3.1.89 \(\int \frac {a+b \tanh ^{-1}(c x^2)}{(d x)^{9/2}} \, dx\) [89]

Optimal. Leaf size=317 \[ -\frac {8 b c}{21 d^3 (d x)^{3/2}}+\frac {2 b c^{7/4} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}+\frac {\sqrt {2} b c^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}-\frac {\sqrt {2} b c^{7/4} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {2 b c^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}+\frac {b c^{7/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} d^{9/2}}-\frac {b c^{7/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} d^{9/2}} \]

[Out]

-8/21*b*c/d^3/(d*x)^(3/2)+2/7*b*c^(7/4)*arctan(c^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(9/2)-2/7*(a+b*arctanh(c*x^2))/d
/(d*x)^(7/2)+2/7*b*c^(7/4)*arctanh(c^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(9/2)+1/14*b*c^(7/4)*ln(d^(1/2)+x*c^(1/2)*d^
(1/2)-c^(1/4)*2^(1/2)*(d*x)^(1/2))/d^(9/2)*2^(1/2)-1/14*b*c^(7/4)*ln(d^(1/2)+x*c^(1/2)*d^(1/2)+c^(1/4)*2^(1/2)
*(d*x)^(1/2))/d^(9/2)*2^(1/2)-1/7*b*c^(7/4)*arctan(-1+c^(1/4)*2^(1/2)*(d*x)^(1/2)/d^(1/2))*2^(1/2)/d^(9/2)-1/7
*b*c^(7/4)*arctan(1+c^(1/4)*2^(1/2)*(d*x)^(1/2)/d^(1/2))*2^(1/2)/d^(9/2)

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Rubi [A]
time = 0.18, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6049, 331, 335, 307, 217, 1179, 642, 1176, 631, 210, 218, 214, 211} \begin {gather*} -\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {2 b c^{7/4} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}+\frac {\sqrt {2} b c^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}-\frac {\sqrt {2} b c^{7/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{7 d^{9/2}}+\frac {b c^{7/4} \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{7 \sqrt {2} d^{9/2}}-\frac {b c^{7/4} \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{7 \sqrt {2} d^{9/2}}+\frac {2 b c^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}-\frac {8 b c}{21 d^3 (d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^2])/(d*x)^(9/2),x]

[Out]

(-8*b*c)/(21*d^3*(d*x)^(3/2)) + (2*b*c^(7/4)*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) + (Sqrt[2]*b*c^(
7/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) - (Sqrt[2]*b*c^(7/4)*ArcTan[1 + (Sqrt[2]*c^(
1/4)*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) - (2*(a + b*ArcTanh[c*x^2]))/(7*d*(d*x)^(7/2)) + (2*b*c^(7/4)*ArcTanh[(c
^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) + (b*c^(7/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*
x]])/(7*Sqrt[2]*d^(9/2)) - (b*c^(7/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(7*Sqrt[2]
*d^(9/2))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 307

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b
, 2]]}, Dist[s/(2*b), Int[x^(m - n/2)/(r + s*x^(n/2)), x], x] - Dist[s/(2*b), Int[x^(m - n/2)/(r - s*x^(n/2)),
x], x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && LtQ[m, n] &&  !GtQ[a/b, 0]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 6049

