3.189 \(\int \frac {\coth ^4(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\)
Optimal. Leaf size=159 \[ \frac {b^{5/2} (7 a+5 b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} d (a+b)^2}-\frac {(2 a+5 b) \coth ^3(c+d x)}{6 a^2 d (a+b)}-\frac {\left (2 a^2-2 a b-5 b^2\right ) \coth (c+d x)}{2 a^3 d (a+b)}+\frac {b \coth ^3(c+d x)}{2 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac {x}{(a+b)^2} \]
[Out]
x/(a+b)^2+1/2*b^(5/2)*(7*a+5*b)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(7/2)/(a+b)^2/d-1/2*(2*a^2-2*a*b-5*b^2)*
coth(d*x+c)/a^3/(a+b)/d-1/6*(2*a+5*b)*coth(d*x+c)^3/a^2/(a+b)/d+1/2*b*coth(d*x+c)^3/a/(a+b)/d/(a+b*tanh(d*x+c)
^2)
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Rubi [A] time = 0.28, antiderivative size = 159, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used =
{3670, 472, 583, 522, 206, 205} \[ -\frac {\left (2 a^2-2 a b-5 b^2\right ) \coth (c+d x)}{2 a^3 d (a+b)}+\frac {b^{5/2} (7 a+5 b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} d (a+b)^2}-\frac {(2 a+5 b) \coth ^3(c+d x)}{6 a^2 d (a+b)}+\frac {b \coth ^3(c+d x)}{2 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac {x}{(a+b)^2} \]
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]
[Out]
x/(a + b)^2 + (b^(5/2)*(7*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*(a + b)^2*d) - ((2*a^2
- 2*a*b - 5*b^2)*Coth[c + d*x])/(2*a^3*(a + b)*d) - ((2*a + 5*b)*Coth[c + d*x]^3)/(6*a^2*(a + b)*d) + (b*Coth[
c + d*x]^3)/(2*a*(a + b)*d*(a + b*Tanh[c + d*x]^2))
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 472
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 522
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 583
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 3670
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))
Rubi steps
\begin {align*} \int \frac {\coth ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 \left (1-x^2\right ) \left (a+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 \tanh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-2 a-5 b+5 b x^2}{x^4 \left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 a (a+b) d}\\ &=-\frac {(2 a+5 b) \coth ^3(c+d x)}{6 a^2 (a+b) d}+\frac {b \coth ^3(c+d x)}{2 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {3 \left (2 a^2-2 a b-5 b^2\right )+3 b (2 a+5 b) x^2}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{6 a^2 (a+b) d}\\ &=-\frac {\left (2 a^2-2 a b-5 b^2\right ) \coth (c+d x)}{2 a^3 (a+b) d}-\frac {(2 a+5 b) \coth ^3(c+d x)}{6 a^2 (a+b) d}+\frac {b \coth ^3(c+d x)}{2 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-3 \left (2 a^3-2 a^2 b+2 a b^2+5 b^3\right )-3 b \left (2 a^2-2 a b-5 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{6 a^3 (a+b) d}\\ &=-\frac {\left (2 a^2-2 a b-5 b^2\right ) \coth (c+d x)}{2 a^3 (a+b) d}-\frac {(2 a+5 b) \coth ^3(c+d x)}{6 a^2 (a+b) d}+\frac {b \coth ^3(c+d x)}{2 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^2 d}+\frac {\left (b^3 (7 a+5 b)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^3 (a+b)^2 d}\\ &=\frac {x}{(a+b)^2}+\frac {b^{5/2} (7 a+5 b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{7/2} (a+b)^2 d}-\frac {\left (2 a^2-2 a b-5 b^2\right ) \coth (c+d x)}{2 a^3 (a+b) d}-\frac {(2 a+5 b) \coth ^3(c+d x)}{6 a^2 (a+b) d}+\frac {b \coth ^3(c+d x)}{2 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.56, size = 139, normalized size = 0.87 \[ \frac {\frac {3 b^{5/2} (7 a+5 b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} (a+b)^2}+\frac {3 b^3 \sinh (2 (c+d x))}{a^3 (a+b) ((a+b) \cosh (2 (c+d x))+a-b)}+\frac {4 (3 b-2 a) \coth (c+d x)}{a^3}-\frac {2 \coth (c+d x) \text {csch}^2(c+d x)}{a^2}+\frac {6 (c+d x)}{(a+b)^2}}{6 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]
