Integrand size = 20, antiderivative size = 157 \[ \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{25 a c \left (c-a^2 c x^2\right )^{5/2}}-\frac {4}{45 a c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {8}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {x \text {arctanh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arctanh}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arctanh}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}} \]
-1/25/a/c/(-a^2*c*x^2+c)^(5/2)-4/45/a/c^2/(-a^2*c*x^2+c)^(3/2)+1/5*x*arcta nh(a*x)/c/(-a^2*c*x^2+c)^(5/2)+4/15*x*arctanh(a*x)/c^2/(-a^2*c*x^2+c)^(3/2 )-8/15/a/c^3/(-a^2*c*x^2+c)^(1/2)+8/15*x*arctanh(a*x)/c^3/(-a^2*c*x^2+c)^( 1/2)
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.51 \[ \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {c-a^2 c x^2} \left (149-260 a^2 x^2+120 a^4 x^4-15 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \text {arctanh}(a x)\right )}{225 a c^4 \left (-1+a^2 x^2\right )^3} \]
(Sqrt[c - a^2*c*x^2]*(149 - 260*a^2*x^2 + 120*a^4*x^4 - 15*a*x*(15 - 20*a^ 2*x^2 + 8*a^4*x^4)*ArcTanh[a*x]))/(225*a*c^4*(-1 + a^2*x^2)^3)
Time = 0.50 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6522, 6522, 6520}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 6522 |
\(\displaystyle \frac {4 \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}+\frac {x \text {arctanh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {1}{25 a c \left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 6522 |
\(\displaystyle \frac {4 \left (\frac {2 \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x \text {arctanh}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x \text {arctanh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {1}{25 a c \left (c-a^2 c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 6520 |
\(\displaystyle \frac {x \text {arctanh}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 \left (\frac {x \text {arctanh}(a x)}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \left (\frac {x \text {arctanh}(a x)}{c \sqrt {c-a^2 c x^2}}-\frac {1}{a c \sqrt {c-a^2 c x^2}}\right )}{3 c}-\frac {1}{9 a c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}-\frac {1}{25 a c \left (c-a^2 c x^2\right )^{5/2}}\) |
-1/25*1/(a*c*(c - a^2*c*x^2)^(5/2)) + (x*ArcTanh[a*x])/(5*c*(c - a^2*c*x^2 )^(5/2)) + (4*(-1/9*1/(a*c*(c - a^2*c*x^2)^(3/2)) + (x*ArcTanh[a*x])/(3*c* (c - a^2*c*x^2)^(3/2)) + (2*(-(1/(a*c*Sqrt[c - a^2*c*x^2])) + (x*ArcTanh[a *x])/(c*Sqrt[c - a^2*c*x^2])))/(3*c)))/(5*c)
3.5.70.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/( 2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x]) /; Fre eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
Time = 0.26 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.59
method | result | size |
default | \(-\frac {\left (a x +1\right )^{2} \left (-1+5 \,\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{800 a \left (a x -1\right )^{3} c^{4}}+\frac {5 \left (a x +1\right ) \left (-1+3 \,\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{288 a \,c^{4} \left (a x -1\right )^{2}}-\frac {5 \left (\operatorname {arctanh}\left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{16 a \,c^{4} \left (a x -1\right )}-\frac {5 \left (1+\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{16 a \left (a x +1\right ) c^{4}}+\frac {5 \left (a x -1\right ) \left (1+3 \,\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{288 a \left (a x +1\right )^{2} c^{4}}-\frac {\left (a x -1\right )^{2} \left (1+5 \,\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{800 \left (a x +1\right )^{3} a \,c^{4}}\) | \(250\) |
-1/800*(a*x+1)^2*(-1+5*arctanh(a*x))*(-(a*x-1)*(a*x+1)*c)^(1/2)/a/(a*x-1)^ 3/c^4+5/288*(a*x+1)*(-1+3*arctanh(a*x))*(-(a*x-1)*(a*x+1)*c)^(1/2)/a/c^4/( a*x-1)^2-5/16*(arctanh(a*x)-1)*(-(a*x-1)*(a*x+1)*c)^(1/2)/a/c^4/(a*x-1)-5/ 16*(1+arctanh(a*x))*(-(a*x-1)*(a*x+1)*c)^(1/2)/a/(a*x+1)/c^4+5/288*(a*x-1) *(1+3*arctanh(a*x))*(-(a*x-1)*(a*x+1)*c)^(1/2)/a/(a*x+1)^2/c^4-1/800*(a*x- 1)^2*(1+5*arctanh(a*x))*(-(a*x-1)*(a*x+1)*c)^(1/2)/(a*x+1)^3/a/c^4
Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.71 \[ \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {{\left (240 \, a^{4} x^{4} - 520 \, a^{2} x^{2} - 15 \, {\left (8 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 298\right )} \sqrt {-a^{2} c x^{2} + c}}{450 \, {\left (a^{7} c^{4} x^{6} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} - a c^{4}\right )}} \]
1/450*(240*a^4*x^4 - 520*a^2*x^2 - 15*(8*a^5*x^5 - 20*a^3*x^3 + 15*a*x)*lo g(-(a*x + 1)/(a*x - 1)) + 298)*sqrt(-a^2*c*x^2 + c)/(a^7*c^4*x^6 - 3*a^5*c ^4*x^4 + 3*a^3*c^4*x^2 - a*c^4)
\[ \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{225} \, a {\left (\frac {120}{\sqrt {-a^{2} c x^{2} + c} a^{2} c^{3}} + \frac {20}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2}} + \frac {9}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{2} c}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {artanh}\left (a x\right ) \]
-1/225*a*(120/(sqrt(-a^2*c*x^2 + c)*a^2*c^3) + 20/((-a^2*c*x^2 + c)^(3/2)* a^2*c^2) + 9/((-a^2*c*x^2 + c)^(5/2)*a^2*c)) + 1/15*(8*x/(sqrt(-a^2*c*x^2 + c)*c^3) + 4*x/((-a^2*c*x^2 + c)^(3/2)*c^2) + 3*x/((-a^2*c*x^2 + c)^(5/2) *c))*arctanh(a*x)
Time = 0.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {\sqrt {-a^{2} c x^{2} + c} {\left (4 \, {\left (\frac {2 \, a^{4} x^{2}}{c} - \frac {5 \, a^{2}}{c}\right )} x^{2} + \frac {15}{c}\right )} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{30 \, {\left (a^{2} c x^{2} - c\right )}^{3}} - \frac {120 \, {\left (a^{2} c x^{2} - c\right )}^{2} - 20 \, {\left (a^{2} c x^{2} - c\right )} c + 9 \, c^{2}}{225 \, {\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} c x^{2} + c} a c^{3}} \]
-1/30*sqrt(-a^2*c*x^2 + c)*(4*(2*a^4*x^2/c - 5*a^2/c)*x^2 + 15/c)*x*log(-( a*x + 1)/(a*x - 1))/(a^2*c*x^2 - c)^3 - 1/225*(120*(a^2*c*x^2 - c)^2 - 20* (a^2*c*x^2 - c)*c + 9*c^2)/((a^2*c*x^2 - c)^2*sqrt(-a^2*c*x^2 + c)*a*c^3)
Timed out. \[ \int \frac {\text {arctanh}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]