3.47 \(\int \frac {\coth ^{-1}(a x)}{(c+d x^2)^{9/2}} \, dx\)

Optimal. Leaf size=283 \[ \frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]

[Out]

1/35*a/c/(a^2*c+d)/(d*x^2+c)^(5/2)+1/105*a*(11*a^2*c+6*d)/c^2/(a^2*c+d)^2/(d*x^2+c)^(3/2)+1/7*x*arccoth(a*x)/c
/(d*x^2+c)^(7/2)+6/35*x*arccoth(a*x)/c^2/(d*x^2+c)^(5/2)+8/35*x*arccoth(a*x)/c^3/(d*x^2+c)^(3/2)-1/35*(35*a^6*
c^3+70*a^4*c^2*d+56*a^2*c*d^2+16*d^3)*arctanh(a*(d*x^2+c)^(1/2)/(a^2*c+d)^(1/2))/c^4/(a^2*c+d)^(7/2)+1/35*a*(1
9*a^4*c^2+22*a^2*c*d+8*d^2)/c^3/(a^2*c+d)^3/(d*x^2+c)^(1/2)+16/35*x*arccoth(a*x)/c^4/(d*x^2+c)^(1/2)

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Rubi [A]  time = 1.34, antiderivative size = 283, 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 = {192, 191, 5977, 6688, 12, 6715, 1619, 63, 208} \[ \frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}-\frac {\left (70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}}+\frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x]/(c + d*x^2)^(9/2),x]

[Out]

a/(35*c*(a^2*c + d)*(c + d*x^2)^(5/2)) + (a*(11*a^2*c + 6*d))/(105*c^2*(a^2*c + d)^2*(c + d*x^2)^(3/2)) + (a*(
19*a^4*c^2 + 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c + d)^3*Sqrt[c + d*x^2]) + (x*ArcCoth[a*x])/(7*c*(c + d*x^2)^(
7/2)) + (6*x*ArcCoth[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcCoth[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*
ArcCoth[a*x])/(35*c^4*Sqrt[c + d*x^2]) - ((35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*ArcTanh[(a*Sqrt[
c + d*x^2])/Sqrt[a^2*c + d]])/(35*c^4*(a^2*c + d)^(7/2))

Rule 12

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 191

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 192

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 208

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

Rule 1619

Int[((Px_)*((c_.) + (d_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[c + d*x],
(Px*(c + d*x)^(n + 1/2))/(a + b*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[n + 1/2, 0] &
& GtQ[Expon[Px, x], 2]

Rule 5977

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 6688

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

Rule 6715

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 )^{9/2}} \, dx &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-a \int \frac {\frac {x}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x}{35 c^4 \sqrt {c+d x^2}}}{1-a^2 x^2} \, dx\\ &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-a \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{35 c^4 \left (1-a^2 x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx\\ &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {a \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\left (1-a^2 x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4}\\ &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {a \operatorname {Subst}\left (\int \frac {35 c^3+70 c^2 d x+56 c d^2 x^2+16 d^3 x^3}{\left (1-a^2 x\right ) (c+d x)^{7/2}} \, dx,x,x^2\right )}{70 c^4}\\ &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {a \operatorname {Subst}\left (\int \left (\frac {5 c^3 d}{\left (a^2 c+d\right ) (c+d x)^{7/2}}+\frac {c^2 d \left (11 a^2 c+6 d\right )}{\left (a^2 c+d\right )^2 (c+d x)^{5/2}}+\frac {c d \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{\left (a^2 c+d\right )^3 (c+d x)^{3/2}}+\frac {-35 a^6 c^3-70 a^4 c^2 d-56 a^2 c d^2-16 d^3}{\left (a^2 c+d\right )^3 \left (-1+a^2 x\right ) \sqrt {c+d x}}\right ) \, dx,x,x^2\right )}{70 c^4}\\ &=\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {\left (a \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+a^2 x\right ) \sqrt {c+d x}} \, dx,x,x^2\right )}{70 c^4 \left (a^2 c+d\right )^3}\\ &=\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {\left (a \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\frac {a^2 c}{d}+\frac {a^2 x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{35 c^4 d \left (a^2 c+d\right )^3}\\ &=\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.94, size = 431, normalized size = 1.52 \[ \frac {6 x \left (a^2 c+d\right )^{7/2} \coth ^{-1}(a x) \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )+2 a c \sqrt {a^2 c+d} \left (c+d x^2\right ) \left (3 c^2 \left (a^2 c+d\right )^2+c \left (11 a^2 c+6 d\right ) \left (a^2 c+d\right ) \left (c+d x^2\right )+3 \left (19 a^4 c^2+22 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^2\right )+3 \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \log (1-a x) \left (c+d x^2\right )^{7/2}+3 \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \log (a x+1) \left (c+d x^2\right )^{7/2}-3 \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \left (c+d x^2\right )^{7/2} \log \left (\sqrt {a^2 c+d} \sqrt {c+d x^2}+a c-d x\right )-3 \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \left (c+d x^2\right )^{7/2} \log \left (\sqrt {a^2 c+d} \sqrt {c+d x^2}+a c+d x\right )}{210 c^4 \left (a^2 c+d\right )^{7/2} \left (c+d x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x]/(c + d*x^2)^(9/2),x]

