3.8 \(\int \frac {1}{(a+b \coth ^2(c+d x))^4} \, dx\)

Optimal. Leaf size=201 \[ \frac {b (11 a+5 b) \coth (c+d x)}{24 a^2 d (a+b)^2 \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \coth (c+d x)}{16 a^3 d (a+b)^3 \left (a+b \coth ^2(c+d x)\right )}-\frac {\sqrt {b} \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{16 a^{7/2} d (a+b)^4}+\frac {b \coth (c+d x)}{6 a d (a+b) \left (a+b \coth ^2(c+d x)\right )^3}+\frac {x}{(a+b)^4} \]

[Out]

x/(a+b)^4+1/6*b*coth(d*x+c)/a/(a+b)/d/(a+b*coth(d*x+c)^2)^3+1/24*b*(11*a+5*b)*coth(d*x+c)/a^2/(a+b)^2/d/(a+b*c
oth(d*x+c)^2)^2+1/16*b*(19*a^2+16*a*b+5*b^2)*coth(d*x+c)/a^3/(a+b)^3/d/(a+b*coth(d*x+c)^2)-1/16*(35*a^3+35*a^2
*b+21*a*b^2+5*b^3)*arctan(a^(1/2)*tanh(d*x+c)/b^(1/2))*b^(1/2)/a^(7/2)/(a+b)^4/d

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Rubi [A]  time = 0.28, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3661, 414, 527, 522, 206, 205} \[ \frac {b \left (19 a^2+16 a b+5 b^2\right ) \coth (c+d x)}{16 a^3 d (a+b)^3 \left (a+b \coth ^2(c+d x)\right )}-\frac {\sqrt {b} \left (35 a^2 b+35 a^3+21 a b^2+5 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{16 a^{7/2} d (a+b)^4}+\frac {b (11 a+5 b) \coth (c+d x)}{24 a^2 d (a+b)^2 \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b \coth (c+d x)}{6 a d (a+b) \left (a+b \coth ^2(c+d x)\right )^3}+\frac {x}{(a+b)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[c + d*x]^2)^(-4),x]

[Out]

x/(a + b)^4 - (Sqrt[b]*(35*a^3 + 35*a^2*b + 21*a*b^2 + 5*b^3)*ArcTan[(Sqrt[a]*Tanh[c + d*x])/Sqrt[b]])/(16*a^(
7/2)*(a + b)^4*d) + (b*Coth[c + d*x])/(6*a*(a + b)*d*(a + b*Coth[c + d*x]^2)^3) + (b*(11*a + 5*b)*Coth[c + d*x
])/(24*a^2*(a + b)^2*d*(a + b*Coth[c + d*x]^2)^2) + (b*(19*a^2 + 16*a*b + 5*b^2)*Coth[c + d*x])/(16*a^3*(a + b
)^3*d*(a + b*Coth[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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, 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 527

