 576*a*b^5)*e^(-2*d*x - 2*c) - (384*a^6 + 1177*a^5*b - 736*a^4*b^2 + 112*a^3*b^3 - 624*a^2*b^4 - 6144*a*
b^5 - 2304*b^6)*e^(-4*d*x - 4*c) - 4*(184*a^6 + 1056*a^5*b + 2993*a^4*b^2 + 4122*a^3*b^3 + 5892*a^2*b^4 + 6144
*a*b^5 + 1728*b^6)*e^(-6*d*x - 6*c) + (176*a^6 + 1337*a^5*b + 7554*a^4*b^2 + 16416*a^3*b^3 + 26880*a^2*b^4 + 2
5344*a*b^5 + 6912*b^6)*e^(-8*d*x - 8*c) + 6*(120*a^6 + 528*a^5*b + 525*a^4*b^2 - 52*a^3*b^3 - 896*a^2*b^4 - 12
16*a*b^5 - 384*b^6)*e^(-10*d*x - 10*c) + 3*(96*a^6 + 111*a^5*b - 220*a^4*b^2 - 776*a^3*b^3 - 832*a^2*b^4 - 256
*a*b^5)*e^(-12*d*x - 12*c))/((a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4 + (a^9 + 12*a^8*b + 38*a^7*b^2 +
52*a^6*b^3 + 33*a^5*b^4 + 8*a^4*b^5)*e^(-2*d*x - 2*c) - (3*a^9 + 20*a^8*b + 34*a^7*b^2 - 4*a^6*b^3 - 61*a^5*b
^4 - 56*a^4*b^5 - 16*a^3*b^6)*e^(-4*d*x - 4*c) - (3*a^9 + 28*a^8*b + 130*a^7*b^2 + 300*a^6*b^3 + 355*a^5*b^4 +
208*a^4*b^5 + 48*a^3*b^6)*e^(-6*d*x - 6*c) + (3*a^9 + 28*a^8*b + 130*a^7*b^2 + 300*a^6*b^3 + 355*a^5*b^4 + 20
8*a^4*b^5 + 48*a^3*b^6)*e^(-8*d*x - 8*c) + (3*a^9 + 20*a^8*b + 34*a^7*b^2 - 4*a^6*b^3 - 61*a^5*b^4 - 56*a^4*b^
5 - 16*a^3*b^6)*e^(-10*d*x - 10*c) - (a^9 + 12*a^8*b + 38*a^7*b^2 + 52*a^6*b^3 + 33*a^5*b^4 + 8*a^4*b^5)*e^(-1
2*d*x - 12*c) - (a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*e^(-14*d*x - 14*c))*d) + 1/48*(32*a^5 + 77*a
^4*b - 72*a^3*b^2 - 12*a^2*b^3 + 3*(32*a^5 + 65*a^4*b + 94*a^3*b^2 + 128*a^2*b^3 + 32*a*b^4)*e^(12*d*x + 12*c)
+ 6*(48*a^5 + 200*a^4*b + 203*a^3*b^2 + 90*a^2...
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11993 vs.
\(2 (220) = 440\).
time = 0.64, size = 24263, normalized size = 104.58 \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)^3,x, algorithm="fricas")
[Out]
[1/48*(48*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^14 + 672*(a^6 + 4*a^5*b + 6*a^4*
b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)*sinh(d*x + c)^13 + 48*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^
2*b^4)*d*x*sinh(d*x + c)^14 - 12*(16*a^6 + 40*a^5*b - 17*a^3*b^3 - 44*a^2*b^4 - 16*a*b^5 - 4*(a^6 + 12*a^5*b +
38*a^4*b^2 + 52*a^3*b^3 + 33*a^2*b^4 + 8*a*b^5)*d*x)*cosh(d*x + c)^12 - 12*(16*a^6 + 40*a^5*b - 17*a^3*b^3 -
44*a^2*b^4 - 16*a*b^5 - 364*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^2 - 4*(a^6 + 1
2*a^5*b + 38*a^4*b^2 + 52*a^3*b^3 + 33*a^2*b^4 + 8*a*b^5)*d*x)*sinh(d*x + c)^12 + 48*(364*(a^6 + 4*a^5*b + 6*a
^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^3 - 3*(16*a^6 + 40*a^5*b - 17*a^3*b^3 - 44*a^2*b^4 - 16*a*b^5
- 4*(a^6 + 12*a^5*b + 38*a^4*b^2 + 52*a^3*b^3 + 33*a^2*b^4 + 8*a*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^11 - 2
4*(24*a^6 + 112*a^5*b + 160*a^4*b^2 - 11*a^2*b^4 - 68*a*b^5 - 24*b^6 + 2*(3*a^6 + 20*a^5*b + 34*a^4*b^2 - 4*a^
3*b^3 - 61*a^2*b^4 - 56*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^10 + 24*(2002*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^
3 + a^2*b^4)*d*x*cosh(d*x + c)^4 - 24*a^6 - 112*a^5*b - 160*a^4*b^2 + 11*a^2*b^4 + 68*a*b^5 + 24*b^6 - 2*(3*a^
6 + 20*a^5*b + 34*a^4*b^2 - 4*a^3*b^3 - 61*a^2*b^4 - 56*a*b^5 - 16*b^6)*d*x - 33*(16*a^6 + 40*a^5*b - 17*a^3*b
^3 - 44*a^2*b^4 - 16*a*b^5 - 4*(a^6 + 12*a^5*b + 38*a^4*b^2 + 52*a^3*b^3 + 33*a^2*b^4 + 8*a*b^5)*d*x)*cosh(d*x