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))*((d_)*(x_))^(m_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcTan
h[c*x^n])/(d*(m + 1))), x] - Dist[b*c*(n/(d^n*(m + 1))), Int[(d*x)^(m + n)/(1 - c^2*x^(2*n)), x], x] /; FreeQ[
{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^2\right )}{(d x)^{9/2}} \, dx &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {(4 b c) \int \frac {x}{(d x)^{7/2} \left (1-c^2 x^4\right )} \, dx}{7 d}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {(4 b c) \int \frac {1}{(d x)^{5/2} \left (1-c^2 x^4\right )} \, dx}{7 d^2}\\ &=-\frac {8 b c}{21 d^3 (d x)^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {\left (4 b c^3\right ) \int \frac {(d x)^{3/2}}{1-c^2 x^4} \, dx}{7 d^6}\\ &=-\frac {8 b c}{21 d^3 (d x)^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {\left (8 b c^3\right ) \text {Subst}\left (\int \frac {x^4}{1-\frac {c^2 x^8}{d^4}} \, dx,x,\sqrt {d x}\right )}{7 d^7}\\ &=-\frac {8 b c}{21 d^3 (d x)^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {\left (4 b c^2\right ) \text {Subst}\left (\int \frac {1}{d^2-c x^4} \, dx,x,\sqrt {d x}\right )}{7 d^3}-\frac {\left (4 b c^2\right ) \text {Subst}\left (\int \frac {1}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{7 d^3}\\ &=-\frac {8 b c}{21 d^3 (d x)^{3/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{7 d^4}+\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{7 d^4}-\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \frac {d-\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{7 d^4}-\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \frac {d+\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{7 d^4}\\ &=-\frac {8 b c}{21 d^3 (d x)^{3/2}}+\frac {2 b c^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {2 b c^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}+\frac {\left (b c^{7/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}+2 x}{-\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{7 \sqrt {2} d^{9/2}}+\frac {\left (b c^{7/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt [4]{c}}-2 x}{-\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {d x}\right )}{7 \sqrt {2} d^{9/2}}-\frac {\left (b c^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{7 d^4}-\frac {\left (b c^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d x}\right )}{7 d^4}\\ &=-\frac {8 b c}{21 d^3 (d x)^{3/2}}+\frac {2 b c^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {2 b c^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}+\frac {b c^{7/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} d^{9/2}}-\frac {b c^{7/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} d^{9/2}}-\frac {\left (\sqrt {2} b c^{7/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}+\frac {\left (\sqrt {2} b c^{7/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}\\ &=-\frac {8 b c}{21 d^3 (d x)^{3/2}}+\frac {2 b c^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}+\frac {\sqrt {2} b c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}-\frac {\sqrt {2} b c^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}-\frac {2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{7 d (d x)^{7/2}}+\frac {2 b c^{7/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{7 d^{9/2}}+\frac {b c^{7/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} d^{9/2}}-\frac {b c^{7/4} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{7 \sqrt {2} d^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 281, normalized size = 0.89 \begin {gather*} \frac {\sqrt {d x} \left (-12 a-16 b c x^2+6 \sqrt {2} b c^{7/4} x^{7/2} \text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-6 \sqrt {2} b c^{7/4} x^{7/2} \text {ArcTan}\left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+12 b c^{7/4} x^{7/2} \text {ArcTan}\left (\sqrt [4]{c} \sqrt {x}\right )-12 b \tanh ^{-1}\left (c x^2\right )-6 b c^{7/4} x^{7/2} \log \left (1-\sqrt [4]{c} \sqrt {x}\right )+6 b c^{7/4} x^{7/2} \log \left (1+\sqrt [4]{c} \sqrt {x}\right )+3 \sqrt {2} b c^{7/4} x^{7/2} \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )-3 \sqrt {2} b c^{7/4} x^{7/2} \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{42 d^5 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d*x]*(-12*a - 16*b*c*x^2 + 6*Sqrt[2]*b*c^(7/4)*x^(7/2)*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] - 6*Sqrt[2]*b
*c^(7/4)*x^(7/2)*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] + 12*b*c^(7/4)*x^(7/2)*ArcTan[c^(1/4)*Sqrt[x]] - 12*b*Arc
Tanh[c*x^2] - 6*b*c^(7/4)*x^(7/2)*Log[1 - c^(1/4)*Sqrt[x]] + 6*b*c^(7/4)*x^(7/2)*Log[1 + c^(1/4)*Sqrt[x]] + 3*
Sqrt[2]*b*c^(7/4)*x^(7/2)*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 3*Sqrt[2]*b*c^(7/4)*x^(7/2)*Log[1 + S
qrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(42*d^5*x^4)

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Maple [A]
time = 0.06, size = 301, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{7 \left (d x \right )^{\frac {7}{2}}}-\frac {2 b \arctanh \left (c \,x^{2}\right )}{7 \left (d x \right )^{\frac {7}{2}}}-\frac {b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{14 d^{4}}-\frac {b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{7 d^{4}}-\frac {b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{7 d^{4}}-\frac {8 b c}{21 d^{2} \left (d x \right )^{\frac {3}{2}}}+\frac {b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7 d^{4}}+\frac {2 b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7 d^{4}}}{d}\)	\(301\)
default	\(\frac {-\frac {2 a}{7 \left (d x \right )^{\frac {7}{2}}}-\frac {2 b \arctanh \left (c \,x^{2}\right )}{7 \left (d x \right )^{\frac {7}{2}}}-\frac {b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{14 d^{4}}-\frac {b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{7 d^{4}}-\frac {b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{7 d^{4}}-\frac {8 b c}{21 d^{2} \left (d x \right )^{\frac {3}{2}}}+\frac {b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7 d^{4}}+\frac {2 b \,c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{7 d^{4}}}{d}\)	\(301\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^2))/(d*x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/d*(-1/7*a/(d*x)^(7/2)-1/7*b/(d*x)^(7/2)*arctanh(c*x^2)-1/28*b/d^4*c^2*(d^2/c)^(1/4)*2^(1/2)*ln((d*x+(d^2/c)^
(1/4)*(d*x)^(1/2)*2^(1/2)+(d^2/c)^(1/2))/(d*x-(d^2/c)^(1/4)*(d*x)^(1/2)*2^(1/2)+(d^2/c)^(1/2)))-1/14*b/d^4*c^2
*(d^2/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(d^2/c)^(1/4)*(d*x)^(1/2)+1)-1/14*b/d^4*c^2*(d^2/c)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(d^2/c)^(1/4)*(d*x)^(1/2)-1)-4/21*b*c/d^2/(d*x)^(3/2)+1/14*b/d^4*c^2*(d^2/c)^(1/4)*ln(((d*x)^(1/2)+(d
^2/c)^(1/4))/((d*x)^(1/2)-(d^2/c)^(1/4)))+1/7*b/d^4*c^2*(d^2/c)^(1/4)*arctan((d*x)^(1/2)/(d^2/c)^(1/4)))

