3.1.84 \(\int \sqrt {d x} (a+b \tanh ^{-1}(c x^2)) \, dx\) [84]

Optimal. Leaf size=301 \[ \frac {2 b \sqrt {d} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}-\frac {\sqrt {2} b \sqrt {d} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\sqrt {2} b \sqrt {d} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}} \]

[Out]

2/3*(d*x)^(3/2)*(a+b*arctanh(c*x^2))/d+2/3*b*arctan(c^(1/4)*(d*x)^(1/2)/d^(1/2))*d^(1/2)/c^(3/4)-2/3*b*arctanh
(c^(1/4)*(d*x)^(1/2)/d^(1/2))*d^(1/2)/c^(3/4)+1/6*b*ln(d^(1/2)+x*c^(1/2)*d^(1/2)-c^(1/4)*2^(1/2)*(d*x)^(1/2))*
d^(1/2)/c^(3/4)*2^(1/2)-1/6*b*ln(d^(1/2)+x*c^(1/2)*d^(1/2)+c^(1/4)*2^(1/2)*(d*x)^(1/2))*d^(1/2)/c^(3/4)*2^(1/2
)+1/3*b*arctan(-1+c^(1/4)*2^(1/2)*(d*x)^(1/2)/d^(1/2))*2^(1/2)*d^(1/2)/c^(3/4)+1/3*b*arctan(1+c^(1/4)*2^(1/2)*
(d*x)^(1/2)/d^(1/2))*2^(1/2)*d^(1/2)/c^(3/4)

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6049, 335, 307, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \begin {gather*} \frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}+\frac {2 b \sqrt {d} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}-\frac {\sqrt {2} b \sqrt {d} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\sqrt {2} b \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{3 c^{3/4}}+\frac {b \sqrt {d} \log \left (\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}+\sqrt {d}\right )}{3 \sqrt {2} c^{3/4}}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*Sqrt[d]*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/4)) - (Sqrt[2]*b*Sqrt[d]*ArcTan[1 - (Sqrt[2]*c^(1/4)
*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/4)) + (Sqrt[2]*b*Sqrt[d]*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*c^(
3/4)) + (2*(d*x)^(3/2)*(a + b*ArcTanh[c*x^2]))/(3*d) - (2*b*Sqrt[d]*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*c
^(3/4)) + (b*Sqrt[d]*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(3*Sqrt[2]*c^(3/4)) - (b*Sq
rt[d]*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(3*Sqrt[2]*c^(3/4))