[Out]
((6*(c + d*x))/(a + b)^2 + (3*b^(5/2)*(7*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(7/2)*(a + b)^2)
+ (4*(-2*a + 3*b)*Coth[c + d*x])/a^3 - (2*Coth[c + d*x]*Csch[c + d*x]^2)/a^2 + (3*b^3*Sinh[2*(c + d*x)])/(a^3
*(a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(6*d)
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fricas [B] time = 0.59, size = 8482, normalized size = 53.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[1/12*(12*(a^4 + a^3*b)*d*x*cosh(d*x + c)^10 + 120*(a^4 + a^3*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^9 + 12*(a^4 +
a^3*b)*d*x*sinh(d*x + c)^10 - 12*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^8 +
12*(45*(a^4 + a^3*b)*d*x*cosh(d*x + c)^2 - 4*a^4 - 8*a^3*b + 7*a*b^3 + 5*b^4 - (a^4 + 5*a^3*b)*d*x)*sinh(d*x +
c)^8 + 96*(15*(a^4 + a^3*b)*d*x*cosh(d*x + c)^3 - (4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*c
osh(d*x + c))*sinh(d*x + c)^7 - 24*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b)*d*x)*cosh
(d*x + c)^6 + 24*(105*(a^4 + a^3*b)*d*x*cosh(d*x + c)^4 - 2*a^4 + 2*a^3*b + 2*a^2*b^2 - 9*a*b^3 - 10*b^4 - (a^
4 - 5*a^3*b)*d*x - 14*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)
^6 + 48*(63*(a^4 + a^3*b)*d*x*cosh(d*x + c)^5 - 14*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*c
osh(d*x + c)^3 - 3*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b)*d*x)*cosh(d*x + c))*sinh(
d*x + c)^5 + 8*(2*a^4 - 30*a^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^4 + 8
*(315*(a^4 + a^3*b)*d*x*cosh(d*x + c)^6 - 105*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d
*x + c)^4 + 2*a^4 - 30*a^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x - 45*(2*a^4 - 2*a^3*b -
2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 32*a^4 - 48*a^3*b + 48*
a^2*b^2 + 124*a*b^3 + 60*b^4 + 32*(45*(a^4 + a^3*b)*d*x*cosh(d*x + c)^7 - 21*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^
4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^5 - 15*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b
)*d*x)*cosh(d*x + c)^3 + (2*a^4 - 30*a^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x)*cosh(d*x
+ c))*sinh(d*x + c)^3 - 12*(a^4 + a^3*b)*d*x - 4*(4*a^4 - 20*a^3*b - 4*a^2*b^2 + 74*a*b^3 + 60*b^4 - 3*(a^4 +
5*a^3*b)*d*x)*cosh(d*x + c)^2 + 4*(135*(a^4 + a^3*b)*d*x*cosh(d*x + c)^8 - 84*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b
^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^6 - 90*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*
b)*d*x)*cosh(d*x + c)^4 - 4*a^4 + 20*a^3*b + 4*a^2*b^2 - 74*a*b^3 - 60*b^4 + 3*(a^4 + 5*a^3*b)*d*x + 12*(2*a^4
- 30*a^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((7
*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^10 + 10*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)*sinh(d*x + c)^
9 + (7*a^2*b^2 + 12*a*b^3 + 5*b^4)*sinh(d*x + c)^10 - (7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^8 - (7*a^2
*b^2 + 40*a*b^3 + 25*b^4 - 45*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(7*a^2*b
^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^3 - (7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(7
*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^6 + 2*(105*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^4 - 7*a^2*
b^2 + 30*a*b^3 + 25*b^4 - 14*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(7*a^2*b
^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^5 - 14*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^3 - 3*(7*a^2*b^2 - 3
0*a*b^3 - 25*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^4 + 2*(105*
(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^6 - 35*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^4 + 7*a^2*b^
