3.1.7 \(\int \frac {1}{(a+b \coth ^2(c+d x))^3} \, dx\) [7]

Optimal. Leaf size=142 \[ \frac {x}{(a+b)^3}-\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{8 a^{5/2} (a+b)^3 d}+\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b (7 a+3 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \coth ^2(c+d x)\right )} \]

[Out]

x/(a+b)^3+1/4*b*coth(d*x+c)/a/(a+b)/d/(a+b*coth(d*x+c)^2)^2+1/8*b*(7*a+3*b)*coth(d*x+c)/a^2/(a+b)^2/d/(a+b*cot
h(d*x+c)^2)-1/8*(15*a^2+10*a*b+3*b^2)*arctan(a^(1/2)*tanh(d*x+c)/b^(1/2))*b^(1/2)/a^(5/2)/(a+b)^3/d

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Rubi [A]
time = 0.12, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3742, 425, 541, 536, 212, 211} \begin {gather*} \frac {b (7 a+3 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left (a+b \coth ^2(c+d x)\right )}-\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{8 a^{5/2} d (a+b)^3}+\frac {b \coth (c+d x)}{4 a d (a+b) \left (a+b \coth ^2(c+d x)\right )^2}+\frac {x}{(a+b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[c + d*x]^2)^(-3),x]

[Out]

x/(a + b)^3 - (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Tanh[c + d*x])/Sqrt[b]])/(8*a^(5/2)*(a + b)^3
*d) + (b*Coth[c + d*x])/(4*a*(a + b)*d*(a + b*Coth[c + d*x]^2)^2) + (b*(7*a + 3*b)*Coth[c + d*x])/(8*a^2*(a +
b)^2*d*(a + b*Coth[c + d*x]^2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 425

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]) && IntBinomi
alQ[a, b, c, d, 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 541

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 3742

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 )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {b-4 (a+b)+3 b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\coth (c+d x)\right )}{4 a (a+b) d}\\ &=\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b (7 a+3 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \coth ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {8 a^2+7 a b+3 b^2-b (7 a+3 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\coth (c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b (7 a+3 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \coth ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{(a+b)^3 d}+\frac {\left (b \left (15 a^2+10 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\coth (c+d x)\right )}{8 a^2 (a+b)^3 d}\\ &=\frac {x}{(a+b)^3}-\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {b}}\right )}{8 a^{5/2} (a+b)^3 d}+\frac {b \coth (c+d x)}{4 a (a+b) d \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b (7 a+3 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \coth ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 147, normalized size = 1.04 \begin {gather*} \frac {\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 b (a+b)^2 \coth (c+d x)}{a \left (a+b \coth ^2(c+d x)\right )^2}+\frac {b (a+b) (7 a+3 b) \coth (c+d x)}{a^2 \left (a+b \coth ^2(c+d x)\right )}-4 \log (1-\coth (c+d x))+4 \log (1+\coth (c+d x))}{8 (a+b)^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Coth[c + d*x]^2)^(-3),x]

[Out]

((Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Coth[c + d*x])/Sqrt[a]])/a^(5/2) + (2*b*(a + b)^2*Coth[c +
d*x])/(a*(a + b*Coth[c + d*x]^2)^2) + (b*(a + b)*(7*a + 3*b)*Coth[c + d*x])/(a^2*(a + b*Coth[c + d*x]^2)) - 4
*Log[1 - Coth[c + d*x]] + 4*Log[1 + Coth[c + d*x]])/(8*(a + b)^3*d)