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 303

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

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[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 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 \sqrt {d x} \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {(4 b c) \int \frac {x (d x)^{3/2}}{1-c^2 x^4} \, dx}{3 d}\\ &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {(4 b c) \int \frac {(d x)^{5/2}}{1-c^2 x^4} \, dx}{3 d^2}\\ &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {(8 b c) \text {Subst}\left (\int \frac {x^6}{1-\frac {c^2 x^8}{d^4}} \, dx,x,\sqrt {d x}\right )}{3 d^3}\\ &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {1}{3} (4 b d) \text {Subst}\left (\int \frac {x^2}{d^2-c x^4} \, dx,x,\sqrt {d x}\right )+\frac {1}{3} (4 b d) \text {Subst}\left (\int \frac {x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )\\ &=\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {(2 b d) \text {Subst}\left (\int \frac {1}{d-\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {c}}+\frac {(2 b d) \text {Subst}\left (\int \frac {1}{d+\sqrt {c} x^2} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {c}}-\frac {(2 b d) \text {Subst}\left (\int \frac {d-\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {c}}+\frac {(2 b d) \text {Subst}\left (\int \frac {d+\sqrt {c} x^2}{d^2+c x^4} \, dx,x,\sqrt {d x}\right )}{3 \sqrt {c}}\\ &=\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\left (b \sqrt {d}\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 )}{3 \sqrt {2} c^{3/4}}+\frac {\left (b \sqrt {d}\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 )}{3 \sqrt {2} c^{3/4}}+\frac {(b d) \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 )}{3 c}+\frac {(b d) \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 )}{3 c}\\ &=\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}+\frac {\left (\sqrt {2} b \sqrt {d}\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 )}{3 c^{3/4}}-\frac {\left (\sqrt {2} b \sqrt {d}\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 )}{3 c^{3/4}}\\ &=\frac {2 b \sqrt {d} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}-\frac {\sqrt {2} b \sqrt {d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\sqrt {2} b \sqrt {d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {2 (d x)^{3/2} \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{3 d}-\frac {2 b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 227, normalized size = 0.75 \begin {gather*} \frac {\sqrt {d x} \left (4 a c^{3/4} x^{3/2}-2 \sqrt {2} b \text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+2 \sqrt {2} b \text {ArcTan}\left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+4 b \text {ArcTan}\left (\sqrt [4]{c} \sqrt {x}\right )+4 b c^{3/4} x^{3/2} \tanh ^{-1}\left (c x^2\right )+2 b \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b \log \left (1+\sqrt [4]{c} \sqrt {x}\right )+\sqrt {2} b \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )-\sqrt {2} b \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{6 c^{3/4} \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d*x]*(4*a*c^(3/4)*x^(3/2) - 2*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]] + 2*Sqrt[2]*b*ArcTan[1 + Sqr
t[2]*c^(1/4)*Sqrt[x]] + 4*b*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*c^(3/4)*x^(3/2)*ArcTanh[c*x^2] + 2*b*Log[1 - c^(1/4)
*Sqrt[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*
Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x]))/(6*c^(3/4)*Sqrt[x])

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 289, normalized size = 0.96

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+\frac {2 b \left (d x \right )^{\frac {3}{2}} \arctanh \left (c \,x^{2}\right )}{3}+\frac {b \,d^{2} \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{6 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {b \,d^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {b \,d^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {2 b \,d^{2} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {b \,d^{2} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}}{d}\)	\(289\)
default	\(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+\frac {2 b \left (d x \right )^{\frac {3}{2}} \arctanh \left (c \,x^{2}\right )}{3}+\frac {b \,d^{2} \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )}{6 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {b \,d^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {b \,d^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+\frac {2 b \,d^{2} \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {b \,d^{2} \ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3 c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}}{d}\)	\(289\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.47, size = 301, normalized size = 1.00 \begin {gather*} \frac {4 \, \left (d x\right )^{\frac {3}{2}} a + {\left (4 \, \left (d x\right )^{\frac {3}{2}} \operatorname {artanh}\left (c x^{2}\right ) + \frac {{\left (\frac {d^{4} {\left (\frac {2 \, \sqrt {2} \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} \sqrt {c}} + \frac {2 \, \sqrt {2} \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} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}} + \frac {\sqrt {2} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}}\right )}}{c} + \frac {2 \, d^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {\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} \sqrt {c}}\right )}}{c}\right )} c}{d^{2}}\right )} b}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.43, size = 400, normalized size = 1.33 \begin {gather*} \frac {1}{3} \, {\left (b x \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a x\right )} \sqrt {d x} - \frac {4}{3} \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \sqrt {d x} b^{3} c d - \sqrt {b^{6} d^{3} x + \sqrt {\frac {b^{4} d^{2}}{c^{3}}} b^{4} c d^{2}} \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} c}{b^{4} d^{2}}\right ) - \frac {4}{3} \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \sqrt {d x} b^{3} c d - \sqrt {b^{6} d^{3} x - \sqrt {-\frac {b^{4} d^{2}}{c^{3}}} b^{4} c d^{2}} \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} c}{b^{4} d^{2}}\right ) - \frac {1}{3} \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d + \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) + \frac {1}{3} \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d - \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) + \frac {1}{3} \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d + \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) - \frac {1}{3} \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d - \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*x)*(b*arctanh(c*x^2) + a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________