2 - 30*a*b^3 - 25*b^4 - 15*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 7*a^2*b^2 - 12*a
*b^3 - 5*b^4 + 8*(15*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^7 - 7*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d
*x + c)^5 - 5*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^3 + (7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c))
*sinh(d*x + c)^3 + (7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^2 + (45*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d
*x + c)^8 - 28*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^6 - 30*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x +
c)^4 + 7*a^2*b^2 + 40*a*b^3 + 25*b^4 + 12*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
2*(5*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^9 - 4*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^7 - 6*(7
*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^5 + 4*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^3 + (7*a^2*b^2
+ 40*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log(((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a
^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x
+ c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2
+ 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*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 - a*b)*sqrt(-b/a))/((a + b)*cosh(
d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a - b)*cosh(d*x + c)^2 + 2
*(3*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sin
h(d*x + c) + a + b)) + 8*(15*(a^4 + a^3*b)*d*x*cosh(d*x + c)^9 - 12*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4
+ 5*a^3*b)*d*x)*cosh(d*x + c)^7 - 18*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b)*d*x)*co
sh(d*x + c)^5 + 4*(2*a^4 - 30*a^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^3
- (4*a^4 - 20*a^3*b - 4*a^2*b^2 + 74*a*b^3 + 60*b^4 - 3*(a^4 + 5*a^3*b)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a
^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^10 + 10*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x +
c)*sinh(d*x + c)^9 + (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*sinh(d*x + c)^10 - (a^6 + 7*a^5*b + 11*a^4*b^2 +
5*a^3*b^3)*d*cosh(d*x + c)^8 + (45*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^2 - (a^6 + 7*a^5*b +
11*a^4*b^2 + 5*a^3*b^3)*d)*sinh(d*x + c)^8 - 2*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^6 + 8*
(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^3 - (a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(
d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^4 - 14*(a^6 + 7*a^5*b
+ 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^2 - (a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d)*sinh(d*x + c)^6 + 2*
(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^4 + 4*(63*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh
(d*x + c)^5 - 14*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^3 - 3*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5
*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^6 -
35*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^4 - 15*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*c
osh(d*x + c)^2 + (a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d)*sinh(d*x + c)^4 + (a^6 + 7*a^5*b + 11*a^4*b^2 + 5*
a^3*b^3)*d*cosh(d*x + c)^2 + 8*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^7 - 7*(a^6 + 7*a^5*b
+ 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^5 - 5*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^3 + (a
^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*
b^3)*d*cosh(d*x + c)^8 - 28*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^6 - 30*(a^6 - 3*a^5*b - 9
*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^4 + 12*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^2 + (a^6