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Maple [A]
time = 0.97, size = 156, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {\frac {b \left (\frac {\frac {b \left (7 a^{2}+10 a b +3 b^{2}\right ) \left (\coth ^{3}\left (d x +c \right )\right )}{8 a^{2}}+\frac {\left (9 a^{2}+14 a b +5 b^{2}\right ) \coth \left (d x +c \right )}{8 a}}{\left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {\left (15 a^{2}+10 a b +3 b^{2}\right ) \arctan \left (\frac {b \coth \left (d x +c \right )}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a +b \right )^{3}}+\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\)	\(156\)
default	\(\frac {\frac {b \left (\frac {\frac {b \left (7 a^{2}+10 a b +3 b^{2}\right ) \left (\coth ^{3}\left (d x +c \right )\right )}{8 a^{2}}+\frac {\left (9 a^{2}+14 a b +5 b^{2}\right ) \coth \left (d x +c \right )}{8 a}}{\left (a +b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {\left (15 a^{2}+10 a b +3 b^{2}\right ) \arctan \left (\frac {b \coth \left (d x +c \right )}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a +b \right )^{3}}+\frac {\ln \left (\coth \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{3}}-\frac {\ln \left (\coth \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\)	\(156\)
risch	\(\frac {x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}+\frac {\left (9 a^{3} {\mathrm e}^{6 d x +6 c}-a^{2} b \,{\mathrm e}^{6 d x +6 c}-13 a \,b^{2} {\mathrm e}^{6 d x +6 c}-3 b^{3} {\mathrm e}^{6 d x +6 c}-27 a^{3} {\mathrm e}^{4 d x +4 c}+9 a^{2} b \,{\mathrm e}^{4 d x +4 c}-21 a \,b^{2} {\mathrm e}^{4 d x +4 c}-9 b^{3} {\mathrm e}^{4 d x +4 c}+27 a^{3} {\mathrm e}^{2 d x +2 c}+13 a^{2} b \,{\mathrm e}^{2 d x +2 c}-23 a \,b^{2} {\mathrm e}^{2 d x +2 c}-9 b^{3} {\mathrm e}^{2 d x +2 c}-9 a^{3}-21 a^{2} b -15 a \,b^{2}-3 b^{3}\right ) b}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}-2 a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} \left (a +b \right )^{3} a^{2} d}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-a b}-b}{a +b}\right )}{16 a \left (a +b \right )^{3} d}+\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-a b}-b}{a +b}\right ) b}{8 a^{2} \left (a +b \right )^{3} d}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {a +2 \sqrt {-a b}-b}{a +b}\right ) b^{2}}{16 a^{3} \left (a +b \right )^{3} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 a \left (a +b \right )^{3} d}-\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{8 a^{2} \left (a +b \right )^{3} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b^{2}}{16 a^{3} \left (a +b \right )^{3} d}\)	\(592\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*coth(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a+b)^3*b*((1/8*b*(7*a^2+10*a*b+3*b^2)/a^2*coth(d*x+c)^3+1/8*(9*a^2+14*a*b+5*b^2)/a*coth(d*x+c))/(a+b*c
oth(d*x+c)^2)^2+1/8*(15*a^2+10*a*b+3*b^2)/a^2/(a*b)^(1/2)*arctan(b*coth(d*x+c)/(a*b)^(1/2)))+1/2/(a+b)^3*ln(co
th(d*x+c)+1)-1/2/(a+b)^3*ln(coth(d*x+c)-1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (128) = 256\).
time = 0.60, size = 509, normalized size = 3.58 \begin {gather*} \frac {{\left (15 \, a^{2} b + 10 \, a b^{2} + 3 \, b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - a + b}{2 \, \sqrt {a b}}\right )}{8 \, {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt {a b} d} + \frac {9 \, a^{3} b + 21 \, a^{2} b^{2} + 15 \, a b^{3} + 3 \, b^{4} - {\left (27 \, a^{3} b + 13 \, a^{2} b^{2} - 23 \, a b^{3} - 9 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (9 \, a^{3} b - 3 \, a^{2} b^{2} + 7 \, a b^{3} + 3 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (9 \, a^{3} b - a^{2} b^{2} - 13 \, a b^{3} - 3 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{4 \, {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5} - 4 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{7} + 7 \, a^{6} b + 6 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 7 \, a^{3} b^{4} + 3 \, a^{2} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/8*(15*a^2*b + 10*a*b^2 + 3*b^3)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) - a + b)/sqrt(a*b))/((a^5 + 3*a^4*b + 3