+ c)^2)*sinh(d*x + c)^10 + 48*(2002*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^5 - 5
5*(16*a^6 + 40*a^5*b - 17*a^3*b^3 - 44*a^2*b^4 - 16*a*b^5 - 4*(a^6 + 12*a^5*b + 38*a^4*b^2 + 52*a^3*b^3 + 33*a
^2*b^4 + 8*a*b^5)*d*x)*cosh(d*x + c)^3 - 5*(24*a^6 + 112*a^5*b + 160*a^4*b^2 - 11*a^2*b^4 - 68*a*b^5 - 24*b^6
+ 2*(3*a^6 + 20*a^5*b + 34*a^4*b^2 - 4*a^3*b^3 - 61*a^2*b^4 - 56*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c))*sinh(d*x
+ c)^9 - 4*(128*a^6 + 440*a^5*b + 1152*a^4*b^2 + 2073*a^3*b^3 + 642*a^2*b^4 + 1416*a*b^5 + 432*b^6 + 12*(3*a^6
+ 28*a^5*b + 130*a^4*b^2 + 300*a^3*b^3 + 355*a^2*b^4 + 208*a*b^5 + 48*b^6)*d*x)*cosh(d*x + c)^8 + 4*(36036*(a
^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^6 - 128*a^6 - 440*a^5*b - 1152*a^4*b^2 - 207
3*a^3*b^3 - 642*a^2*b^4 - 1416*a*b^5 - 432*b^6 - 1485*(16*a^6 + 40*a^5*b - 17*a^3*b^3 - 44*a^2*b^4 - 16*a*b^5
- 4*(a^6 + 12*a^5*b + 38*a^4*b^2 + 52*a^3*b^3 + 33*a^2*b^4 + 8*a*b^5)*d*x)*cosh(d*x + c)^4 - 12*(3*a^6 + 28*a^
5*b + 130*a^4*b^2 + 300*a^3*b^3 + 355*a^2*b^4 + 208*a*b^5 + 48*b^6)*d*x - 270*(24*a^6 + 112*a^5*b + 160*a^4*b^
2 - 11*a^2*b^4 - 68*a*b^5 - 24*b^6 + 2*(3*a^6 + 20*a^5*b + 34*a^4*b^2 - 4*a^3*b^3 - 61*a^2*b^4 - 56*a*b^5 - 16
*b^6)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 32*(5148*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*c
osh(d*x + c)^7 - 297*(16*a^6 + 40*a^5*b - 17*a^3*b^3 - 44*a^2*b^4 - 16*a*b^5 - 4*(a^6 + 12*a^5*b + 38*a^4*b^2
+ 52*a^3*b^3 + 33*a^2*b^4 + 8*a*b^5)*d*x)*cosh(d*x + c)^5 - 90*(24*a^6 + 112*a^5*b + 160*a^4*b^2 - 11*a^2*b^4
- 68*a*b^5 - 24*b^6 + 2*(3*a^6 + 20*a^5*b + 34*a^4*b^2 - 4*a^3*b^3 - 61*a^2*b^4 - 56*a*b^5 - 16*b^6)*d*x)*cosh
(d*x + c)^3 - (128*a^6 + 440*a^5*b + 1152*a^4*b^2 + 2073*a^3*b^3 + 642*a^2*b^4 + 1416*a*b^5 + 432*b^6 + 12*(3*
a^6 + 28*a^5*b + 130*a^4*b^2 + 300*a^3*b^3 + 355*a^2*b^4 + 208*a*b^5 + 48*b^6)*d*x)*cosh(d*x + c))*sinh(d*x +
c)^7 - 16*(8*a^6 - 96*a^5*b - 512*a^4*b^2 - 768*a^3*b^3 - 57*a^2*b^4 - 330*a*b^5 - 108*b^6 - 3*(3*a^6 + 28*a^5
*b + 130*a^4*b^2 + 300*a^3*b^3 + 355*a^2*b^4 + 208*a*b^5 + 48*b^6)*d*x)*cosh(d*x + c)^6 + 16*(9009*(a^6 + 4*a^
5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^8 - 693*(16*a^6 + 40*a^5*b - 17*a^3*b^3 - 44*a^2*b^4
- 16*a*b^5 - 4*(a^6 + 12*a^5*b + 38*a^4*b^2 + 52*a^3*b^3 + 33*a^2*b^4 + 8*a*b^5)*d*x)*cosh(d*x + c)^6 - 8*a^6
+ 96*a^5*b + 512*a^4*b^2 + 768*a^3*b^3 + 57*a^2*b^4 + 330*a*b^5 + 108*b^6 - 315*(24*a^6 + 112*a^5*b + 160*a^4*
b^2 - 11*a^2*b^4 - 68*a*b^5 - 24*b^6 + 2*(3*a^6 + 20*a^5*b + 34*a^4*b^2 - 4*a^3*b^3 - 61*a^2*b^4 - 56*a*b^5 -
16*b^6)*d*x)*cosh(d*x + c)^4 + 3*(3*a^6 + 28*a^5*b + 130*a^4*b^2 + 300*a^3*b^3 + 355*a^2*b^4 + 208*a*b^5 + 48*
b^6)*d*x - 7*(128*a^6 + 440*a^5*b + 1152*a^4*b^2 + 2073*a^3*b^3 + 642*a^2*b^4 + 1416*a*b^5 + 432*b^6 + 12*(3*a
^6 + 28*a^5*b + 130*a^4*b^2 + 300*a^3*b^3 + 355*a^2*b^4 + 208*a*b^5 + 48*b^6)*d*x)*cosh(d*x + c)^2)*sinh(d*x +
c)^6 - 128*a^6 - 416*a^5*b - 204*a^3*b^3 - 72*a^2*b^4 + 32*(3003*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2
*b^4)*d*x*cosh(d*x + c)^9 - 297*(16*a^6 + 40*a^5*b - 17*a^3*b^3 - 44*a^2*b^4 - 16*a*b^5 - 4*(a^6 + 12*a^5*b +
38*a^4*b^2 + 52*a^3*b^3 + 33*a^2*b^4 + 8*a*b^5)*d*x)*cosh(d*x + c)^7 - 189*(24*a^6 + 112*a^5*b + 160*a^4*b^2 -