+ 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d)*sinh(d*x + c)^2 - (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d + 2*(5*(a^6 +
3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^9 - 4*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)
^7 - 6*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^5 + 4*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d
*cosh(d*x + c)^3 + (a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)), 1/6*(6*(a^4 + a^3
*b)*d*x*cosh(d*x + c)^10 + 60*(a^4 + a^3*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^9 + 6*(a^4 + a^3*b)*d*x*sinh(d*x +
c)^10 - 6*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^8 + 6*(45*(a^4 + a^3*b)*d*x
*cosh(d*x + c)^2 - 4*a^4 - 8*a^3*b + 7*a*b^3 + 5*b^4 - (a^4 + 5*a^3*b)*d*x)*sinh(d*x + c)^8 + 48*(15*(a^4 + a^
3*b)*d*x*cosh(d*x + c)^3 - (4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c))*sinh(d*x +
c)^7 - 12*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^6 + 12*(105*(a
^4 + a^3*b)*d*x*cosh(d*x + c)^4 - 2*a^4 + 2*a^3*b + 2*a^2*b^2 - 9*a*b^3 - 10*b^4 - (a^4 - 5*a^3*b)*d*x - 14*(4
*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(63*(a^4 + a^3*b
)*d*x*cosh(d*x + c)^5 - 14*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^3 - 3*(2*a^
4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(2*a^4 -
30*a^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^4 + 4*(315*(a^4 + a^3*b)*d*x*
cosh(d*x + c)^6 - 105*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^4 + 2*a^4 - 30*a
^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x - 45*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10
*b^4 + (a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 16*a^4 - 24*a^3*b + 24*a^2*b^2 + 62*a*b^3 + 30*
b^4 + 16*(45*(a^4 + a^3*b)*d*x*cosh(d*x + c)^7 - 21*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*
cosh(d*x + c)^5 - 15*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^3 +
(2*a^4 - 30*a^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 6
*(a^4 + a^3*b)*d*x - 2*(4*a^4 - 20*a^3*b - 4*a^2*b^2 + 74*a*b^3 + 60*b^4 - 3*(a^4 + 5*a^3*b)*d*x)*cosh(d*x + c
)^2 + 2*(135*(a^4 + a^3*b)*d*x*cosh(d*x + c)^8 - 84*(4*a^4 + 8*a^3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*
cosh(d*x + c)^6 - 90*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*b^4 + (a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^4 -
4*a^4 + 20*a^3*b + 4*a^2*b^2 - 74*a*b^3 - 60*b^4 + 3*(a^4 + 5*a^3*b)*d*x + 12*(2*a^4 - 30*a^3*b - 30*a^2*b^2 +
38*a*b^3 + 45*b^4 + 3*(a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((7*a^2*b^2 + 12*a*b^3 + 5*b^
4)*cosh(d*x + c)^10 + 10*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)*sinh(d*x + c)^9 + (7*a^2*b^2 + 12*a*b^3
+ 5*b^4)*sinh(d*x + c)^10 - (7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^8 - (7*a^2*b^2 + 40*a*b^3 + 25*b^4 -
45*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cos
h(d*x + c)^3 - (7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(7*a^2*b^2 - 30*a*b^3 - 25*b
^4)*cosh(d*x + c)^6 + 2*(105*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^4 - 7*a^2*b^2 + 30*a*b^3 + 25*b^4 -
14*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cos
h(d*x + c)^5 - 14*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^3 - 3*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x
+ c))*sinh(d*x + c)^5 + 2*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^4 + 2*(105*(7*a^2*b^2 + 12*a*b^3 + 5*
b^4)*cosh(d*x + c)^6 - 35*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^4 + 7*a^2*b^2 - 30*a*b^3 - 25*b^4 - 15
*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 7*a^2*b^2 - 12*a*b^3 - 5*b^4 + 8*(15*(7*a^