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

Rule 3661

Int[((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[(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, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \coth ^2(c+d x)\right )^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a+b x^2\right )^4} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {b \coth (c+d x)}{6 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^3}-\frac {\operatorname {Subst}\left (\int \frac {b-6 (a+b)+5 b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\coth (c+d x)\right )}{6 a (a+b) d}\\ &=\frac {b \coth (c+d x)}{6 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \coth (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \coth ^2(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {3 \left (8 a^2+11 a b+5 b^2\right )-3 b (11 a+5 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\coth (c+d x)\right )}{24 a^2 (a+b)^2 d}\\ &=\frac {b \coth (c+d x)}{6 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \coth (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \coth (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \coth ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-3 \left (16 a^3+19 a^2 b+16 a b^2+5 b^3\right )+3 b \left (19 a^2+16 a b+5 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\coth (c+d x)\right )}{48 a^3 (a+b)^3 d}\\ &=\frac {b \coth (c+d x)}{6 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \coth (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \coth (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \coth ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{(a+b)^4 d}+\frac {\left (b \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\coth (c+d x)\right )}{16 a^3 (a+b)^4 d}\\ &=\frac {x}{(a+b)^4}-\frac {\sqrt {b} \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{16 a^{7/2} (a+b)^4 d}+\frac {b \coth (c+d x)}{6 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^3}+\frac {b (11 a+5 b) \coth (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b \left (19 a^2+16 a b+5 b^2\right ) \coth (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \coth ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 203, normalized size = 1.01 \[ \frac {\frac {2 b (11 a+5 b) (a+b)^2 \coth (c+d x)}{a^2 \left (a+b \coth ^2(c+d x)\right )^2}+\frac {3 b \left (19 a^2+16 a b+5 b^2\right ) (a+b) \coth (c+d x)}{a^3 \left (a+b \coth ^2(c+d x)\right )}+\frac {3 \sqrt {b} \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a}}\right )}{a^{7/2}}+\frac {8 b (a+b)^3 \coth (c+d x)}{a \left (a+b \coth ^2(c+d x)\right )^3}-24 \log (1-\coth (c+d x))+24 \log (\coth (c+d x)+1)}{48 d (a+b)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*Sqrt[b]*(35*a^3 + 35*a^2*b + 21*a*b^2 + 5*b^3)*ArcTan[(Sqrt[b]*Coth[c + d*x])/Sqrt[a]])/a^(7/2) + (8*b*(a
+ b)^3*Coth[c + d*x])/(a*(a + b*Coth[c + d*x]^2)^3) + (2*b*(a + b)^2*(11*a + 5*b)*Coth[c + d*x])/(a^2*(a + b*C
oth[c + d*x]^2)^2) + (3*b*(a + b)*(19*a^2 + 16*a*b + 5*b^2)*Coth[c + d*x])/(a^3*(a + b*Coth[c + d*x]^2)) - 24*
Log[1 - Coth[c + d*x]] + 24*Log[1 + Coth[c + d*x]])/(48*(a + b)^4*d)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c)^2)^4,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.50, size = 1356, normalized size = 6.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c)^2)^4,x, algorithm="giac")

[Out]