[Out]

(2*a*c*Sqrt[a^2*c + d]*(c + d*x^2)*(3*c^2*(a^2*c + d)^2 + c*(a^2*c + d)*(11*a^2*c + 6*d)*(c + d*x^2) + 3*(19*a
^4*c^2 + 22*a^2*c*d + 8*d^2)*(c + d*x^2)^2) + 6*(a^2*c + d)^(7/2)*x*(35*c^3 + 70*c^2*d*x^2 + 56*c*d^2*x^4 + 16
*d^3*x^6)*ArcCoth[a*x] + 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[1 - a*x]
+ 3*(35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[1 + a*x] - 3*(35*a^6*c^3 + 70*a^
4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]] - 3*(35*a^
6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*(c + d*x^2)^(7/2)*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2
]])/(210*c^4*(a^2*c + d)^(7/2)*(c + d*x^2)^(7/2))

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fricas [B]  time = 0.93, size = 2004, normalized size = 7.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/(d*x^2+c)^(9/2),x, algorithm="fricas")

[Out]

[1/420*(3*(35*a^6*c^7 + 70*a^4*c^6*d + 56*a^2*c^5*d^2 + (35*a^6*c^3*d^4 + 70*a^4*c^2*d^5 + 56*a^2*c*d^6 + 16*d
^7)*x^8 + 16*c^4*d^3 + 4*(35*a^6*c^4*d^3 + 70*a^4*c^3*d^4 + 56*a^2*c^2*d^5 + 16*c*d^6)*x^6 + 6*(35*a^6*c^5*d^2
+ 70*a^4*c^4*d^3 + 56*a^2*c^3*d^4 + 16*c^2*d^5)*x^4 + 4*(35*a^6*c^6*d + 70*a^4*c^5*d^2 + 56*a^2*c^4*d^3 + 16*
c^3*d^4)*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*(142*a^7*c^7 +
320*a^5*c^6*d + 244*a^3*c^5*d^2 + 66*a*c^4*d^3 + 6*(19*a^7*c^4*d^3 + 41*a^5*c^3*d^4 + 30*a^3*c^2*d^5 + 8*a*c*d
^6)*x^6 + 2*(182*a^7*c^5*d^2 + 397*a^5*c^4*d^3 + 293*a^3*c^3*d^4 + 78*a*c^2*d^5)*x^4 + 2*(196*a^7*c^6*d + 434*
a^5*c^5*d^2 + 325*a^3*c^4*d^3 + 87*a*c^3*d^4)*x^2 + 3*(16*(a^8*c^4*d^3 + 4*a^6*c^3*d^4 + 6*a^4*c^2*d^5 + 4*a^2
*c*d^6 + d^7)*x^7 + 56*(a^8*c^5*d^2 + 4*a^6*c^4*d^3 + 6*a^4*c^3*d^4 + 4*a^2*c^2*d^5 + c*d^6)*x^5 + 70*(a^8*c^6
*d + 4*a^6*c^5*d^2 + 6*a^4*c^4*d^3 + 4*a^2*c^3*d^4 + c^2*d^5)*x^3 + 35*(a^8*c^7 + 4*a^6*c^6*d + 6*a^4*c^5*d^2
+ 4*a^2*c^4*d^3 + c^3*d^4)*x)*log((a*x + 1)/(a*x - 1)))*sqrt(d*x^2 + c))/(a^8*c^12 + 4*a^6*c^11*d + 6*a^4*c^10