2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^7 - 7*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^5 - 5*(7*a^2*b^2 -
30*a*b^3 - 25*b^4)*cosh(d*x + c)^3 + (7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + (7*a^2*
b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^2 + (45*(7*a^2*b^2 + 12*a*b^3 + 5*b^4)*cosh(d*x + c)^8 - 28*(7*a^2*b^2
+ 40*a*b^3 + 25*b^4)*cosh(d*x + c)^6 - 30*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^4 + 7*a^2*b^2 + 40*a*b
^3 + 25*b^4 + 12*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(7*a^2*b^2 + 12*a*b^3
+ 5*b^4)*cosh(d*x + c)^9 - 4*(7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh(d*x + c)^7 - 6*(7*a^2*b^2 - 30*a*b^3 - 25*b
^4)*cosh(d*x + c)^5 + 4*(7*a^2*b^2 - 30*a*b^3 - 25*b^4)*cosh(d*x + c)^3 + (7*a^2*b^2 + 40*a*b^3 + 25*b^4)*cosh
(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c
) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(b/a)/b) + 4*(15*(a^4 + a^3*b)*d*x*cosh(d*x + c)^9 - 12*(4*a^4 + 8*a^
3*b - 7*a*b^3 - 5*b^4 + (a^4 + 5*a^3*b)*d*x)*cosh(d*x + c)^7 - 18*(2*a^4 - 2*a^3*b - 2*a^2*b^2 + 9*a*b^3 + 10*
b^4 + (a^4 - 5*a^3*b)*d*x)*cosh(d*x + c)^5 + 4*(2*a^4 - 30*a^3*b - 30*a^2*b^2 + 38*a*b^3 + 45*b^4 + 3*(a^4 - 5
*a^3*b)*d*x)*cosh(d*x + c)^3 - (4*a^4 - 20*a^3*b - 4*a^2*b^2 + 74*a*b^3 + 60*b^4 - 3*(a^4 + 5*a^3*b)*d*x)*cosh
(d*x + c))*sinh(d*x + c))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^10 + 10*(a^6 + 3*a^5*b + 3*a^
4*b^2 + a^3*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*sinh(d*x + c)^10 -
(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^8 + (45*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(
d*x + c)^2 - (a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d)*sinh(d*x + c)^8 - 2*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^
3*b^3)*d*cosh(d*x + c)^6 + 8*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^3 - (a^6 + 7*a^5*b + 11
*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d
*x + c)^4 - 14*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^2 - (a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3
*b^3)*d)*sinh(d*x + c)^6 + 2*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^4 + 4*(63*(a^6 + 3*a^5*b
+ 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^5 - 14*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^3 - 3*(
a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a^6 + 3*a^5*b + 3*a^4*b^2 +
a^3*b^3)*d*cosh(d*x + c)^6 - 35*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^4 - 15*(a^6 - 3*a^5*b
- 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^2 + (a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d)*sinh(d*x + c)^4 + (a^
6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^2 + 8*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d
*x + c)^7 - 7*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c)^5 - 5*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^
3*b^3)*d*cosh(d*x + c)^3 + (a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^6
+ 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^8 - 28*(a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x +
c)^6 - 30*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^4 + 12*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b
^3)*d*cosh(d*x + c)^2 + (a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d)*sinh(d*x + c)^2 - (a^6 + 3*a^5*b + 3*a^4*b
^2 + a^3*b^3)*d + 2*(5*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*cosh(d*x + c)^9 - 4*(a^6 + 7*a^5*b + 11*a^4*b^2
+ 5*a^3*b^3)*d*cosh(d*x + c)^7 - 6*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^5 + 4*(a^6 - 3*a^5
*b - 9*a^4*b^2 - 5*a^3*b^3)*d*cosh(d*x + c)^3 + (a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*d*cosh(d*x + c))*sinh
(d*x + c))]