-1/48*(3*(35*a^3*b + 35*a^2*b^2 + 21*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^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*sqrt(a*b)) - 48*(d*x + c)/(a^4 + 4*a^3*b + 6*a^
2*b^2 + 4*a*b^3 + b^4) - 2*(87*a^11*b*e^(10*d*x + 10*c) + 591*a^10*b^2*e^(10*d*x + 10*c) + 1533*a^9*b^3*e^(10*
d*x + 10*c) + 1413*a^8*b^4*e^(10*d*x + 10*c) - 1674*a^7*b^5*e^(10*d*x + 10*c) - 6426*a^6*b^6*e^(10*d*x + 10*c)
- 8694*a^5*b^7*e^(10*d*x + 10*c) - 6822*a^4*b^8*e^(10*d*x + 10*c) - 3357*a^3*b^9*e^(10*d*x + 10*c) - 1029*a^2
*b^10*e^(10*d*x + 10*c) - 183*a*b^11*e^(10*d*x + 10*c) - 15*b^12*e^(10*d*x + 10*c) - 435*a^11*b*e^(8*d*x + 8*c
) - 2661*a^10*b^2*e^(8*d*x + 8*c) - 6657*a^9*b^3*e^(8*d*x + 8*c) - 8871*a^8*b^4*e^(8*d*x + 8*c) - 7950*a^7*b^5
*e^(8*d*x + 8*c) - 8610*a^6*b^6*e^(8*d*x + 8*c) - 12306*a^5*b^7*e^(8*d*x + 8*c) - 13182*a^4*b^8*e^(8*d*x + 8*c
) - 8751*a^3*b^9*e^(8*d*x + 8*c) - 3465*a^2*b^10*e^(8*d*x + 8*c) - 765*a*b^11*e^(8*d*x + 8*c) - 75*b^12*e^(8*d
*x + 8*c) + 870*a^11*b*e^(6*d*x + 6*c) + 5278*a^10*b^2*e^(6*d*x + 6*c) + 13722*a^9*b^3*e^(6*d*x + 6*c) + 19602
*a^8*b^4*e^(6*d*x + 6*c) + 14908*a^7*b^5*e^(6*d*x + 6*c) + 300*a^6*b^6*e^(6*d*x + 6*c) - 14412*a^5*b^7*e^(6*d*
x + 6*c) - 19228*a^4*b^8*e^(6*d*x + 6*c) - 13698*a^3*b^9*e^(6*d*x + 6*c) - 5802*a^2*b^10*e^(6*d*x + 6*c) - 139
0*a*b^11*e^(6*d*x + 6*c) - 150*b^12*e^(6*d*x + 6*c) - 870*a^11*b*e^(4*d*x + 4*c) - 5778*a^10*b^2*e^(4*d*x + 4*
c) - 16362*a^9*b^3*e^(4*d*x + 4*c) - 26190*a^8*b^4*e^(4*d*x + 4*c) - 27996*a^7*b^5*e^(4*d*x + 4*c) - 25620*a^6
*b^6*e^(4*d*x + 4*c) - 24948*a^5*b^7*e^(4*d*x + 4*c) - 22332*a^4*b^8*e^(4*d*x + 4*c) - 14430*a^3*b^9*e^(4*d*x
+ 4*c) - 5946*a^2*b^10*e^(4*d*x + 4*c) - 1410*a*b^11*e^(4*d*x + 4*c) - 150*b^12*e^(4*d*x + 4*c) + 435*a^11*b*e
^(2*d*x + 2*c) + 3411*a^10*b^2*e^(2*d*x + 2*c) + 11433*a^9*b^3*e^(2*d*x + 2*c) + 20793*a^8*b^4*e^(2*d*x + 2*c)
+ 20526*a^7*b^5*e^(2*d*x + 2*c) + 6510*a^6*b^6*e^(2*d*x + 2*c) - 9534*a^5*b^7*e^(2*d*x + 2*c) - 14622*a^4*b^8
*e^(2*d*x + 2*c) - 9777*a^3*b^9*e^(2*d*x + 2*c) - 3729*a^2*b^10*e^(2*d*x + 2*c) - 795*a*b^11*e^(2*d*x + 2*c) -
75*b^12*e^(2*d*x + 2*c) + 549755813673*a^11*b - 841*a^10*b^2 - 3669*a^9*b^3 - 9531*a^8*b^4 - 16374*a^7*b^5 -
19530*a^6*b^6 - 16506*a^5*b^7 - 9894*a^4*b^8 - 4131*a^3*b^9 - 1149*a^2*b^10 - 193*a*b^11 - 15*b^12)/((a^13 + 1
0*a^12*b + 45*a^11*b^2 + 120*a^10*b^3 + 210*a^9*b^4 + 252*a^8*b^5 + 210*a^7*b^6 + 120*a^6*b^7 + 45*a^5*b^8 + 1
0*a^4*b^9 + a^3*b^10)*(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)^3))/d