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Maxima [A]
time = 0.48, size = 297, normalized size = 0.94 \begin {gather*} -\frac {b {\left (\frac {c {\left (\frac {6 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} d} + \frac {6 \, \sqrt {2} c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} d} + \frac {3 \, \sqrt {2} c^{\frac {3}{4}} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{d^{\frac {3}{2}}} - \frac {3 \, \sqrt {2} c^{\frac {3}{4}} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{d^{\frac {3}{2}}} - \frac {12 \, c \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} d} + \frac {6 \, c \log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} d} + \frac {16}{\left (d x\right )^{\frac {3}{2}}}\right )}}{d^{2}} + \frac {12 \, \operatorname {artanh}\left (c x^{2}\right )}{\left (d x\right )^{\frac {7}{2}}}\right )} + \frac {12 \, a}{\left (d x\right )^{\frac {7}{2}}}}{42 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/42*(b*(c*(6*sqrt(2)*c*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*sqrt(d) + 2*sqrt(d*x)*sqrt(c))/sqrt(sqrt(c)*d))/(
sqrt(sqrt(c)*d)*d) + 6*sqrt(2)*c*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*sqrt(d) - 2*sqrt(d*x)*sqrt(c))/sqrt(sqrt
(c)*d))/(sqrt(sqrt(c)*d)*d) + 3*sqrt(2)*c^(3/4)*log(sqrt(c)*d*x + sqrt(2)*sqrt(d*x)*c^(1/4)*sqrt(d) + d)/d^(3/
2) - 3*sqrt(2)*c^(3/4)*log(sqrt(c)*d*x - sqrt(2)*sqrt(d*x)*c^(1/4)*sqrt(d) + d)/d^(3/2) - 12*c*arctan(sqrt(d*x
)*sqrt(c)/sqrt(sqrt(c)*d))/(sqrt(sqrt(c)*d)*d) + 6*c*log((sqrt(d*x)*sqrt(c) - sqrt(sqrt(c)*d))/(sqrt(d*x)*sqrt
(c) + sqrt(sqrt(c)*d)))/(sqrt(sqrt(c)*d)*d) + 16/(d*x)^(3/2))/d^2 + 12*arctanh(c*x^2)/(d*x)^(7/2)) + 12*a/(d*x
)^(7/2))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (212) = 424\).
time = 0.45, size = 450, normalized size = 1.42 \begin {gather*} -\frac {12 \, d^{5} x^{4} \left (\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} b c^{2} d^{13} \left (\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {3}{4}} - \sqrt {d^{10} \sqrt {\frac {b^{4} c^{7}}{d^{18}}} + b^{2} c^{4} d x} d^{13} \left (\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {3}{4}}}{b^{4} c^{7}}\right ) + 12 \, d^{5} x^{4} \left (-\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} b c^{2} d^{13} \left (-\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {3}{4}} - \sqrt {d^{10} \sqrt {-\frac {b^{4} c^{7}}{d^{18}}} + b^{2} c^{4} d x} d^{13} \left (-\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {3}{4}}}{b^{4} c^{7}}\right ) - 3 \, d^{5} x^{4} \left (\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (d^{5} \left (\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} + \sqrt {d x} b c^{2}\right ) + 3 \, d^{5} x^{4} \left (\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (-d^{5} \left (\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} + \sqrt {d x} b c^{2}\right ) + 3 \, d^{5} x^{4} \left (-\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (d^{5} \left (-\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} + \sqrt {d x} b c^{2}\right ) - 3 \, d^{5} x^{4} \left (-\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (-d^{5} \left (-\frac {b^{4} c^{7}}{d^{18}}\right )^{\frac {1}{4}} + \sqrt {d x} b c^{2}\right ) + {\left (8 \, b c x^{2} + 3 \, b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 6 \, a\right )} \sqrt {d x}}{21 \, d^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(12*d^5*x^4*(b^4*c^7/d^18)^(1/4)*arctan(-(sqrt(d*x)*b*c^2*d^13*(b^4*c^7/d^18)^(3/4) - sqrt(d^10*sqrt(b^4
*c^7/d^18) + b^2*c^4*d*x)*d^13*(b^4*c^7/d^18)^(3/4))/(b^4*c^7)) + 12*d^5*x^4*(-b^4*c^7/d^18)^(1/4)*arctan(-(sq
rt(d*x)*b*c^2*d^13*(-b^4*c^7/d^18)^(3/4) - sqrt(d^10*sqrt(-b^4*c^7/d^18) + b^2*c^4*d*x)*d^13*(-b^4*c^7/d^18)^(
3/4))/(b^4*c^7)) - 3*d^5*x^4*(b^4*c^7/d^18)^(1/4)*log(d^5*(b^4*c^7/d^18)^(1/4) + sqrt(d*x)*b*c^2) + 3*d^5*x^4*
(b^4*c^7/d^18)^(1/4)*log(-d^5*(b^4*c^7/d^18)^(1/4) + sqrt(d*x)*b*c^2) + 3*d^5*x^4*(-b^4*c^7/d^18)^(1/4)*log(d^
5*(-b^4*c^7/d^18)^(1/4) + sqrt(d*x)*b*c^2) - 3*d^5*x^4*(-b^4*c^7/d^18)^(1/4)*log(-d^5*(-b^4*c^7/d^18)^(1/4) +
sqrt(d*x)*b*c^2) + (8*b*c*x^2 + 3*b*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 6*a)*sqrt(d*x))/(d^5*x^4)