*a^3*b^2 + a^2*b^3)*sqrt(a*b)*d) + 1/4*(9*a^3*b + 21*a^2*b^2 + 15*a*b^3 + 3*b^4 - (27*a^3*b + 13*a^2*b^2 - 23*
a*b^3 - 9*b^4)*e^(-2*d*x - 2*c) + 3*(9*a^3*b - 3*a^2*b^2 + 7*a*b^3 + 3*b^4)*e^(-4*d*x - 4*c) - (9*a^3*b - a^2*
b^2 - 13*a*b^3 - 3*b^4)*e^(-6*d*x - 6*c))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 - 4*
(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^
2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*e^(-4*d*x - 4*c) - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4
- a^2*b^5)*e^(-6*d*x - 6*c) + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*e^(-8*d*x - 8*c
))*d) + (d*x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3592 vs. \(2 (128) = 256\).
time = 0.49, size = 7508, normalized size = 52.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(16*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^8 + 128*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)*sinh
(d*x + c)^7 + 16*(a^4 + 2*a^3*b + a^2*b^2)*d*x*sinh(d*x + c)^8 + 4*(9*a^3*b - a^2*b^2 - 13*a*b^3 - 3*b^4 - 16*
(a^4 - a^2*b^2)*d*x)*cosh(d*x + c)^6 + 4*(112*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^2 + 9*a^3*b - a^2*b^
2 - 13*a*b^3 - 3*b^4 - 16*(a^4 - a^2*b^2)*d*x)*sinh(d*x + c)^6 + 8*(112*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x
+ c)^3 + 3*(9*a^3*b - a^2*b^2 - 13*a*b^3 - 3*b^4 - 16*(a^4 - a^2*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 4
*(27*a^3*b - 9*a^2*b^2 + 21*a*b^3 + 9*b^4 - 8*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*d*x)*cosh(d*x + c)^4 + 4*(280*(a^4
+ 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^4 - 27*a^3*b + 9*a^2*b^2 - 21*a*b^3 - 9*b^4 + 8*(3*a^4 - 2*a^3*b + 3*a
^2*b^2)*d*x + 15*(9*a^3*b - a^2*b^2 - 13*a*b^3 - 3*b^4 - 16*(a^4 - a^2*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c
)^4 - 36*a^3*b - 84*a^2*b^2 - 60*a*b^3 - 12*b^4 + 16*(56*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^5 + 5*(9*
a^3*b - a^2*b^2 - 13*a*b^3 - 3*b^4 - 16*(a^4 - a^2*b^2)*d*x)*cosh(d*x + c)^3 - (27*a^3*b - 9*a^2*b^2 + 21*a*b^
3 + 9*b^4 - 8*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 16*(a^4 + 2*a^3*b + a^2*b^2)
*d*x + 4*(27*a^3*b + 13*a^2*b^2 - 23*a*b^3 - 9*b^4 - 16*(a^4 - a^2*b^2)*d*x)*cosh(d*x + c)^2 + 4*(112*(a^4 + 2
*a^3*b + a^2*b^2)*d*x*cosh(d*x + c)^6 + 15*(9*a^3*b - a^2*b^2 - 13*a*b^3 - 3*b^4 - 16*(a^4 - a^2*b^2)*d*x)*cos
h(d*x + c)^4 + 27*a^3*b + 13*a^2*b^2 - 23*a*b^3 - 9*b^4 - 16*(a^4 - a^2*b^2)*d*x - 6*(27*a^3*b - 9*a^2*b^2 + 2
1*a*b^3 + 9*b^4 - 8*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((15*a^4 + 40*a^3*b
+ 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(15*a^4 + 40*a^3*b + 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*cosh(
d*x + c)*sinh(d*x + c)^7 + (15*a^4 + 40*a^3*b + 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*sinh(d*x + c)^8 - 4*(15*a^4 + 1
0*a^3*b - 12*a^2*b^2 - 10*a*b^3 - 3*b^4)*cosh(d*x + c)^6 - 4*(15*a^4 + 10*a^3*b - 12*a^2*b^2 - 10*a*b^3 - 3*b^
4 - 7*(15*a^4 + 40*a^3*b + 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^4 + 40
*a^3*b + 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*cosh(d*x + c)^3 - 3*(15*a^4 + 10*a^3*b - 12*a^2*b^2 - 10*a*b^3 - 3*b^4
)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(45*a^4 + 34*a^2*b^2 + 24*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^4
