3.1.46 \(\int \frac {\coth ^{-1}(a x)}{(c+d x^2)^{7/2}} \, dx\) [46]

Optimal. Leaf size=200 \[ \frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}} \]

[Out]

1/15*a/c/(a^2*c+d)/(d*x^2+c)^(3/2)+1/5*x*arccoth(a*x)/c/(d*x^2+c)^(5/2)+4/15*x*arccoth(a*x)/c^2/(d*x^2+c)^(3/2
)-1/15*(15*a^4*c^2+20*a^2*c*d+8*d^2)*arctanh(a*(d*x^2+c)^(1/2)/(a^2*c+d)^(1/2))/c^3/(a^2*c+d)^(5/2)+1/15*a*(7*
a^2*c+4*d)/c^2/(a^2*c+d)^2/(d*x^2+c)^(1/2)+8/15*x*arccoth(a*x)/c^3/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.72, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {198, 197, 6124, 6820, 12, 6847, 911, 1275, 214} \begin {gather*} \frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/(c + d*x^2)^(7/2),x]

[Out]

a/(15*c*(a^2*c + d)*(c + d*x^2)^(3/2)) + (a*(7*a^2*c + 4*d))/(15*c^2*(a^2*c + d)^2*Sqrt[c + d*x^2]) + (x*ArcCo
th[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCoth[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcCoth[a*x])/(15*c^3*
Sqrt[c + d*x^2]) - ((15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]])/(15*c^3*(a
^2*c + d)^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

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 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 6124

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^
2)^q, x]}, Dist[a + b*ArcCoth[c*x], u, x] - Dist[b*c, Int[u/(1 - c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x
] && (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-a \int \frac {\frac {x}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {c+d x^2}}}{1-a^2 x^2} \, dx\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{15 c^3 \left (1-a^2 x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\left (1-a^2 x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3}\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \frac {15 c^2+20 c d x+8 d^2 x^2}{\left (1-a^2 x\right ) (c+d x)^{5/2}} \, dx,x,x^2\right )}{30 c^3}\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \frac {3 c^2+4 c x^2+8 x^4}{x^4 \left (\frac {a^2 c+d}{d}-\frac {a^2 x^2}{d}\right )} \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 d}\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \left (\frac {3 c^2 d}{\left (a^2 c+d\right ) x^4}+\frac {c d \left (7 a^2 c+4 d\right )}{\left (a^2 c+d\right )^2 x^2}+\frac {d \left (15 a^4 c^2+20 a^2 c d+8 d^2\right )}{\left (a^2 c+d\right )^2 \left (a^2 c+d-a^2 x^2\right )}\right ) \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 d}\\ &=\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (a \left (15 a^4 c^2+20 a^2 c d+8 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^2 c+d-a^2 x^2} \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 \left (a^2 c+d\right )^2}\\ &=\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 329, normalized size = 1.64 \begin {gather*} \frac {2 a c \sqrt {a^2 c+d} \left (c+d x^2\right ) \left (d \left (5 c+4 d x^2\right )+a^2 c \left (8 c+7 d x^2\right )\right )+2 \left (a^2 c+d\right )^{5/2} x \left (15 c^2+20 c d x^2+8 d^2 x^4\right ) \coth ^{-1}(a x)+\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log (1-a x)+\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log (1+a x)-\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log \left (a c-d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )-\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log \left (a c+d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )}{30 c^3 \left (a^2 c+d\right )^{5/2} \left (c+d x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x]/(c + d*x^2)^(7/2),x]

[Out]

(2*a*c*Sqrt[a^2*c + d]*(c + d*x^2)*(d*(5*c + 4*d*x^2) + a^2*c*(8*c + 7*d*x^2)) + 2*(a^2*c + d)^(5/2)*x*(15*c^2
+ 20*c*d*x^2 + 8*d^2*x^4)*ArcCoth[a*x] + (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[1 - a*x] + (
15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)*Log[1 + a*x] - (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2
)^(5/2)*Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]] - (15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*(c + d*x^2)^(5/2)
*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(30*c^3*(a^2*c + d)^(5/2)*(c + d*x^2)^(5/2))

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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)/(d*x^2+c)^(7/2),x)

[Out]

int(arccoth(a*x)/(d*x^2+c)^(7/2),x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (172) = 344\).
time = 0.48, size = 401, normalized size = 2.00 \begin {gather*} \frac {1}{30} \, a {\left (\frac {\frac {3 \, a^{3} d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} \sqrt {a^{2} c + d}} + \frac {2 \, {\left (3 \, {\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}\right )}}{{\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}}{d} + \frac {4 \, {\left (\frac {a d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{2} c^{3} + c^{2} d\right )} \sqrt {a^{2} c + d}} + \frac {2 \, d}{{\left (a^{2} c^{3} + c^{2} d\right )} \sqrt {d x^{2} + c}}\right )}}{d} + \frac {8 \, \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{\sqrt {a^{2} c + d} a c^{3}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {d x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {arcoth}\left (a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/(d*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

