Integrand size = 16, antiderivative size = 85 \[ \int \frac {a+b \text {arctanh}(c x)}{(d x)^{3/2}} \, dx=\frac {2 b \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 (a+b \text {arctanh}(c x))}{d \sqrt {d x}}+\frac {2 b \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}} \]
2*b*arctan(c^(1/2)*(d*x)^(1/2)/d^(1/2))*c^(1/2)/d^(3/2)+2*b*arctanh(c^(1/2 )*(d*x)^(1/2)/d^(1/2))*c^(1/2)/d^(3/2)-2*(a+b*arctanh(c*x))/d/(d*x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \text {arctanh}(c x)}{(d x)^{3/2}} \, dx=\frac {x \left (-2 a+2 b \sqrt {c} \sqrt {x} \arctan \left (\sqrt {c} \sqrt {x}\right )-2 b \text {arctanh}(c x)-b \sqrt {c} \sqrt {x} \log \left (1-\sqrt {c} \sqrt {x}\right )+b \sqrt {c} \sqrt {x} \log \left (1+\sqrt {c} \sqrt {x}\right )\right )}{(d x)^{3/2}} \]
(x*(-2*a + 2*b*Sqrt[c]*Sqrt[x]*ArcTan[Sqrt[c]*Sqrt[x]] - 2*b*ArcTanh[c*x] - b*Sqrt[c]*Sqrt[x]*Log[1 - Sqrt[c]*Sqrt[x]] + b*Sqrt[c]*Sqrt[x]*Log[1 + S qrt[c]*Sqrt[x]]))/(d*x)^(3/2)
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6464, 266, 756, 218, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{(d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6464 |
\(\displaystyle \frac {2 b c \int \frac {1}{\sqrt {d x} \left (1-c^2 x^2\right )}dx}{d}-\frac {2 (a+b \text {arctanh}(c x))}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {4 b c \int \frac {1}{1-c^2 x^2}d\sqrt {d x}}{d^2}-\frac {2 (a+b \text {arctanh}(c x))}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {4 b c \left (\frac {1}{2} d \int \frac {1}{d-c d x}d\sqrt {d x}+\frac {1}{2} d \int \frac {1}{c x d+d}d\sqrt {d x}\right )}{d^2}-\frac {2 (a+b \text {arctanh}(c x))}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {4 b c \left (\frac {1}{2} d \int \frac {1}{d-c d x}d\sqrt {d x}+\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {c}}\right )}{d^2}-\frac {2 (a+b \text {arctanh}(c x))}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {4 b c \left (\frac {\sqrt {d} \arctan \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {c}}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt {c}}\right )}{d^2}-\frac {2 (a+b \text {arctanh}(c x))}{d \sqrt {d x}}\) |
(-2*(a + b*ArcTanh[c*x]))/(d*Sqrt[d*x]) + (4*b*c*((Sqrt[d]*ArcTan[(Sqrt[c] *Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[c]) + (Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d*x])/S qrt[d]])/(2*Sqrt[c])))/d^2
3.1.39.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))*((d_)*(x_))^(m_), x_Symbol] : > Simp[(d*x)^(m + 1)*((a + b*ArcTanh[c*x^n])/(d*(m + 1))), x] - Simp[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]
Time = 0.59 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{\sqrt {d x}}-\frac {2 b \,\operatorname {arctanh}\left (c x \right )}{\sqrt {d x}}+\frac {2 b c \,\operatorname {arctanh}\left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{\sqrt {c d}}+\frac {2 b c \arctan \left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{\sqrt {c d}}}{d}\) | \(69\) |
default | \(\frac {-\frac {2 a}{\sqrt {d x}}-\frac {2 b \,\operatorname {arctanh}\left (c x \right )}{\sqrt {d x}}+\frac {2 b c \,\operatorname {arctanh}\left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{\sqrt {c d}}+\frac {2 b c \arctan \left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{\sqrt {c d}}}{d}\) | \(69\) |