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maple [B]  time = 0.13, size = 608, normalized size = 3.02 \[ -\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 d \left (a +b \right )^{4}}+\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 d \left (a +b \right )^{4}}+\frac {19 b^{3} \left (\coth ^{5}\left (d x +c \right )\right )}{16 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3}}+\frac {35 b^{4} \left (\coth ^{5}\left (d x +c \right )\right )}{16 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3} a}+\frac {21 b^{5} \left (\coth ^{5}\left (d x +c \right )\right )}{16 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3} a^{2}}+\frac {5 b^{6} \left (\coth ^{5}\left (d x +c \right )\right )}{16 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3} a^{3}}+\frac {17 b^{2} a \left (\coth ^{3}\left (d x +c \right )\right )}{6 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3}}+\frac {11 b^{3} \left (\coth ^{3}\left (d x +c \right )\right )}{2 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3}}+\frac {7 b^{4} \left (\coth ^{3}\left (d x +c \right )\right )}{2 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3} a}+\frac {5 b^{5} \left (\coth ^{3}\left (d x +c \right )\right )}{6 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3} a^{2}}+\frac {29 b \,a^{2} \coth \left (d x +c \right )}{16 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3}}+\frac {61 b^{2} a \coth \left (d x +c \right )}{16 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3}}+\frac {43 b^{3} \coth \left (d x +c \right )}{16 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3}}+\frac {11 b^{4} \coth \left (d x +c \right )}{16 d \left (a +b \right )^{4} \left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{3} a}+\frac {35 b \arctan \left (\frac {\coth \left (d x +c \right ) b}{\sqrt {a b}}\right )}{16 d \left (a +b \right )^{4} \sqrt {a b}}+\frac {35 b^{2} \arctan \left (\frac {\coth \left (d x +c \right ) b}{\sqrt {a b}}\right )}{16 d \left (a +b \right )^{4} a \sqrt {a b}}+\frac {21 b^{3} \arctan \left (\frac {\coth \left (d x +c \right ) b}{\sqrt {a b}}\right )}{16 d \left (a +b \right )^{4} a^{2} \sqrt {a b}}+\frac {5 b^{4} \arctan \left (\frac {\coth \left (d x +c \right ) b}{\sqrt {a b}}\right )}{16 d \left (a +b \right )^{4} a^{3} \sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*coth(d*x+c)^2)^4,x)

[Out]

-1/2/d/(a+b)^4*ln(coth(d*x+c)-1)+1/2/d/(a+b)^4*ln(coth(d*x+c)+1)+19/16/d/(a+b)^4*b^3/(a+b*coth(d*x+c)^2)^3*cot
h(d*x+c)^5+35/16/d/(a+b)^4*b^4/(a+b*coth(d*x+c)^2)^3/a*coth(d*x+c)^5+21/16/d/(a+b)^4*b^5/(a+b*coth(d*x+c)^2)^3
/a^2*coth(d*x+c)^5+5/16/d/(a+b)^4*b^6/(a+b*coth(d*x+c)^2)^3/a^3*coth(d*x+c)^5+17/6/d/(a+b)^4*b^2/(a+b*coth(d*x
+c)^2)^3*a*coth(d*x+c)^3+11/2/d/(a+b)^4*b^3/(a+b*coth(d*x+c)^2)^3*coth(d*x+c)^3+7/2/d/(a+b)^4*b^4/(a+b*coth(d*
x+c)^2)^3/a*coth(d*x+c)^3+5/6/d/(a+b)^4*b^5/(a+b*coth(d*x+c)^2)^3/a^2*coth(d*x+c)^3+29/16/d/(a+b)^4*b/(a+b*cot
h(d*x+c)^2)^3*a^2*coth(d*x+c)+61/16/d/(a+b)^4*b^2/(a+b*coth(d*x+c)^2)^3*a*coth(d*x+c)+43/16/d/(a+b)^4*b^3/(a+b
*coth(d*x+c)^2)^3*coth(d*x+c)+11/16/d/(a+b)^4*b^4/(a+b*coth(d*x+c)^2)^3/a*coth(d*x+c)+35/16/d/(a+b)^4*b/(a*b)^
(1/2)*arctan(coth(d*x+c)*b/(a*b)^(1/2))+35/16/d/(a+b)^4*b^2/a/(a*b)^(1/2)*arctan(coth(d*x+c)*b/(a*b)^(1/2))+21
/16/d/(a+b)^4*b^3/a^2/(a*b)^(1/2)*arctan(coth(d*x+c)*b/(a*b)^(1/2))+5/16/d/(a+b)^4*b^4/a^3/(a*b)^(1/2)*arctan(
coth(d*x+c)*b/(a*b)^(1/2))