1/30*a*((3*a^3*d*log((sqrt(d*x^2 + c)*a^2 - sqrt(a^2*c + d)*a)/(sqrt(d*x^2 + c)*a^2 + sqrt(a^2*c + d)*a))/((a^
4*c^3 + 2*a^2*c^2*d + c*d^2)*sqrt(a^2*c + d)) + 2*(3*(d*x^2 + c)*a^2*d + a^2*c*d + d^2)/((a^4*c^3 + 2*a^2*c^2*
d + c*d^2)*(d*x^2 + c)^(3/2)))/d + 4*(a*d*log((sqrt(d*x^2 + c)*a^2 - sqrt(a^2*c + d)*a)/(sqrt(d*x^2 + c)*a^2 +
sqrt(a^2*c + d)*a))/((a^2*c^3 + c^2*d)*sqrt(a^2*c + d)) + 2*d/((a^2*c^3 + c^2*d)*sqrt(d*x^2 + c)))/d + 8*log(
(sqrt(d*x^2 + c)*a^2 - sqrt(a^2*c + d)*a)/(sqrt(d*x^2 + c)*a^2 + sqrt(a^2*c + d)*a))/(sqrt(a^2*c + d)*a*c^3))
+ 1/15*(8*x/(sqrt(d*x^2 + c)*c^3) + 4*x/((d*x^2 + c)^(3/2)*c^2) + 3*x/((d*x^2 + c)^(5/2)*c))*arccoth(a*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (172) = 344\).
time = 0.43, size = 1278, normalized size = 6.39 \begin {gather*} \left [\frac {{\left (15 \, a^{4} c^{5} + 20 \, a^{2} c^{4} d + {\left (15 \, a^{4} c^{2} d^{3} + 20 \, a^{2} c d^{4} + 8 \, d^{5}\right )} x^{6} + 8 \, c^{3} d^{2} + 3 \, {\left (15 \, a^{4} c^{3} d^{2} + 20 \, a^{2} c^{2} d^{3} + 8 \, c d^{4}\right )} x^{4} + 3 \, {\left (15 \, a^{4} c^{4} d + 20 \, a^{2} c^{3} d^{2} + 8 \, c^{2} d^{3}\right )} x^{2}\right )} \sqrt {a^{2} c + d} \log \left (\frac {a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} + 8 \, a^{2} c d + 2 \, {\left (4 \, a^{4} c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} d x^{2} + 2 \, a^{3} c + a d\right )} \sqrt {a^{2} c + d} \sqrt {d x^{2} + c} + d^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}\right ) + 2 \, {\left (16 \, a^{5} c^{5} + 26 \, a^{3} c^{4} d + 10 \, a c^{3} d^{2} + 2 \, {\left (7 \, a^{5} c^{3} d^{2} + 11 \, a^{3} c^{2} d^{3} + 4 \, a c d^{4}\right )} x^{4} + 6 \, {\left (5 \, a^{5} c^{4} d + 8 \, a^{3} c^{3} d^{2} + 3 \, a c^{2} d^{3}\right )} x^{2} + {\left (8 \, {\left (a^{6} c^{3} d^{2} + 3 \, a^{4} c^{2} d^{3} + 3 \, a^{2} c d^{4} + d^{5}\right )} x^{5} + 20 \, {\left (a^{6} c^{4} d + 3 \, a^{4} c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{3} + 15 \, {\left (a^{6} c^{5} + 3 \, a^{4} c^{4} d + 3 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\right )} x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {d x^{2} + c}}{60 \, {\left (a^{6} c^{9} + 3 \, a^{4} c^{8} d + 3 \, a^{2} c^{7} d^{2} + c^{6} d^{3} + {\left (a^{6} c^{6} d^{3} + 3 \, a^{4} c^{5} d^{4} + 3 \, a^{2} c^{4} d^{5} + c^{3} d^{6}\right )} x^{6} + 3 \, {\left (a^{6} c^{7} d^{2} + 3 \, a^{4} c^{6} d^{3} + 3 \, a^{2} c^{5} d^{4} + c^{4} d^{5}\right )} x^{4} + 3 \, {\left (a^{6} c^{8} d + 3 \, a^{4} c^{7} d^{2} + 3 \, a^{2} c^{6} d^{3} + c^{5} d^{4}\right )} x^{2}\right )}}, \frac {{\left (15 \, a^{4} c^{5} + 20 \, a^{2} c^{4} d + {\left (15 \, a^{4} c^{2} d^{3} + 20 \, a^{2} c d^{4} + 8 \, d^{5}\right )} x^{6} + 8 \, c^{3} d^{2} + 3 \, {\left (15 \, a^{4} c^{3} d^{2} + 20 \, a^{2} c^{2} d^{3} + 8 \, c d^{4}\right )} x^{4} + 3 \, {\left (15 \, a^{4} c^{4} d + 20 \, a^{2} c^{3} d^{2} + 8 \, c^{2} d^{3}\right )} x^{2}\right )} \sqrt {-a^{2} c - d} \arctan \left (\frac {{\left (a^{2} d x^{2} + 2 \, a^{2} c + d\right )} \sqrt {-a^{2} c - d} \sqrt {d x^{2} + c}}{2 \, {\left (a^{3} c^{2} + a c d + {\left (a^{3} c d + a d^{2}\right )} x^{2}\right )}}\right ) + {\left (16 \, a^{5} c^{5} + 26 \, a^{3} c^{4} d + 10 \, a c^{3} d^{2} + 2 \, {\left (7 \, a^{5} c^{3} d^{2} + 11 \, a^{3} c^{2} d^{3} + 4 \, a c d^{4}\right )} x^{4} + 6 \, {\left (5 \, a^{5} c^{4} d + 8 \, a^{3} c^{3} d^{2} + 3 \, a c^{2} d^{3}\right )} x^{2} + {\left (8 \, {\left (a^{6} c^{3} d^{2} + 3 \, a^{4} c^{2} d^{3} + 3 \, a^{2} c d^{4} + d^{5}\right )} x^{5} + 20 \, {\left (a^{6} c^{4} d + 3 \, a^{4} c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{3} + 15 \, {\left (a^{6} c^{5} + 3 \, a^{4} c^{4} d + 3 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\right )} x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {d x^{2} + c}}{30 \, {\left (a^{6} c^{9} + 3 \, a^{4} c^{8} d + 3 \, a^{2} c^{7} d^{2} + c^{6} d^{3} + {\left (a^{6} c^{6} d^{3} + 3 \, a^{4} c^{5} d^{4} + 3 \, a^{2} c^{4} d^{5} + c^{3} d^{6}\right )} x^{6} + 3 \, {\left (a^{6} c^{7} d^{2} + 3 \, a^{4} c^{6} d^{3} + 3 \, a^{2} c^{5} d^{4} + c^{4} d^{5}\right )} x^{4} + 3 \, {\left (a^{6} c^{8} d + 3 \, a^{4} c^{7} d^{2} + 3 \, a^{2} c^{6} d^{3} + c^{5} d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/(d*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