*d^2 + 4*a^2*c^9*d^3 + c^8*d^4 + (a^8*c^8*d^4 + 4*a^6*c^7*d^5 + 6*a^4*c^6*d^6 + 4*a^2*c^5*d^7 + c^4*d^8)*x^8 +
4*(a^8*c^9*d^3 + 4*a^6*c^8*d^4 + 6*a^4*c^7*d^5 + 4*a^2*c^6*d^6 + c^5*d^7)*x^6 + 6*(a^8*c^10*d^2 + 4*a^6*c^9*d
^3 + 6*a^4*c^8*d^4 + 4*a^2*c^7*d^5 + c^6*d^6)*x^4 + 4*(a^8*c^11*d + 4*a^6*c^10*d^2 + 6*a^4*c^9*d^3 + 4*a^2*c^8
*d^4 + c^7*d^5)*x^2), 1/210*(3*(35*a^6*c^7 + 70*a^4*c^6*d + 56*a^2*c^5*d^2 + (35*a^6*c^3*d^4 + 70*a^4*c^2*d^5
+ 56*a^2*c*d^6 + 16*d^7)*x^8 + 16*c^4*d^3 + 4*(35*a^6*c^4*d^3 + 70*a^4*c^3*d^4 + 56*a^2*c^2*d^5 + 16*c*d^6)*x^
6 + 6*(35*a^6*c^5*d^2 + 70*a^4*c^4*d^3 + 56*a^2*c^3*d^4 + 16*c^2*d^5)*x^4 + 4*(35*a^6*c^6*d + 70*a^4*c^5*d^2 +
56*a^2*c^4*d^3 + 16*c^3*d^4)*x^2)*sqrt(-a^2*c - d)*arctan(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)) + (142*a^7*c^7 + 320*a^5*c^6*d + 244*a^3*c^5*d^2 + 66*a
*c^4*d^3 + 6*(19*a^7*c^4*d^3 + 41*a^5*c^3*d^4 + 30*a^3*c^2*d^5 + 8*a*c*d^6)*x^6 + 2*(182*a^7*c^5*d^2 + 397*a^5
*c^4*d^3 + 293*a^3*c^3*d^4 + 78*a*c^2*d^5)*x^4 + 2*(196*a^7*c^6*d + 434*a^5*c^5*d^2 + 325*a^3*c^4*d^3 + 87*a*c
^3*d^4)*x^2 + 3*(16*(a^8*c^4*d^3 + 4*a^6*c^3*d^4 + 6*a^4*c^2*d^5 + 4*a^2*c*d^6 + d^7)*x^7 + 56*(a^8*c^5*d^2 +
4*a^6*c^4*d^3 + 6*a^4*c^3*d^4 + 4*a^2*c^2*d^5 + c*d^6)*x^5 + 70*(a^8*c^6*d + 4*a^6*c^5*d^2 + 6*a^4*c^4*d^3 + 4
*a^2*c^3*d^4 + c^2*d^5)*x^3 + 35*(a^8*c^7 + 4*a^6*c^6*d + 6*a^4*c^5*d^2 + 4*a^2*c^4*d^3 + c^3*d^4)*x)*log((a*x
+ 1)/(a*x - 1)))*sqrt(d*x^2 + c))/(a^8*c^12 + 4*a^6*c^11*d + 6*a^4*c^10*d^2 + 4*a^2*c^9*d^3 + c^8*d^4 + (a^8*
c^8*d^4 + 4*a^6*c^7*d^5 + 6*a^4*c^6*d^6 + 4*a^2*c^5*d^7 + c^4*d^8)*x^8 + 4*(a^8*c^9*d^3 + 4*a^6*c^8*d^4 + 6*a^
4*c^7*d^5 + 4*a^2*c^6*d^6 + c^5*d^7)*x^6 + 6*(a^8*c^10*d^2 + 4*a^6*c^9*d^3 + 6*a^4*c^8*d^4 + 4*a^2*c^7*d^5 + c
^6*d^6)*x^4 + 4*(a^8*c^11*d + 4*a^6*c^10*d^2 + 6*a^4*c^9*d^3 + 4*a^2*c^8*d^4 + c^7*d^5)*x^2)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/(d*x^2+c)^(9/2),x, algorithm="giac")