+ 40*a^3*b + 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 45*a^4 + 34*a^2*b^2 + 24*a*b^3 + 9*b^4 - 30*(15*
a^4 + 10*a^3*b - 12*a^2*b^2 - 10*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 15*a^4 + 40*a^3*b + 38*a^2*
b^2 + 16*a*b^3 + 3*b^4 + 8*(7*(15*a^4 + 40*a^3*b + 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*cosh(d*x + c)^5 - 10*(15*a^4
+ 10*a^3*b - 12*a^2*b^2 - 10*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (45*a^4 + 34*a^2*b^2 + 24*a*b^3 + 9*b^4)*cosh(d
*x + c))*sinh(d*x + c)^3 - 4*(15*a^4 + 10*a^3*b - 12*a^2*b^2 - 10*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(15*a^
4 + 40*a^3*b + 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*cosh(d*x + c)^6 - 15*(15*a^4 + 10*a^3*b - 12*a^2*b^2 - 10*a*b^3
- 3*b^4)*cosh(d*x + c)^4 - 15*a^4 - 10*a^3*b + 12*a^2*b^2 + 10*a*b^3 + 3*b^4 + 3*(45*a^4 + 34*a^2*b^2 + 24*a*b
^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((15*a^4 + 40*a^3*b + 38*a^2*b^2 + 16*a*b^3 + 3*b^4)*cosh(d*x
+ c)^7 - 3*(15*a^4 + 10*a^3*b - 12*a^2*b^2 - 10*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (45*a^4 + 34*a^2*b^2 + 24*a*
b^3 + 9*b^4)*cosh(d*x + c)^3 - (15*a^4 + 10*a^3*b - 12*a^2*b^2 - 10*a*b^3 - 3*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))*sinh(d*x + c) + a + b)) + 8*(16*(a^4 + 2*a^3*b + a^2
*b^2)*d*x*cosh(d*x + c)^7 + 3*(9*a^3*b - a^2*b^2 - 13*a*b^3 - 3*b^4 - 16*(a^4 - a^2*b^2)*d*x)*cosh(d*x + c)^5
- 2*(27*a^3*b - 9*a^2*b^2 + 21*a*b^3 + 9*b^4 - 8*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*d*x)*cosh(d*x + c)^3 + (27*a^3*
b + 13*a^2*b^2 - 23*a*b^3 - 9*b^4 - 16*(a^4 - a^2*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^7 + 5*a^6*b + 10
*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^8 + 8*(a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 +
5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 +
a^2*b^5)*d*sinh(d*x + c)^8 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^6
+ 4*(7*(a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^2 - (a^7 + 3*a^6*b + 2
*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*d)*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (128) = 256\).
time = 0.42, size = 409, normalized size = 2.88 \begin {gather*} -\frac {\frac {{\left (15 \, a^{2} b + 10 \, a b^{2} + 3 \, b^{3}\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} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt {a b}} - \frac {8 \, {\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 13 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} - 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 21 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 23 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, a^{3} b - 21 \, a^{2} b^{2} - 15 \, a b^{3} - 3 \, b^{4}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 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 )}^{2}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*coth(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/8*((15*a^2*b + 10*a*b^2 + 3*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) - a + b)/sqrt(a*b))/((a^
5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a*b)) - 8*(d*x + c)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 2*(9*a^3*b*e^(6*
d*x + 6*c) - a^2*b^2*e^(6*d*x + 6*c) - 13*a*b^3*e^(6*d*x + 6*c) - 3*b^4*e^(6*d*x + 6*c) - 27*a^3*b*e^(4*d*x +
4*c) + 9*a^2*b^2*e^(4*d*x + 4*c) - 21*a*b^3*e^(4*d*x + 4*c) - 9*b^4*e^(4*d*x + 4*c) + 27*a^3*b*e^(2*d*x + 2*c)
+ 13*a^2*b^2*e^(2*d*x + 2*c) - 23*a*b^3*e^(2*d*x + 2*c) - 9*b^4*e^(2*d*x + 2*c) - 9*a^3*b - 21*a^2*b^2 - 15*a
*b^3 - 3*b^4)/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*(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)^2))/d