[1/60*((15*a^4*c^5 + 20*a^2*c^4*d + (15*a^4*c^2*d^3 + 20*a^2*c*d^4 + 8*d^5)*x^6 + 8*c^3*d^2 + 3*(15*a^4*c^3*d^
2 + 20*a^2*c^2*d^3 + 8*c*d^4)*x^4 + 3*(15*a^4*c^4*d + 20*a^2*c^3*d^2 + 8*c^2*d^3)*x^2)*sqrt(a^2*c + d)*log((a^
4*d^2*x^4 + 8*a^4*c^2 + 8*a^2*c*d + 2*(4*a^4*c*d + 3*a^2*d^2)*x^2 - 4*(a^3*d*x^2 + 2*a^3*c + a*d)*sqrt(a^2*c +
d)*sqrt(d*x^2 + c) + d^2)/(a^4*x^4 - 2*a^2*x^2 + 1)) + 2*(16*a^5*c^5 + 26*a^3*c^4*d + 10*a*c^3*d^2 + 2*(7*a^5
*c^3*d^2 + 11*a^3*c^2*d^3 + 4*a*c*d^4)*x^4 + 6*(5*a^5*c^4*d + 8*a^3*c^3*d^2 + 3*a*c^2*d^3)*x^2 + (8*(a^6*c^3*d
^2 + 3*a^4*c^2*d^3 + 3*a^2*c*d^4 + d^5)*x^5 + 20*(a^6*c^4*d + 3*a^4*c^3*d^2 + 3*a^2*c^2*d^3 + c*d^4)*x^3 + 15*
(a^6*c^5 + 3*a^4*c^4*d + 3*a^2*c^3*d^2 + c^2*d^3)*x)*log((a*x + 1)/(a*x - 1)))*sqrt(d*x^2 + c))/(a^6*c^9 + 3*a
^4*c^8*d + 3*a^2*c^7*d^2 + c^6*d^3 + (a^6*c^6*d^3 + 3*a^4*c^5*d^4 + 3*a^2*c^4*d^5 + c^3*d^6)*x^6 + 3*(a^6*c^7*
d^2 + 3*a^4*c^6*d^3 + 3*a^2*c^5*d^4 + c^4*d^5)*x^4 + 3*(a^6*c^8*d + 3*a^4*c^7*d^2 + 3*a^2*c^6*d^3 + c^5*d^4)*x
^2), 1/30*((15*a^4*c^5 + 20*a^2*c^4*d + (15*a^4*c^2*d^3 + 20*a^2*c*d^4 + 8*d^5)*x^6 + 8*c^3*d^2 + 3*(15*a^4*c^
3*d^2 + 20*a^2*c^2*d^3 + 8*c*d^4)*x^4 + 3*(15*a^4*c^4*d + 20*a^2*c^3*d^2 + 8*c^2*d^3)*x^2)*sqrt(-a^2*c - d)*ar
ctan(1/2*(a^2*d*x^2 + 2*a^2*c + d)*sqrt(-a^2*c - d)*sqrt(d*x^2 + c)/(a^3*c^2 + a*c*d + (a^3*c*d + a*d^2)*x^2))
+ (16*a^5*c^5 + 26*a^3*c^4*d + 10*a*c^3*d^2 + 2*(7*a^5*c^3*d^2 + 11*a^3*c^2*d^3 + 4*a*c*d^4)*x^4 + 6*(5*a^5*c
^4*d + 8*a^3*c^3*d^2 + 3*a*c^2*d^3)*x^2 + (8*(a^6*c^3*d^2 + 3*a^4*c^2*d^3 + 3*a^2*c*d^4 + d^5)*x^5 + 20*(a^6*c
^4*d + 3*a^4*c^3*d^2 + 3*a^2*c^2*d^3 + c*d^4)*x^3 + 15*(a^6*c^5 + 3*a^4*c^4*d + 3*a^2*c^3*d^2 + c^2*d^3)*x)*lo
g((a*x + 1)/(a*x - 1)))*sqrt(d*x^2 + c))/(a^6*c^9 + 3*a^4*c^8*d + 3*a^2*c^7*d^2 + c^6*d^3 + (a^6*c^6*d^3 + 3*a
^4*c^5*d^4 + 3*a^2*c^4*d^5 + c^3*d^6)*x^6 + 3*(a^6*c^7*d^2 + 3*a^4*c^6*d^3 + 3*a^2*c^5*d^4 + c^4*d^5)*x^4 + 3*
(a^6*c^8*d + 3*a^4*c^7*d^2 + 3*a^2*c^6*d^3 + c^5*d^4)*x^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(a*x)/(d*x**2+c)**(7/2),x)

