Integrand size = 16, antiderivative size = 80 \[ \int \frac {a+b \text {arctanh}(c x)}{(1+c x)^4} \, dx=-\frac {b}{18 c (1+c x)^3}-\frac {b}{24 c (1+c x)^2}-\frac {b}{24 c (1+c x)}+\frac {b \text {arctanh}(c x)}{24 c}-\frac {a+b \text {arctanh}(c x)}{3 c (1+c x)^3} \] Output:
-1/18*b/c/(c*x+1)^3-1/24*b/c/(c*x+1)^2-1/24*b/c/(c*x+1)+1/24*b*arctanh(c*x )/c-1/3*(a+b*arctanh(c*x))/c/(c*x+1)^3
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \text {arctanh}(c x)}{(1+c x)^4} \, dx=-\frac {48 a+2 b \left (10+9 c x+3 c^2 x^2\right )+48 b \text {arctanh}(c x)+3 b (1+c x)^3 \log (1-c x)-3 b (1+c x)^3 \log (1+c x)}{144 c (1+c x)^3} \] Input:
Integrate[(a + b*ArcTanh[c*x])/(1 + c*x)^4,x]
Output:
-1/144*(48*a + 2*b*(10 + 9*c*x + 3*c^2*x^2) + 48*b*ArcTanh[c*x] + 3*b*(1 + c*x)^3*Log[1 - c*x] - 3*b*(1 + c*x)^3*Log[1 + c*x])/(c*(1 + c*x)^3)
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6478, 456, 54, 2009}
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.
\(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{(c x+1)^4} \, dx\) |
\(\Big \downarrow \) 6478 |
\(\displaystyle \frac {1}{3} b \int \frac {1}{(c x+1)^3 \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{3 c (c x+1)^3}\) |
\(\Big \downarrow \) 456 |
\(\displaystyle \frac {1}{3} b \int \frac {1}{(1-c x) (c x+1)^4}dx-\frac {a+b \text {arctanh}(c x)}{3 c (c x+1)^3}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{3} b \int \left (\frac {1}{8 (c x+1)^2}+\frac {1}{4 (c x+1)^3}+\frac {1}{2 (c x+1)^4}-\frac {1}{8 \left (c^2 x^2-1\right )}\right )dx-\frac {a+b \text {arctanh}(c x)}{3 c (c x+1)^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} b \left (\frac {\text {arctanh}(c x)}{8 c}-\frac {1}{8 c (c x+1)}-\frac {1}{8 c (c x+1)^2}-\frac {1}{6 c (c x+1)^3}\right )-\frac {a+b \text {arctanh}(c x)}{3 c (c x+1)^3}\) |
Input:
Int[(a + b*ArcTanh[c*x])/(1 + c*x)^4,x]
Output:
-1/3*(a + b*ArcTanh[c*x])/(c*(1 + c*x)^3) + (b*(-1/6*1/(c*(1 + c*x)^3) - 1 /(8*c*(1 + c*x)^2) - 1/(8*c*(1 + c*x)) + ArcTanh[c*x]/(8*c)))/3
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ (c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !Integ erQ[n]))
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol ] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])/(e*(q + 1))), x] - Simp[b *(c/(e*(q + 1))) Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {-\frac {a}{3 \left (c x +1\right )^{3}}+b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 \left (c x +1\right )^{3}}-\frac {\ln \left (c x -1\right )}{48}-\frac {1}{18 \left (c x +1\right )^{3}}-\frac {1}{24 \left (c x +1\right )^{2}}-\frac {1}{24 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{48}\right )}{c}\) | \(75\) |
default | \(\frac {-\frac {a}{3 \left (c x +1\right )^{3}}+b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 \left (c x +1\right )^{3}}-\frac {\ln \left (c x -1\right )}{48}-\frac {1}{18 \left (c x +1\right )^{3}}-\frac {1}{24 \left (c x +1\right )^{2}}-\frac {1}{24 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{48}\right )}{c}\) | \(75\) |
parts | \(-\frac {a}{3 \left (c x +1\right )^{3} c}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 \left (c x +1\right )^{3}}-\frac {\ln \left (c x -1\right )}{48}-\frac {1}{18 \left (c x +1\right )^{3}}-\frac {1}{24 \left (c x +1\right )^{2}}-\frac {1}{24 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{48}\right )}{c}\) | \(77\) |