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giac [A] time = 0.44, size = 281, normalized size = 1.77 \[ \frac {\frac {3 \, {\left (7 \, a b^{3} + 5 \, b^{4}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {a b}} + \frac {6 \, {\left (d x + c\right )}}{a^{2} + 2 \, a b + b^{2}} - \frac {6 \, {\left (a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - b^{4} e^{\left (2 \, d x + 2 \, c\right )} + a b^{3} + b^{4}\right )}}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}} - \frac {8 \, {\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - 3 \, b\right )}}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
[Out]
1/6*(3*(7*a*b^3 + 5*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^5 + 2*a^4*b
+ a^3*b^2)*sqrt(a*b)) + 6*(d*x + c)/(a^2 + 2*a*b + b^2) - 6*(a*b^3*e^(2*d*x + 2*c) - b^4*e^(2*d*x + 2*c) + a*
b^3 + b^4)/((a^5 + 2*a^4*b + a^3*b^2)*(a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*
d*x + 2*c) + a + b)) - 8*(3*a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) - 3*a*e^(2*d*x + 2*c) + 6*b*e^(2*d*x + 2*c
) + 2*a - 3*b)/(a^3*(e^(2*d*x + 2*c) - 1)^3))/d
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maple [B] time = 0.47, size = 1137, normalized size = 7.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x)
[Out]
-1/24/d/a^2*tanh(1/2*d*x+1/2*c)^3-5/8/d/a^2*tanh(1/2*d*x+1/2*c)+1/d/a^3*tanh(1/2*d*x+1/2*c)*b-1/d/(a+b)^2*ln(t
anh(1/2*d*x+1/2*c)-1)+1/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c)+1)+1/d*b^3/(a+b)^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tan
h(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)^3+1/d*b^4/(a+b)^2/a^3/(tanh(1/2*d*x+1/2*
c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)^3+1/d*b^3/(a+b)^2/a^2/(tanh(
1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)+1/d*b^4/(a+b)^2/
a^3/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)-7/2/d*
b^3/(a+b)^2/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^
(1/2)-a-2*b)*a)^(1/2))+7/2/d*b^3/(a+b)^2/a^2/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)
/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))-6/d*b^4/(a+b)^2/a^2/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*
arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))-7/2/d*b^3/(a+b)^2/a/(b*(a+b))^(1/2)/((2*(b*
(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-7/2/d*b^3/(a+b)
^2/a^2/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-6
/d*b^4/(a+b)^2/a^2/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+
b))^(1/2)+a+2*b)*a)^(1/2))+5/2/d*b^4/(a+b)^2/a^3/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/
2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))-5/2/d*b^5/(a+b)^2/a^3/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^
(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))-5/2/d*b^4/(a+b)^2/a^3/((2*(b*(a+b))^(
1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-5/2/d*b^5/(a+b)^2/a^3/(
b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^
(1/2))-1/24/d/a^2/tanh(1/2*d*x+1/2*c)^3-5/8/d/a^2/tanh(1/2*d*x+1/2*c)+1/d/a^3/tanh(1/2*d*x+1/2*c)*b
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maxima [B] time = 1.08, size = 2345, normalized size = 14.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
-1/4*(a^2*b - a*b^2 - b^3)*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a^5 + 2*a^4*b +
a^3*b^2)*d) + 1/4*(a^2*b - a*b^2 - b^3)*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a
^5 + 2*a^4*b + a^3*b^2)*d) + 1/32*(3*a^3*b - 29*a^2*b^2 - 11*a*b^3 + 5*b^4)*arctan(1/2*((a + b)*e^(2*d*x + 2*c
) + a - b)/sqrt(a*b))/((a^5 + 2*a^4*b + a^3*b^2)*sqrt(a*b)*d) - 1/32*(3*a^3*b - 29*a^2*b^2 - 11*a*b^3 + 5*b^4)
*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^5 + 2*a^4*b + a^3*b^2)*sqrt(a*b)*d) + 1/48*(44*a
^4 + 117*a^3*b + 111*a^2*b^2 + 23*a*b^3 - 15*b^4 + 3*(24*a^4 + 69*a^3*b + 45*a^2*b^2 + 27*a*b^3 - 5*b^4)*e^(8*