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maxima [B]  time = 0.73, size = 925, normalized size = 4.60 \[ \frac {{\left (35 \, a^{3} b + 35 \, a^{2} b^{2} + 21 \, a b^{3} + 5 \, b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - a + b}{2 \, \sqrt {a b}}\right )}{16 \, {\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} d} + \frac {87 \, a^{5} b + 319 \, a^{4} b^{2} + 450 \, a^{3} b^{3} + 306 \, a^{2} b^{4} + 103 \, a b^{5} + 15 \, b^{6} - 3 \, {\left (145 \, a^{5} b + 267 \, a^{4} b^{2} + 34 \, a^{3} b^{3} - 178 \, a^{2} b^{4} - 115 \, a b^{5} - 25 \, b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, {\left (145 \, a^{5} b + 93 \, a^{4} b^{2} - 6 \, a^{3} b^{3} + 106 \, a^{2} b^{4} + 85 \, a b^{5} + 25 \, b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, {\left (435 \, a^{5} b + 29 \, a^{4} b^{2} + 162 \, a^{3} b^{3} - 306 \, a^{2} b^{4} - 245 \, a b^{5} - 75 \, b^{6}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (145 \, a^{5} b + 17 \, a^{4} b^{2} - 58 \, a^{3} b^{3} + 150 \, a^{2} b^{4} + 105 \, a b^{5} + 25 \, b^{6}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - 3 \, {\left (29 \, a^{5} b + 23 \, a^{4} b^{2} - 62 \, a^{3} b^{3} - 82 \, a^{2} b^{4} - 31 \, a b^{5} - 5 \, b^{6}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{24 \, {\left (a^{10} + 7 \, a^{9} b + 21 \, a^{8} b^{2} + 35 \, a^{7} b^{3} + 35 \, a^{6} b^{4} + 21 \, a^{5} b^{5} + 7 \, a^{4} b^{6} + a^{3} b^{7} - 6 \, {\left (a^{10} + 5 \, a^{9} b + 9 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{6} b^{4} - 9 \, a^{5} b^{5} - 5 \, a^{4} b^{6} - a^{3} b^{7}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (5 \, a^{10} + 19 \, a^{9} b + 25 \, a^{8} b^{2} + 15 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 25 \, a^{5} b^{5} + 19 \, a^{4} b^{6} + 5 \, a^{3} b^{7}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (5 \, a^{10} + 17 \, a^{9} b + 21 \, a^{8} b^{2} + 9 \, a^{7} b^{3} - 9 \, a^{6} b^{4} - 21 \, a^{5} b^{5} - 17 \, a^{4} b^{6} - 5 \, a^{3} b^{7}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (5 \, a^{10} + 19 \, a^{9} b + 25 \, a^{8} b^{2} + 15 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 25 \, a^{5} b^{5} + 19 \, a^{4} b^{6} + 5 \, a^{3} b^{7}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - 6 \, {\left (a^{10} + 5 \, a^{9} b + 9 \, a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{6} b^{4} - 9 \, a^{5} b^{5} - 5 \, a^{4} b^{6} - a^{3} b^{7}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + {\left (a^{10} + 7 \, a^{9} b + 21 \, a^{8} b^{2} + 35 \, a^{7} b^{3} + 35 \, a^{6} b^{4} + 21 \, a^{5} b^{5} + 7 \, a^{4} b^{6} + a^{3} b^{7}\right )} e^{\left (-12 \, d x - 12 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c)^2)^4,x, algorithm="maxima")

[Out]