parallelrisch | \(-\frac {-72 a \,c^{2} x^{2}-21 b c x -9 \,\operatorname {arctanh}\left (c x \right ) b c x -9 \,\operatorname {arctanh}\left (c x \right ) b \,c^{2} x^{2}-24 a \,c^{3} x^{3}-27 b \,c^{2} x^{2}-72 a c x +21 b \,\operatorname {arctanh}\left (c x \right )-10 b \,c^{3} x^{3}-3 x^{3} \operatorname {arctanh}\left (c x \right ) b \,c^{3}}{72 \left (c x +1\right )^{3} c}\) | \(102\) |
orering | \(-\frac {\left (40 c^{4} x^{4}+65 x^{3} c^{3}-33 c^{2} x^{2}-93 c x +21\right ) \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )}{72 \left (c x +1\right )^{3} c}-\frac {x \left (10 c^{2} x^{2}+27 c x +21\right ) \left (c x -1\right ) \left (c x +1\right )^{2} \left (\frac {b c}{\left (-c^{2} x^{2}+1\right ) \left (c x +1\right )^{4}}-\frac {4 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) c}{\left (c x +1\right )^{5}}\right )}{72 c}\) | \(125\) |
risch | \(-\frac {b \ln \left (c x +1\right )}{6 c \left (c x +1\right )^{3}}-\frac {3 \ln \left (c x -1\right ) b \,c^{3} x^{3}-3 \ln \left (-c x -1\right ) b \,c^{3} x^{3}+9 \ln \left (c x -1\right ) b \,c^{2} x^{2}-9 \ln \left (-c x -1\right ) b \,c^{2} x^{2}+6 b \,c^{2} x^{2}+9 \ln \left (c x -1\right ) b c x -9 \ln \left (-c x -1\right ) b c x +18 b c x +3 b \ln \left (c x -1\right )-3 b \ln \left (-c x -1\right )-24 b \ln \left (-c x +1\right )+48 a +20 b}{144 \left (c x +1\right )^{3} c}\) | \(168\) |
Input:
int((a+b*arctanh(c*x))/(c*x+1)^4,x,method=_RETURNVERBOSE)
Output:
1/c*(-1/3*a/(c*x+1)^3+b*(-1/3/(c*x+1)^3*arctanh(c*x)-1/48*ln(c*x-1)-1/18/( c*x+1)^3-1/24/(c*x+1)^2-1/24/(c*x+1)+1/48*ln(c*x+1)))
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.14 \[ \int \frac {a+b \text {arctanh}(c x)}{(1+c x)^4} \, dx=-\frac {6 \, b c^{2} x^{2} + 18 \, b c x - 3 \, {\left (b c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b c x - 7 \, b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) + 48 \, a + 20 \, b}{144 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} \] Input:
integrate((a+b*arctanh(c*x))/(c*x+1)^4,x, algorithm="fricas")
Output:
-1/144*(6*b*c^2*x^2 + 18*b*c*x - 3*(b*c^3*x^3 + 3*b*c^2*x^2 + 3*b*c*x - 7* b)*log(-(c*x + 1)/(c*x - 1)) + 48*a + 20*b)/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c)
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (63) = 126\).
Time = 1.19 (sec) , antiderivative size = 294, normalized size of antiderivative = 3.68 \[ \int \frac {a+b \text {arctanh}(c x)}{(1+c x)^4} \, dx=\begin {cases} - \frac {24 a}{72 c^{4} x^{3} + 216 c^{3} x^{2} + 216 c^{2} x + 72 c} + \frac {3 b c^{3} x^{3} \operatorname {atanh}{\left (c x \right )}}{72 c^{4} x^{3} + 216 c^{3} x^{2} + 216 c^{2} x + 72 c} + \frac {9 b c^{2} x^{2} \operatorname {atanh}{\left (c x \right )}}{72 c^{4} x^{3} + 216 c^{3} x^{2} + 216 c^{2} x + 72 c} - \frac {3 b c^{2} x^{2}}{72 c^{4} x^{3} + 216 c^{3} x^{2} + 216 c^{2} x + 72 c} + \frac {9 b c x \operatorname {atanh}{\left (c x \right )}}{72 c^{4} x^{3} + 216 c^{3} x^{2} + 216 c^{2} x + 72 c} - \frac {9 b c x}{72 c^{4} x^{3} + 216 c^{3} x^{2} + 216 c^{2} x + 72 c} - \frac {21 b \operatorname {atanh}{\left (c x \right )}}{72 c^{4} x^{3} + 216 c^{3} x^{2} + 216 c^{2} x + 72 c} - \frac {10 b}{72 c^{4} x^{3} + 216 c^{3} x^{2} + 216 c^{2} x + 72 c} & \text {for}\: c \neq 0 \\a x & \text {otherwise} \end {cases} \] Input:
