3.85 \(\int \frac{a+b \tanh ^{-1}(c x^2)}{\sqrt{d x}} \, dx\)

Optimal. Leaf size=285 \[ \frac{2 \sqrt{d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac{b \log \left (\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{\sqrt{2} \sqrt [4]{c} \sqrt{d}}+\frac{b \log \left (\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{\sqrt{2} \sqrt [4]{c} \sqrt{d}}-\frac{2 b \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{c} \sqrt{d}}-\frac{\sqrt{2} b \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{c} \sqrt{d}}+\frac{\sqrt{2} b \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}+1\right )}{\sqrt [4]{c} \sqrt{d}}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{c} \sqrt{d}} \]

[Out]

(-2*b*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) - (Sqrt[2]*b*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x
])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) + (Sqrt[2]*b*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d])
+ (2*Sqrt[d*x]*(a + b*ArcTanh[c*x^2]))/d - (2*b*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) - (b*
Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(Sqrt[2]*c^(1/4)*Sqrt[d]) + (b*Log[Sqrt[d] + Sqr
t[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(Sqrt[2]*c^(1/4)*Sqrt[d])

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Rubi [A]  time = 0.24214, antiderivative size = 285, normalized size of antiderivative = 1., 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 = {6097, 16, 329, 301, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ \frac{2 \sqrt{d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{d}-\frac{b \log \left (\sqrt{c} \sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{\sqrt{2} \sqrt [4]{c} \sqrt{d}}+\frac{b \log \left (\sqrt{c} \sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt{d x}+\sqrt{d}\right )}{\sqrt{2} \sqrt [4]{c} \sqrt{d}}-\frac{2 b \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{c} \sqrt{d}}-\frac{\sqrt{2} b \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{c} \sqrt{d}}+\frac{\sqrt{2} b \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}+1\right )}{\sqrt [4]{c} \sqrt{d}}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d x}}{\sqrt{d}}\right )}{\sqrt [4]{c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) - (Sqrt[2]*b*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x
])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) + (Sqrt[2]*b*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d])
+ (2*Sqrt[d*x]*(a + b*ArcTanh[c*x^2]))/d - (2*b*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) - (b*
Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(Sqrt[2]*c^(1/4)*Sqrt[d]) + (b*Log[Sqrt[d] + Sqr
t[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(Sqrt[2]*c^(1/4)*Sqrt[d])

Rule 6097

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

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 329

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 301

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 211

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 1165

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 628

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 1162

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 617

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 204

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

Rule 212

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] &&
!GtQ[a/b, 0]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0586972, size = 227, normalized size = 0.8 \[ \frac{\sqrt{x} \left (4 a \sqrt [4]{c} \sqrt{x}+4 b \sqrt [4]{c} \sqrt{x} \tanh ^{-1}\left (c x^2\right )+2 b \log \left (1-\sqrt [4]{c} \sqrt{x}\right )-2 b \log \left (\sqrt [4]{c} \sqrt{x}+1\right )-\sqrt{2} b \log \left (\sqrt{c} x-\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )+\sqrt{2} b \log \left (\sqrt{c} x+\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )-2 \sqrt{2} b \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{c} \sqrt{x}\right )+2 \sqrt{2} b \tan ^{-1}\left (\sqrt{2} \sqrt [4]{c} \sqrt{x}+1\right )-4 b \tan ^{-1}\left (\sqrt [4]{c} \sqrt{x}\right )\right )}{2 \sqrt [4]{c} \sqrt{d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(4*a*c^(1/4)*Sqrt[x] - 2*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] + 2*Sqrt[2]*b*ArcTan[1 + Sqrt[
2]*c^(1/4)*Sqrt[x]] - 4*b*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*c^(1/4)*Sqrt[x]*ArcTanh[c*x^2] + 2*b*Log[1 - c^(1/4)*S
qrt[x]] - 2*b*Log[1 + c^(1/4)*Sqrt[x]] - Sqrt[2]*b*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + Sqrt[2]*b*Lo
g[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(2*c^(1/4)*Sqrt[d*x])