1/16*(35*a^3*b + 35*a^2*b^2 + 21*a*b^3 + 5*b^4)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) - a + b)/sqrt(a*b))/((a^7
+ 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*sqrt(a*b)*d) + 1/24*(87*a^5*b + 319*a^4*b^2 + 450*a^3*b^3 + 306*
a^2*b^4 + 103*a*b^5 + 15*b^6 - 3*(145*a^5*b + 267*a^4*b^2 + 34*a^3*b^3 - 178*a^2*b^4 - 115*a*b^5 - 25*b^6)*e^(
-2*d*x - 2*c) + 6*(145*a^5*b + 93*a^4*b^2 - 6*a^3*b^3 + 106*a^2*b^4 + 85*a*b^5 + 25*b^6)*e^(-4*d*x - 4*c) - 2*
(435*a^5*b + 29*a^4*b^2 + 162*a^3*b^3 - 306*a^2*b^4 - 245*a*b^5 - 75*b^6)*e^(-6*d*x - 6*c) + 3*(145*a^5*b + 17
*a^4*b^2 - 58*a^3*b^3 + 150*a^2*b^4 + 105*a*b^5 + 25*b^6)*e^(-8*d*x - 8*c) - 3*(29*a^5*b + 23*a^4*b^2 - 62*a^3
*b^3 - 82*a^2*b^4 - 31*a*b^5 - 5*b^6)*e^(-10*d*x - 10*c))/((a^10 + 7*a^9*b + 21*a^8*b^2 + 35*a^7*b^3 + 35*a^6*
b^4 + 21*a^5*b^5 + 7*a^4*b^6 + a^3*b^7 - 6*(a^10 + 5*a^9*b + 9*a^8*b^2 + 5*a^7*b^3 - 5*a^6*b^4 - 9*a^5*b^5 - 5
*a^4*b^6 - a^3*b^7)*e^(-2*d*x - 2*c) + 3*(5*a^10 + 19*a^9*b + 25*a^8*b^2 + 15*a^7*b^3 + 15*a^6*b^4 + 25*a^5*b^
5 + 19*a^4*b^6 + 5*a^3*b^7)*e^(-4*d*x - 4*c) - 4*(5*a^10 + 17*a^9*b + 21*a^8*b^2 + 9*a^7*b^3 - 9*a^6*b^4 - 21*
a^5*b^5 - 17*a^4*b^6 - 5*a^3*b^7)*e^(-6*d*x - 6*c) + 3*(5*a^10 + 19*a^9*b + 25*a^8*b^2 + 15*a^7*b^3 + 15*a^6*b
^4 + 25*a^5*b^5 + 19*a^4*b^6 + 5*a^3*b^7)*e^(-8*d*x - 8*c) - 6*(a^10 + 5*a^9*b + 9*a^8*b^2 + 5*a^7*b^3 - 5*a^6
*b^4 - 9*a^5*b^5 - 5*a^4*b^6 - a^3*b^7)*e^(-10*d*x - 10*c) + (a^10 + 7*a^9*b + 21*a^8*b^2 + 35*a^7*b^3 + 35*a^
6*b^4 + 21*a^5*b^5 + 7*a^4*b^6 + a^3*b^7)*e^(-12*d*x - 12*c))*d) + (d*x + c)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a
*b^3 + b^4)*d)

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mupad [B]  time = 2.07, size = 3685, normalized size = 18.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*coth(c + d*x)^2)^4,x)

[Out]