parts | \(-\frac {2 a}{\sqrt {d x}\, d}-\frac {2 b \,\operatorname {arctanh}\left (c x \right )}{d \sqrt {d x}}+\frac {2 b c \arctan \left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{d \sqrt {c d}}+\frac {2 b c \,\operatorname {arctanh}\left (\frac {c \sqrt {d x}}{\sqrt {c d}}\right )}{d \sqrt {c d}}\) | \(78\) |
2/d*(-a/(d*x)^(1/2)-b/(d*x)^(1/2)*arctanh(c*x)+b*c/(c*d)^(1/2)*arctanh(c*( d*x)^(1/2)/(c*d)^(1/2))+b*c/(c*d)^(1/2)*arctan(c*(d*x)^(1/2)/(c*d)^(1/2)))
Time = 0.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.60 \[ \int \frac {a+b \text {arctanh}(c x)}{(d x)^{3/2}} \, dx=\left [-\frac {2 \, b d x \sqrt {\frac {c}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {\frac {c}{d}}}{c x}\right ) - b d x \sqrt {\frac {c}{d}} \log \left (\frac {c x + 2 \, \sqrt {d x} \sqrt {\frac {c}{d}} + 1}{c x - 1}\right ) + \sqrt {d x} {\left (b \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a\right )}}{d^{2} x}, -\frac {2 \, b d x \sqrt {-\frac {c}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {c}{d}}}{c x}\right ) - b d x \sqrt {-\frac {c}{d}} \log \left (\frac {c x + 2 \, \sqrt {d x} \sqrt {-\frac {c}{d}} - 1}{c x + 1}\right ) + \sqrt {d x} {\left (b \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a\right )}}{d^{2} x}\right ] \]
[-(2*b*d*x*sqrt(c/d)*arctan(sqrt(d*x)*sqrt(c/d)/(c*x)) - b*d*x*sqrt(c/d)*l og((c*x + 2*sqrt(d*x)*sqrt(c/d) + 1)/(c*x - 1)) + sqrt(d*x)*(b*log(-(c*x + 1)/(c*x - 1)) + 2*a))/(d^2*x), -(2*b*d*x*sqrt(-c/d)*arctan(sqrt(d*x)*sqrt (-c/d)/(c*x)) - b*d*x*sqrt(-c/d)*log((c*x + 2*sqrt(d*x)*sqrt(-c/d) - 1)/(c *x + 1)) + sqrt(d*x)*(b*log(-(c*x + 1)/(c*x - 1)) + 2*a))/(d^2*x)]
\[ \int \frac {a+b \text {arctanh}(c x)}{(d x)^{3/2}} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \text {arctanh}(c x)}{(d x)^{3/2}} \, dx=\frac {b {\left (\frac {{\left (\frac {2 \, d \arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d}} - \frac {d \log \left (\frac {\sqrt {d x} c - \sqrt {c d}}{\sqrt {d x} c + \sqrt {c d}}\right )}{\sqrt {c d}}\right )} c}{d} - \frac {2 \, \operatorname {artanh}\left (c x\right )}{\sqrt {d x}}\right )} - \frac {2 \, a}{\sqrt {d x}}}{d} \]
(b*((2*d*arctan(sqrt(d*x)*c/sqrt(c*d))/sqrt(c*d) - d*log((sqrt(d*x)*c - sq rt(c*d))/(sqrt(d*x)*c + sqrt(c*d)))/sqrt(c*d))*c/d - 2*arctanh(c*x)/sqrt(d *x)) - 2*a/sqrt(d*x))/d
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \text {arctanh}(c x)}{(d x)^{3/2}} \, dx=\frac {2 \, b c d {\left (\frac {\arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} d} - \frac {\arctan \left (\frac {\sqrt {d x} c}{\sqrt {-c d}}\right )}{\sqrt {-c d} d}\right )} - \frac {b \log \left (-\frac {c d x + d}{c d x - d}\right )}{\sqrt {d x}} - \frac {2 \, a}{\sqrt {d x}}}{d} \]
(2*b*c*d*(arctan(sqrt(d*x)*c/sqrt(c*d))/(sqrt(c*d)*d) - arctan(sqrt(d*x)*c /sqrt(-c*d))/(sqrt(-c*d)*d)) - b*log(-(c*d*x + d)/(c*d*x - d))/sqrt(d*x) - 2*a/sqrt(d*x))/d
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{(d x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]