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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((a+b*atanh(c*x**2))/(d*x)**(9/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (212) = 424\).
time = 5.28, size = 519, normalized size = 1.64 \begin {gather*} -\frac {\frac {6 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} b c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{d^{4}} + \frac {6 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} b c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{d^{4}} - \frac {6 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} b c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{d^{4}} - \frac {6 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} b c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{d^{4}} + \frac {3 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} b c \log \left (d x + \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{d^{4}} - \frac {3 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} b c \log \left (d x - \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{d^{4}} - \frac {3 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} b c \log \left (d x + \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{d^{4}} + \frac {3 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {1}{4}} b c \log \left (d x - \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{d^{4}} + \frac {6 \, b \log \left (-\frac {c d^{2} x^{2} + d^{2}}{c d^{2} x^{2} - d^{2}}\right )}{\sqrt {d x} d^{3} x^{3}} + \frac {4 \, {\left (4 \, b c d^{2} x^{2} + 3 \, a d^{2}\right )}}{\sqrt {d x} d^{5} x^{3}}}{42 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/42*(6*sqrt(2)*(c^3*d^2)^(1/4)*b*c*arctan(1/2*sqrt(2)*(sqrt(2)*(d^2/c)^(1/4) + 2*sqrt(d*x))/(d^2/c)^(1/4))/d
^4 + 6*sqrt(2)*(c^3*d^2)^(1/4)*b*c*arctan(-1/2*sqrt(2)*(sqrt(2)*(d^2/c)^(1/4) - 2*sqrt(d*x))/(d^2/c)^(1/4))/d^
4 - 6*sqrt(2)*(-c^3*d^2)^(1/4)*b*c*arctan(1/2*sqrt(2)*(sqrt(2)*(-d^2/c)^(1/4) + 2*sqrt(d*x))/(-d^2/c)^(1/4))/d
^4 - 6*sqrt(2)*(-c^3*d^2)^(1/4)*b*c*arctan(-1/2*sqrt(2)*(sqrt(2)*(-d^2/c)^(1/4) - 2*sqrt(d*x))/(-d^2/c)^(1/4))
/d^4 + 3*sqrt(2)*(c^3*d^2)^(1/4)*b*c*log(d*x + sqrt(2)*sqrt(d*x)*(d^2/c)^(1/4) + sqrt(d^2/c))/d^4 - 3*sqrt(2)*
(c^3*d^2)^(1/4)*b*c*log(d*x - sqrt(2)*sqrt(d*x)*(d^2/c)^(1/4) + sqrt(d^2/c))/d^4 - 3*sqrt(2)*(-c^3*d^2)^(1/4)*
b*c*log(d*x + sqrt(2)*sqrt(d*x)*(-d^2/c)^(1/4) + sqrt(-d^2/c))/d^4 + 3*sqrt(2)*(-c^3*d^2)^(1/4)*b*c*log(d*x -
sqrt(2)*sqrt(d*x)*(-d^2/c)^(1/4) + sqrt(-d^2/c))/d^4 + 6*b*log(-(c*d^2*x^2 + d^2)/(c*d^2*x^2 - d^2))/(sqrt(d*x
)*d^3*x^3) + 4*(4*b*c*d^2*x^2 + 3*a*d^2)/(sqrt(d*x)*d^5*x^3))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x^2\right )}{{\left (d\,x\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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