log(coth(c + d*x) + 1)/(2*a^4*d + 2*b^4*d + 12*a^2*b^2*d + 8*a*b^3*d + 8*a^3*b*d) + ((coth(c + d*x)^3*(16*a*b^
3 + 5*b^4 + 17*a^2*b^2))/(6*a^2*(3*a*b^2 + 3*a^2*b + a^3 + b^3)) + (coth(c + d*x)*(32*a*b^2 + 29*a^2*b + 11*b^
3))/(16*a*(3*a*b^2 + 3*a^2*b + a^3 + b^3)) + (b^2*coth(c + d*x)^5*(16*a*b^2 + 19*a^2*b + 5*b^3))/(16*a^2*(a*b^
3 + 3*a^3*b + a^4 + 3*a^2*b^2)))/(a^3*d + b^3*d*coth(c + d*x)^6 + 3*a^2*b*d*coth(c + d*x)^2 + 3*a*b^2*d*coth(c
+ d*x)^4) - log(coth(c + d*x) - 1)/(2*d*(a + b)^4) - (atan((((-a^7*b)^(1/2)*((coth(c + d*x)*(210*a*b^8 + 25*b
^9 + 791*a^2*b^7 + 1820*a^3*b^6 + 2695*a^4*b^5 + 2450*a^5*b^4 + 1481*a^6*b^3))/(128*(a^12*d^2 + 6*a^11*b*d^2 +
a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)) + ((((5*a^3*b^13*d^2)/4 + 1
4*a^4*b^12*d^2 + (287*a^5*b^11*d^2)/4 + 224*a^6*b^10*d^2 + (953*a^7*b^9*d^2)/2 + 728*a^8*b^8*d^2 + (1631*a^9*b
^7*d^2)/2 + 668*a^10*b^6*d^2 + (1561*a^11*b^5*d^2)/4 + 154*a^12*b^4*d^2 + (147*a^13*b^3*d^2)/4 + 4*a^14*b^2*d^
2)/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b^6*d^3 + 126*a^10*b^5*d^3
+ 126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) - (coth(c + d*x)*(-a^7*b)^(1/2)*(21*a*b^2 + 35*a^2*b
+ 35*a^3 + 5*b^3)*(1024*a^6*b^11*d^2 + 7168*a^7*b^10*d^2 + 20480*a^8*b^9*d^2 + 28672*a^9*b^8*d^2 + 14336*a^10*
b^7*d^2 - 14336*a^11*b^6*d^2 - 28672*a^12*b^5*d^2 - 20480*a^13*b^4*d^2 - 7168*a^14*b^3*d^2 - 1024*a^15*b^2*d^2
))/(4096*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)*(a^12*d^2 + 6*a^11*b*d^2 + a^6*b^6*d^2
+ 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)))*(-a^7*b)^(1/2)*(21*a*b^2 + 35*a^2*b + 3
5*a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))*(21*a*b^2 + 35*a^2*b + 35*
a^3 + 5*b^3)*1i)/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)) + ((-a^7*b)^(1/2)*((coth(c
+ d*x)*(210*a*b^8 + 25*b^9 + 791*a^2*b^7 + 1820*a^3*b^6 + 2695*a^4*b^5 + 2450*a^5*b^4 + 1481*a^6*b^3))/(128*(
a^12*d^2 + 6*a^11*b*d^2 + a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)) -
((((5*a^3*b^13*d^2)/4 + 14*a^4*b^12*d^2 + (287*a^5*b^11*d^2)/4 + 224*a^6*b^10*d^2 + (953*a^7*b^9*d^2)/2 + 728*
a^8*b^8*d^2 + (1631*a^9*b^7*d^2)/2 + 668*a^10*b^6*d^2 + (1561*a^11*b^5*d^2)/4 + 154*a^12*b^4*d^2 + (147*a^13*b
^3*d^2)/4 + 4*a^14*b^2*d^2)/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b
^6*d^3 + 126*a^10*b^5*d^3 + 126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) + (coth(c + d*x)*(-a^7*b)^(1
/2)*(21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3)*(1024*a^6*b^11*d^2 + 7168*a^7*b^10*d^2 + 20480*a^8*b^9*d^2 + 28672*
a^9*b^8*d^2 + 14336*a^10*b^7*d^2 - 14336*a^11*b^6*d^2 - 28672*a^12*b^5*d^2 - 20480*a^13*b^4*d^2 - 7168*a^14*b^
3*d^2 - 1024*a^15*b^2*d^2))/(4096*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)*(a^12*d^2 + 6*
a^11*b*d^2 + a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)))*(-a^7*b)^(1/2)
*(21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))*(
21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3)*1i)/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))/