[Out]

integrate(arccoth(a*x)/(d*x^2 + c)^(9/2), x)

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maple [F]  time = 0.94, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x)/(d*x^2+c)^(9/2),x)

[Out]

int(arccoth(a*x)/(d*x^2+c)^(9/2),x)

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maxima [B]  time = 0.43, size = 639, normalized size = 2.26 \[ \frac {1}{210} \, a {\left (\frac {\frac {15 \, a^{5} d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{6} c^{4} + 3 \, a^{4} c^{3} d + 3 \, a^{2} c^{2} d^{2} + c d^{3}\right )} \sqrt {a^{2} c + d}} + \frac {2 \, {\left (15 \, {\left (d x^{2} + c\right )}^{2} a^{4} d + 3 \, a^{4} c^{2} d + 6 \, a^{2} c d^{2} + 3 \, d^{3} + 5 \, {\left (a^{4} c d + a^{2} d^{2}\right )} {\left (d x^{2} + c\right )}\right )}}{{\left (a^{6} c^{4} + 3 \, a^{4} c^{3} d + 3 \, a^{2} c^{2} d^{2} + c d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}}{d} + \frac {6 \, {\left (\frac {3 \, a^{3} d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{4} c^{4} + 2 \, a^{2} c^{3} d + c^{2} d^{2}\right )} \sqrt {a^{2} c + d}} + \frac {2 \, {\left (3 \, {\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}\right )}}{{\left (a^{4} c^{4} + 2 \, a^{2} c^{3} d + c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\right )}}{d} + \frac {24 \, {\left (\frac {a d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{2} c^{4} + c^{3} d\right )} \sqrt {a^{2} c + d}} + \frac {2 \, d}{{\left (a^{2} c^{4} + c^{3} d\right )} \sqrt {d x^{2} + c}}\right )}}{d} + \frac {48 \, \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{\sqrt {a^{2} c + d} a c^{4}}\right )} + \frac {1}{35} \, {\left (\frac {16 \, x}{\sqrt {d x^{2} + c} c^{4}} + \frac {8 \, x}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{3}} + \frac {6 \, x}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} c^{2}} + \frac {5 \, x}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} c}\right )} \operatorname {arcoth}\left (a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x)/(d*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

1/210*a*((15*a^5*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^6*c^4 + 3*a^4*c^3*d + 3*a^2*c^2*d^2 + c*d^3)*sqrt(a^2*c + d)) + 2*(15*(d*x^2 + c)^2*a^4*d + 3*a^4*c^2*d + 6*
a^2*c*d^2 + 3*d^3 + 5*(a^4*c*d + a^2*d^2)*(d*x^2 + c))/((a^6*c^4 + 3*a^4*c^3*d + 3*a^2*c^2*d^2 + c*d^3)*(d*x^2
+ c)^(5/2)))/d + 6*(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^4 + 2*a^2*c^3*d + c^2*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^
4 + 2*a^2*c^3*d + c^2*d^2)*(d*x^2 + c)^(3/2)))/d + 24*(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^4 + c^3*d)*sqrt(a^2*c + d)) + 2*d/((a^2*c^4 + c^3*d)*sqrt(d*x^2
+ c)))/d + 48*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^4)) + 1/35*(16*x/(sqrt(d*x^2 + c)*c^4) + 8*x/((d*x^2 + c)^(3/2)*c^3) + 6*x/((d*x^2 + c)^(5/2)*c^
2) + 5*x/((d*x^2 + c)^(7/2)*c))*arccoth(a*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acoth(a*x)/(c + d*x^2)^(9/2),x)

[Out]

int(acoth(a*x)/(c + d*x^2)^(9/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {9}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(a*x)/(d*x**2+c)**(9/2),x)

[Out]

Integral(acoth(a*x)/(c + d*x**2)**(9/2), x)

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