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Maple [A]  time = 0.013, size = 273, normalized size = 1. \begin{align*} 2\,{\frac{a\sqrt{dx}}{d}}+2\,{\frac{b\sqrt{dx}{\it Artanh} \left ( c{x}^{2} \right ) }{d}}+{\frac{b\sqrt{2}}{2\,d}\sqrt [4]{{\frac{{d}^{2}}{c}}}\ln \left ({ \left ( dx+\sqrt [4]{{\frac{{d}^{2}}{c}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{{d}^{2}}{c}}} \right ) \left ( dx-\sqrt [4]{{\frac{{d}^{2}}{c}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{{d}^{2}}{c}}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{2}}{d}\sqrt [4]{{\frac{{d}^{2}}{c}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}+1 \right ) }+{\frac{b\sqrt{2}}{d}\sqrt [4]{{\frac{{d}^{2}}{c}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}}-1 \right ) }-{\frac{b}{d}\sqrt [4]{{\frac{{d}^{2}}{c}}}\ln \left ({ \left ( \sqrt{dx}+\sqrt [4]{{\frac{{d}^{2}}{c}}} \right ) \left ( \sqrt{dx}-\sqrt [4]{{\frac{{d}^{2}}{c}}} \right ) ^{-1}} \right ) }-2\,{\frac{b}{d}\sqrt [4]{{\frac{{d}^{2}}{c}}}\arctan \left ({\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{c}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*a*(d*x)^(1/2)+2/d*b*(d*x)^(1/2)*arctanh(c*x^2)+1/2/d*b*(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/d*b*(d^2/c)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(d^2/c)^(1/4)*(d*x)^(1/2)+1)+1/d*b*(d^2/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(d^2/c)^(1/4)*(d*x)^(1
/2)-1)-1/d*b*(d^2/c)^(1/4)*ln(((d*x)^(1/2)+(d^2/c)^(1/4))/((d*x)^(1/2)-(d^2/c)^(1/4)))-2/d*b*(d^2/c)^(1/4)*arc
tan((d*x)^(1/2)/(d^2/c)^(1/4))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.22864, size = 72, normalized size = 0.25 \begin{align*} \frac{\sqrt{d x}{\left (b \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(d*x)*(b*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a)/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{atanh}{\left (c x^{2} \right )}}{\sqrt{d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.24348, size = 666, normalized size = 2.34 \begin{align*} \frac{{\left (c d^{2}{\left (\frac{2 \, \sqrt{2} \left (c^{3} d^{2}\right )^{\frac{1}{4}} \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 )}{c^{2} d^{2}} + \frac{2 \, \sqrt{2} \left (c^{3} d^{2}\right )^{\frac{1}{4}} \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 )}{c^{2} d^{2}} - \frac{2 \, \sqrt{2} \left (-c^{3} d^{2}\right )^{\frac{1}{4}} \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 )}{c^{2} d^{2}} - \frac{2 \, \sqrt{2} \left (-c^{3} d^{2}\right )^{\frac{1}{4}} \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 )}{c^{2} d^{2}} + \frac{\sqrt{2} \left (c^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x + \sqrt{2} \sqrt{d x} \left (\frac{d^{2}}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{d^{2}}{c}}\right )}{c^{2} d^{2}} - \frac{\sqrt{2} \left (c^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x - \sqrt{2} \sqrt{d x} \left (\frac{d^{2}}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{d^{2}}{c}}\right )}{c^{2} d^{2}} - \frac{\sqrt{2} \left (-c^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x + \sqrt{2} \sqrt{d x} \left (-\frac{d^{2}}{c}\right )^{\frac{1}{4}} + \sqrt{-\frac{d^{2}}{c}}\right )}{c^{2} d^{2}} + \frac{\sqrt{2} \left (-c^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x - \sqrt{2} \sqrt{d x} \left (-\frac{d^{2}}{c}\right )^{\frac{1}{4}} + \sqrt{-\frac{d^{2}}{c}}\right )}{c^{2} d^{2}}\right )} + 2 \, \sqrt{d x} \log \left (-\frac{c x^{2} + 1}{c x^{2} - 1}\right )\right )} b + 4 \, \sqrt{d x} a}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((c*d^2*(2*sqrt(2)*(c^3*d^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(d^2/c)^(1/4) + 2*sqrt(d*x))/(d^2/c)^(1/4))
/(c^2*d^2) + 2*sqrt(2)*(c^3*d^2)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d^2/c)^(1/4) - 2*sqrt(d*x))/(d^2/c)^(1/4)
)/(c^2*d^2) - 2*sqrt(2)*(-c^3*d^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-d^2/c)^(1/4) + 2*sqrt(d*x))/(-d^2/c)^(1
/4))/(c^2*d^2) - 2*sqrt(2)*(-c^3*d^2)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-d^2/c)^(1/4) - 2*sqrt(d*x))/(-d^2/c
)^(1/4))/(c^2*d^2) + sqrt(2)*(c^3*d^2)^(1/4)*log(d*x + sqrt(2)*sqrt(d*x)*(d^2/c)^(1/4) + sqrt(d^2/c))/(c^2*d^2
) - sqrt(2)*(c^3*d^2)^(1/4)*log(d*x - sqrt(2)*sqrt(d*x)*(d^2/c)^(1/4) + sqrt(d^2/c))/(c^2*d^2) - sqrt(2)*(-c^3
*d^2)^(1/4)*log(d*x + sqrt(2)*sqrt(d*x)*(-d^2/c)^(1/4) + sqrt(-d^2/c))/(c^2*d^2) + sqrt(2)*(-c^3*d^2)^(1/4)*lo
g(d*x - sqrt(2)*sqrt(d*x)*(-d^2/c)^(1/4) + sqrt(-d^2/c))/(c^2*d^2)) + 2*sqrt(d*x)*log(-(c*x^2 + 1)/(c*x^2 - 1)
))*b + 4*sqrt(d*x)*a)/d