d*x + 8*c) + 6*(6*a^4 - 31*a^3*b - 50*a^2*b^2 - 51*a*b^3 + 10*b^4)*e^(6*d*x + 6*c) - 2*(50*a^4 - 78*a^3*b - 22
5*a^2*b^2 - 196*a*b^3 + 45*b^4)*e^(4*d*x + 4*c) - 2*(10*a^4 + 115*a^3*b + 182*a^2*b^2 + 95*a*b^3 - 30*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 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*e^(8*d*x + 8*c) + 2*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*e^(6*d*
x + 6*c) - 2*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*e^(4*d*x + 4*c) - (a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3
)*e^(2*d*x + 2*c))*d) - 1/48*(44*a^4 + 117*a^3*b + 111*a^2*b^2 + 23*a*b^3 - 15*b^4 - 2*(10*a^4 + 115*a^3*b + 1
82*a^2*b^2 + 95*a*b^3 - 30*b^4)*e^(-2*d*x - 2*c) - 2*(50*a^4 - 78*a^3*b - 225*a^2*b^2 - 196*a*b^3 + 45*b^4)*e^
(-4*d*x - 4*c) + 6*(6*a^4 - 31*a^3*b - 50*a^2*b^2 - 51*a*b^3 + 10*b^4)*e^(-6*d*x - 6*c) + 3*(24*a^4 + 69*a^3*b
+ 45*a^2*b^2 + 27*a*b^3 - 5*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 + 7*a^5*b + 1
1*a^4*b^2 + 5*a^3*b^3)*e^(-2*d*x - 2*c) - 2*(a^6 - 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*e^(-4*d*x - 4*c) + 2*(a^6
- 3*a^5*b - 9*a^4*b^2 - 5*a^3*b^3)*e^(-6*d*x - 6*c) + (a^6 + 7*a^5*b + 11*a^4*b^2 + 5*a^3*b^3)*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 + 7*a^2*b - 16*a*b^2 - 15*b^3 +
3*(8*a^3 + 11*a^2*b + 6*a*b^2 - 5*b^3)*e^(8*d*x + 8*c) + 6*(4*a^3 - 7*a^2*b - 13*a*b^2 + 10*b^3)*e^(6*d*x + 6
*c) - 2*(8*a^3 - 44*a^2*b - 43*a*b^2 + 45*b^3)*e^(4*d*x + 4*c) - 2*(4*a^3 + 27*a^2*b + 5*a*b^2 - 30*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^(10*d*x + 10*c) + (a^5 + 6*a^4*b + 5*a^3*b
^2)*e^(8*d*x + 8*c) + 2*(a^5 - 4*a^4*b - 5*a^3*b^2)*e^(6*d*x + 6*c) - 2*(a^5 - 4*a^4*b - 5*a^3*b^2)*e^(4*d*x +
4*c) - (a^5 + 6*a^4*b + 5*a^3*b^2)*e^(2*d*x + 2*c))*d) - 1/12*(8*a^3 + 7*a^2*b - 16*a*b^2 - 15*b^3 - 2*(4*a^3
+ 27*a^2*b + 5*a*b^2 - 30*b^3)*e^(-2*d*x - 2*c) - 2*(8*a^3 - 44*a^2*b - 43*a*b^2 + 45*b^3)*e^(-4*d*x - 4*c) +
6*(4*a^3 - 7*a^2*b - 13*a*b^2 + 10*b^3)*e^(-6*d*x - 6*c) + 3*(8*a^3 + 11*a^2*b + 6*a*b^2 - 5*b^3)*e^(-8*d*x -
8*c))/((a^5 + 2*a^4*b + a^3*b^2 - (a^5 + 6*a^4*b + 5*a^3*b^2)*e^(-2*d*x - 2*c) - 2*(a^5 - 4*a^4*b - 5*a^3*b^2
)*e^(-4*d*x - 4*c) + 2*(a^5 - 4*a^4*b - 5*a^3*b^2)*e^(-6*d*x - 6*c) + (a^5 + 6*a^4*b + 5*a^3*b^2)*e^(-8*d*x -
8*c) - (a^5 + 2*a^4*b + a^3*b^2)*e^(-10*d*x - 10*c))*d) + 1/8*(4*a^2 + 19*a*b + 15*b^2 - 2*(2*a^2 + 13*a*b + 3
0*b^2)*e^(-2*d*x - 2*c) - 2*(10*a^2 - 2*a*b - 45*b^2)*e^(-4*d*x - 4*c) - 6*(2*a^2 + a*b + 10*b^2)*e^(-6*d*x -
6*c) + 3*(3*a*b + 5*b^2)*e^(-8*d*x - 8*c))/((a^4 + a^3*b - (a^4 + 5*a^3*b)*e^(-2*d*x - 2*c) - 2*(a^4 - 5*a^3*b
)*e^(-4*d*x - 4*c) + 2*(a^4 - 5*a^3*b)*e^(-6*d*x - 6*c) + (a^4 + 5*a^3*b)*e^(-8*d*x - 8*c) - (a^4 + a^3*b)*e^(
-10*d*x - 10*c))*d) + 1/2*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/(a^3*d) - 1/2*b*l
og(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/(a^3*d) + 1/2*(a - b)*log(e^(2*d*x + 2*c) -
1)/(a^3*d) - b*log(e^(2*d*x + 2*c) - 1)/(a^3*d) - 1/2*(a - b)*log(e^(-2*d*x - 2*c) - 1)/(a^3*d) + b*log(e^(-2*
d*x - 2*c) - 1)/(a^3*d) - 1/8*(3*a*b - 5*b^2)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(sqrt(a*
b)*a^3*d) - 3/16*(3*a*b + 5*b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^3*d) +
1/8*(3*a*b - 5*b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^3*d)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(c + d*x)^4/(a + b*tanh(c + d*x)^2)^2,x)
[Out]
int(coth(c + d*x)^4/(a + b*tanh(c + d*x)^2)^2, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{4}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)**4/(a+b*tanh(d*x+c)**2)**2,x)
[Out]
Integral(coth(c + d*x)**4/(a + b*tanh(c + d*x)**2)**2, x)
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