[Out]

Integral(acoth(a*x)/(c + d*x**2)**(7/2), x)

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Giac [A]
time = 0.43, size = 226, normalized size = 1.13 \begin {gather*} \frac {1}{15} \, a {\left (\frac {{\left (15 \, a^{4} c^{2} + 20 \, a^{2} c d + 8 \, d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c - d}}\right )}{{\left (a^{4} c^{5} + 2 \, a^{2} c^{4} d + c^{3} d^{2}\right )} \sqrt {-a^{2} c - d} a} + \frac {7 \, {\left (d x^{2} + c\right )} a^{2} c + a^{2} c^{2} + 4 \, {\left (d x^{2} + c\right )} d + c d}{{\left (a^{4} c^{4} + 2 \, a^{2} c^{3} d + c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\right )} + \frac {{\left (4 \, x^{2} {\left (\frac {2 \, d^{2} x^{2}}{c^{3}} + \frac {5 \, d}{c^{2}}\right )} + \frac {15}{c}\right )} x \log \left (-\frac {\frac {1}{a x} + 1}{\frac {1}{a x} - 1}\right )}{30 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/(d*x^2+c)^(7/2),x, algorithm="giac")

[Out]

1/15*a*((15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*arctan(sqrt(d*x^2 + c)*a/sqrt(-a^2*c - d))/((a^4*c^5 + 2*a^2*c^4*d +
c^3*d^2)*sqrt(-a^2*c - d)*a) + (7*(d*x^2 + c)*a^2*c + a^2*c^2 + 4*(d*x^2 + c)*d + c*d)/((a^4*c^4 + 2*a^2*c^3*
d + c^2*d^2)*(d*x^2 + c)^(3/2))) + 1/30*(4*x^2*(2*d^2*x^2/c^3 + 5*d/c^2) + 15/c)*x*log(-(1/(a*x) + 1)/(1/(a*x)
- 1))/(d*x^2 + c)^(5/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a*x)/(c + d*x^2)^(7/2),x)

[Out]

int(acoth(a*x)/(c + d*x^2)^(7/2), x)

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