(((185*a*b^7)/128 + (25*b^8)/128 + (303*a^2*b^6)/64 + (567*a^3*b^5)/64 + (1225*a^4*b^4)/128 + (665*a^5*b^3)/12
8)/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b^6*d^3 + 126*a^10*b^5*d^3
+ 126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) + ((-a^7*b)^(1/2)*((coth(c + d*x)*(210*a*b^8 + 25*b^9
+ 791*a^2*b^7 + 1820*a^3*b^6 + 2695*a^4*b^5 + 2450*a^5*b^4 + 1481*a^6*b^3))/(128*(a^12*d^2 + 6*a^11*b*d^2 + a
^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)) + ((((5*a^3*b^13*d^2)/4 + 14*
a^4*b^12*d^2 + (287*a^5*b^11*d^2)/4 + 224*a^6*b^10*d^2 + (953*a^7*b^9*d^2)/2 + 728*a^8*b^8*d^2 + (1631*a^9*b^7
*d^2)/2 + 668*a^10*b^6*d^2 + (1561*a^11*b^5*d^2)/4 + 154*a^12*b^4*d^2 + (147*a^13*b^3*d^2)/4 + 4*a^14*b^2*d^2)
/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b^6*d^3 + 126*a^10*b^5*d^3 +
126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) - (coth(c + d*x)*(-a^7*b)^(1/2)*(21*a*b^2 + 35*a^2*b +
35*a^3 + 5*b^3)*(1024*a^6*b^11*d^2 + 7168*a^7*b^10*d^2 + 20480*a^8*b^9*d^2 + 28672*a^9*b^8*d^2 + 14336*a^10*b^
7*d^2 - 14336*a^11*b^6*d^2 - 28672*a^12*b^5*d^2 - 20480*a^13*b^4*d^2 - 7168*a^14*b^3*d^2 - 1024*a^15*b^2*d^2))
/(4096*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)*(a^12*d^2 + 6*a^11*b*d^2 + a^6*b^6*d^2 +
6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)))*(-a^7*b)^(1/2)*(21*a*b^2 + 35*a^2*b + 35*
a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))*(21*a*b^2 + 35*a^2*b + 35*a^
3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)) - ((-a^7*b)^(1/2)*((coth(c + d*
x)*(210*a*b^8 + 25*b^9 + 791*a^2*b^7 + 1820*a^3*b^6 + 2695*a^4*b^5 + 2450*a^5*b^4 + 1481*a^6*b^3))/(128*(a^12*
d^2 + 6*a^11*b*d^2 + a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)) - ((((5
*a^3*b^13*d^2)/4 + 14*a^4*b^12*d^2 + (287*a^5*b^11*d^2)/4 + 224*a^6*b^10*d^2 + (953*a^7*b^9*d^2)/2 + 728*a^8*b
^8*d^2 + (1631*a^9*b^7*d^2)/2 + 668*a^10*b^6*d^2 + (1561*a^11*b^5*d^2)/4 + 154*a^12*b^4*d^2 + (147*a^13*b^3*d^
2)/4 + 4*a^14*b^2*d^2)/(a^15*d^3 + 9*a^14*b*d^3 + a^6*b^9*d^3 + 9*a^7*b^8*d^3 + 36*a^8*b^7*d^3 + 84*a^9*b^6*d^
3 + 126*a^10*b^5*d^3 + 126*a^11*b^4*d^3 + 84*a^12*b^3*d^3 + 36*a^13*b^2*d^3) + (coth(c + d*x)*(-a^7*b)^(1/2)*(
21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3)*(1024*a^6*b^11*d^2 + 7168*a^7*b^10*d^2 + 20480*a^8*b^9*d^2 + 28672*a^9*b
^8*d^2 + 14336*a^10*b^7*d^2 - 14336*a^11*b^6*d^2 - 28672*a^12*b^5*d^2 - 20480*a^13*b^4*d^2 - 7168*a^14*b^3*d^2
- 1024*a^15*b^2*d^2))/(4096*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)*(a^12*d^2 + 6*a^11*
b*d^2 + a^6*b^6*d^2 + 6*a^7*b^5*d^2 + 15*a^8*b^4*d^2 + 20*a^9*b^3*d^2 + 15*a^10*b^2*d^2)))*(-a^7*b)^(1/2)*(21*
a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d)))*(21*a*
b^2 + 35*a^2*b + 35*a^3 + 5*b^3))/(32*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^10*b*d))))*(-a^7*b
)^(1/2)*(21*a*b^2 + 35*a^2*b + 35*a^3 + 5*b^3)*1i)/(16*(a^11*d + a^7*b^4*d + 4*a^8*b^3*d + 6*a^9*b^2*d + 4*a^1
0*b*d))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c)**2)**4,x)

[Out]

Timed out

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