11*a^2*b^4 - 68*a*b^5 - 24*b^6 + 2*(3*a^6 + 20*a^5*b + 34*a^4*b^2 - 4*a^3*b^3 - 61*a^2*b^4 - 56*a*b^5 - 16*b^
6)*d*x)*cosh(d*x + c)^5 - 7*(128*a^6 + 440*a^5*b + 1152*a^4*b^2 + 2073*a^3*b^3 + 642*a^2*b^4 + 1416*a*b^5 + 43
2*b^6 + 12*(3*a^6 + 28*a^5*b + 130*a^4*b^2 + 300*a^3*b^3 + 355*a^2*b^4 + 208*a*b^5 + 48*b^6)*d*x)*cosh(d*x + c
)^3 - 3*(8*a^6 - 96*a^5*b - 512*a^4*b^2 - 768*a...
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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 )^{3}}\, 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)**3,x)
[Out]
Integral(coth(c + d*x)**4/(a + b*sech(c + d*x)**2)**3, x)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 469 vs.
\(2 (220) = 440\).
time = 2.51, size = 469, normalized size = 2.02 \begin {gather*} -\frac {\frac {3 \, {\left (63 \, a^{2} b^{3} + 36 \, a b^{4} + 8 \, b^{5}\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^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \sqrt {-a b - b^{2}}} - \frac {6 \, {\left (17 \, a^{3} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 44 \, a^{2} b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 51 \, a^{3} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 154 \, a^{2} b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 184 \, a b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{6} e^{\left (4 \, d x + 4 \, c\right )} + 51 \, a^{3} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 116 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 17 \, a^{3} b^{3} + 6 \, a^{2} b^{4}\right )}}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\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 )}^{2}} - \frac {24 \, {\left (d x + c\right )}}{a^{3}} + \frac {16 \, {\left (6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 15 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 24 \, b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a + 13 \, b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, 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)^3,x, algorithm="giac")
[Out]
-1/24*(3*(63*a^2*b^3 + 36*a*b^4 + 8*b^5)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^7 + 4*
a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*sqrt(-a*b - b^2)) - 6*(17*a^3*b^3*e^(6*d*x + 6*c) + 44*a^2*b^4*e^(6*d
*x + 6*c) + 16*a*b^5*e^(6*d*x + 6*c) + 51*a^3*b^3*e^(4*d*x + 4*c) + 154*a^2*b^4*e^(4*d*x + 4*c) + 184*a*b^5*e^
(4*d*x + 4*c) + 48*b^6*e^(4*d*x + 4*c) + 51*a^3*b^3*e^(2*d*x + 2*c) + 116*a^2*b^4*e^(2*d*x + 2*c) + 32*a*b^5*e
^(2*d*x + 2*c) + 17*a^3*b^3 + 6*a^2*b^4)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*(a*e^(4*d*x + 4*c)
+ 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2) - 24*(d*x + c)/a^3 + 16*(6*a*e^(4*d*x + 4*c) + 15*b*e^(4*
d*x + 4*c) - 6*a*e^(2*d*x + 2*c) - 24*b*e^(2*d*x + 2*c) + 4*a + 13*b)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 +
b^4)*(e^(2*d*x + 2*c) - 1)^3))/d
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {coth}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,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)^3,x)
[Out]
int((cosh(c + d*x)^6*coth(c + d*x)^4)/(b + a*cosh(c + d*x)^2)^3, x)
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