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Mupad [B]
time = 1.91, size = 2719, normalized size = 19.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*coth(c + d*x)^2)^3,x)

[Out]

log(coth(c + d*x) + 1)/(2*a^3*d + 2*b^3*d + 6*a*b^2*d + 6*a^2*b*d) + ((coth(c + d*x)*(9*a*b + 5*b^2))/(8*a*(2*
a*b + a^2 + b^2)) + (b*coth(c + d*x)^3*(7*a*b + 3*b^2))/(8*a*(a*b^2 + 2*a^2*b + a^3)))/(a^2*d + b^2*d*coth(c +
d*x)^4 + 2*a*b*d*coth(c + d*x)^2) - log(coth(c + d*x) - 1)/(2*d*(a + b)^3) - (atan((((-a^5*b)^(1/2)*((coth(c
+ d*x)*(60*a*b^6 + 9*b^7 + 190*a^2*b^5 + 300*a^3*b^4 + 289*a^4*b^3))/(32*(a^8*d^2 + 4*a^7*b*d^2 + a^4*b^4*d^2
+ 4*a^5*b^3*d^2 + 6*a^6*b^2*d^2)) + (((96*a^2*b^10*d^2 + 800*a^3*b^9*d^2 + 3040*a^4*b^8*d^2 + 6816*a^5*b^7*d^2
+ 9760*a^6*b^6*d^2 + 9056*a^7*b^5*d^2 + 5280*a^8*b^4*d^2 + 1760*a^9*b^3*d^2 + 256*a^10*b^2*d^2)/(64*(a^10*d^3
+ 6*a^9*b*d^3 + a^4*b^6*d^3 + 6*a^5*b^5*d^3 + 15*a^6*b^4*d^3 + 20*a^7*b^3*d^3 + 15*a^8*b^2*d^3)) - (coth(c +
d*x)*(-a^5*b)^(1/2)*(10*a*b + 15*a^2 + 3*b^2)*(256*a^4*b^9*d^2 + 1280*a^5*b^8*d^2 + 2304*a^6*b^7*d^2 + 1280*a^
7*b^6*d^2 - 1280*a^8*b^5*d^2 - 2304*a^9*b^4*d^2 - 1280*a^10*b^3*d^2 - 256*a^11*b^2*d^2))/(512*(a^8*d + a^5*b^3
*d + 3*a^6*b^2*d + 3*a^7*b*d)*(a^8*d^2 + 4*a^7*b*d^2 + a^4*b^4*d^2 + 4*a^5*b^3*d^2 + 6*a^6*b^2*d^2)))*(-a^5*b)
^(1/2)*(10*a*b + 15*a^2 + 3*b^2))/(16*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*a^7*b*d)))*(10*a*b + 15*a^2 + 3*b^2
)*1i)/(16*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*a^7*b*d)) + ((-a^5*b)^(1/2)*((coth(c + d*x)*(60*a*b^6 + 9*b^7 +
190*a^2*b^5 + 300*a^3*b^4 + 289*a^4*b^3))/(32*(a^8*d^2 + 4*a^7*b*d^2 + a^4*b^4*d^2 + 4*a^5*b^3*d^2 + 6*a^6*b^
2*d^2)) - (((96*a^2*b^10*d^2 + 800*a^3*b^9*d^2 + 3040*a^4*b^8*d^2 + 6816*a^5*b^7*d^2 + 9760*a^6*b^6*d^2 + 9056
*a^7*b^5*d^2 + 5280*a^8*b^4*d^2 + 1760*a^9*b^3*d^2 + 256*a^10*b^2*d^2)/(64*(a^10*d^3 + 6*a^9*b*d^3 + a^4*b^6*d
^3 + 6*a^5*b^5*d^3 + 15*a^6*b^4*d^3 + 20*a^7*b^3*d^3 + 15*a^8*b^2*d^3)) + (coth(c + d*x)*(-a^5*b)^(1/2)*(10*a*
b + 15*a^2 + 3*b^2)*(256*a^4*b^9*d^2 + 1280*a^5*b^8*d^2 + 2304*a^6*b^7*d^2 + 1280*a^7*b^6*d^2 - 1280*a^8*b^5*d
^2 - 2304*a^9*b^4*d^2 - 1280*a^10*b^3*d^2 - 256*a^11*b^2*d^2))/(512*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*a^7*b
*d)*(a^8*d^2 + 4*a^7*b*d^2 + a^4*b^4*d^2 + 4*a^5*b^3*d^2 + 6*a^6*b^2*d^2)))*(-a^5*b)^(1/2)*(10*a*b + 15*a^2 +
3*b^2))/(16*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*a^7*b*d)))*(10*a*b + 15*a^2 + 3*b^2)*1i)/(16*(a^8*d + a^5*b^3
*d + 3*a^6*b^2*d + 3*a^7*b*d)))/((51*a*b^5 + 9*b^6 + 115*a^2*b^4 + 105*a^3*b^3)/(32*(a^10*d^3 + 6*a^9*b*d^3 +
a^4*b^6*d^3 + 6*a^5*b^5*d^3 + 15*a^6*b^4*d^3 + 20*a^7*b^3*d^3 + 15*a^8*b^2*d^3)) + ((-a^5*b)^(1/2)*((coth(c +
d*x)*(60*a*b^6 + 9*b^7 + 190*a^2*b^5 + 300*a^3*b^4 + 289*a^4*b^3))/(32*(a^8*d^2 + 4*a^7*b*d^2 + a^4*b^4*d^2 +
4*a^5*b^3*d^2 + 6*a^6*b^2*d^2)) + (((96*a^2*b^10*d^2 + 800*a^3*b^9*d^2 + 3040*a^4*b^8*d^2 + 6816*a^5*b^7*d^2 +
9760*a^6*b^6*d^2 + 9056*a^7*b^5*d^2 + 5280*a^8*b^4*d^2 + 1760*a^9*b^3*d^2 + 256*a^10*b^2*d^2)/(64*(a^10*d^3 +
6*a^9*b*d^3 + a^4*b^6*d^3 + 6*a^5*b^5*d^3 + 15*a^6*b^4*d^3 + 20*a^7*b^3*d^3 + 15*a^8*b^2*d^3)) - (coth(c + d*
x)*(-a^5*b)^(1/2)*(10*a*b + 15*a^2 + 3*b^2)*(256*a^4*b^9*d^2 + 1280*a^5*b^8*d^2 + 2304*a^6*b^7*d^2 + 1280*a^7*
b^6*d^2 - 1280*a^8*b^5*d^2 - 2304*a^9*b^4*d^2 - 1280*a^10*b^3*d^2 - 256*a^11*b^2*d^2))/(512*(a^8*d + a^5*b^3*d
+ 3*a^6*b^2*d + 3*a^7*b*d)*(a^8*d^2 + 4*a^7*b*d^2 + a^4*b^4*d^2 + 4*a^5*b^3*d^2 + 6*a^6*b^2*d^2)))*(-a^5*b)^(
1/2)*(10*a*b + 15*a^2 + 3*b^2))/(16*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*a^7*b*d)))*(10*a*b + 15*a^2 + 3*b^2))
/(16*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*a^7*b*d)) - ((-a^5*b)^(1/2)*((coth(c + d*x)*(60*a*b^6 + 9*b^7 + 190*
a^2*b^5 + 300*a^3*b^4 + 289*a^4*b^3))/(32*(a^8*d^2 + 4*a^7*b*d^2 + a^4*b^4*d^2 + 4*a^5*b^3*d^2 + 6*a^6*b^2*d^2
)) - (((96*a^2*b^10*d^2 + 800*a^3*b^9*d^2 + 3040*a^4*b^8*d^2 + 6816*a^5*b^7*d^2 + 9760*a^6*b^6*d^2 + 9056*a^7*
b^5*d^2 + 5280*a^8*b^4*d^2 + 1760*a^9*b^3*d^2 + 256*a^10*b^2*d^2)/(64*(a^10*d^3 + 6*a^9*b*d^3 + a^4*b^6*d^3 +
6*a^5*b^5*d^3 + 15*a^6*b^4*d^3 + 20*a^7*b^3*d^3 + 15*a^8*b^2*d^3)) + (coth(c + d*x)*(-a^5*b)^(1/2)*(10*a*b + 1
5*a^2 + 3*b^2)*(256*a^4*b^9*d^2 + 1280*a^5*b^8*d^2 + 2304*a^6*b^7*d^2 + 1280*a^7*b^6*d^2 - 1280*a^8*b^5*d^2 -
2304*a^9*b^4*d^2 - 1280*a^10*b^3*d^2 - 256*a^11*b^2*d^2))/(512*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*a^7*b*d)*(
a^8*d^2 + 4*a^7*b*d^2 + a^4*b^4*d^2 + 4*a^5*b^3*d^2 + 6*a^6*b^2*d^2)))*(-a^5*b)^(1/2)*(10*a*b + 15*a^2 + 3*b^2
))/(16*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*a^7*b*d)))*(10*a*b + 15*a^2 + 3*b^2))/(16*(a^8*d + a^5*b^3*d + 3*a
^6*b^2*d + 3*a^7*b*d))))*(-a^5*b)^(1/2)*(10*a*b + 15*a^2 + 3*b^2)*1i)/(8*(a^8*d + a^5*b^3*d + 3*a^6*b^2*d + 3*
a^7*b*d))

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