integrate((a+b*atanh(c*x))/(c*x+1)**4,x)
Output:
Piecewise((-24*a/(72*c**4*x**3 + 216*c**3*x**2 + 216*c**2*x + 72*c) + 3*b* c**3*x**3*atanh(c*x)/(72*c**4*x**3 + 216*c**3*x**2 + 216*c**2*x + 72*c) + 9*b*c**2*x**2*atanh(c*x)/(72*c**4*x**3 + 216*c**3*x**2 + 216*c**2*x + 72*c ) - 3*b*c**2*x**2/(72*c**4*x**3 + 216*c**3*x**2 + 216*c**2*x + 72*c) + 9*b *c*x*atanh(c*x)/(72*c**4*x**3 + 216*c**3*x**2 + 216*c**2*x + 72*c) - 9*b*c *x/(72*c**4*x**3 + 216*c**3*x**2 + 216*c**2*x + 72*c) - 21*b*atanh(c*x)/(7 2*c**4*x**3 + 216*c**3*x**2 + 216*c**2*x + 72*c) - 10*b/(72*c**4*x**3 + 21 6*c**3*x**2 + 216*c**2*x + 72*c), Ne(c, 0)), (a*x, True))
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.65 \[ \int \frac {a+b \text {arctanh}(c x)}{(1+c x)^4} \, dx=-\frac {1}{144} \, {\left (c {\left (\frac {2 \, {\left (3 \, c^{2} x^{2} + 9 \, c x + 10\right )}}{c^{5} x^{3} + 3 \, c^{4} x^{2} + 3 \, c^{3} x + c^{2}} - \frac {3 \, \log \left (c x + 1\right )}{c^{2}} + \frac {3 \, \log \left (c x - 1\right )}{c^{2}}\right )} + \frac {48 \, \operatorname {artanh}\left (c x\right )}{c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c}\right )} b - \frac {a}{3 \, {\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} \] Input:
integrate((a+b*arctanh(c*x))/(c*x+1)^4,x, algorithm="maxima")
Output:
-1/144*(c*(2*(3*c^2*x^2 + 9*c*x + 10)/(c^5*x^3 + 3*c^4*x^2 + 3*c^3*x + c^2 ) - 3*log(c*x + 1)/c^2 + 3*log(c*x - 1)/c^2) + 48*arctanh(c*x)/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c))*b - 1/3*a/(c^4*x^3 + 3*c^3*x^2 + 3*c^2*x + c)
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (70) = 140\).
Time = 0.12 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.01 \[ \int \frac {a+b \text {arctanh}(c x)}{(1+c x)^4} \, dx=\frac {1}{288} \, c {\left (\frac {6 \, {\left (c x - 1\right )}^{3} {\left (\frac {3 \, {\left (c x + 1\right )}^{2} b}{{\left (c x - 1\right )}^{2}} - \frac {3 \, {\left (c x + 1\right )} b}{c x - 1} + b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{3} c^{2}} + \frac {{\left (c x - 1\right )}^{3} {\left (\frac {36 \, {\left (c x + 1\right )}^{2} a}{{\left (c x - 1\right )}^{2}} - \frac {36 \, {\left (c x + 1\right )} a}{c x - 1} + 12 \, a + \frac {18 \, {\left (c x + 1\right )}^{2} b}{{\left (c x - 1\right )}^{2}} - \frac {9 \, {\left (c x + 1\right )} b}{c x - 1} + 2 \, b\right )}}{{\left (c x + 1\right )}^{3} c^{2}}\right )} \] Input:
integrate((a+b*arctanh(c*x))/(c*x+1)^4,x, algorithm="giac")
Output:
1/288*c*(6*(c*x - 1)^3*(3*(c*x + 1)^2*b/(c*x - 1)^2 - 3*(c*x + 1)*b/(c*x - 1) + b)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^3*c^2) + (c*x - 1)^3*(36*(c* x + 1)^2*a/(c*x - 1)^2 - 36*(c*x + 1)*a/(c*x - 1) + 12*a + 18*(c*x + 1)^2* b/(c*x - 1)^2 - 9*(c*x + 1)*b/(c*x - 1) + 2*b)/((c*x + 1)^3*c^2))
Time = 3.86 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.74 \[ \int \frac {a+b \text {arctanh}(c x)}{(1+c x)^4} \, dx=\frac {\frac {b\,c^2\,x^3}{8}-\frac {b\,x}{8}-\frac {b\,\mathrm {atanh}\left (c\,x\right )}{3\,c}-\frac {12\,a+5\,b}{36\,c}+\frac {b\,c^3\,x^4}{24}+\frac {c\,x^2\,\left (24\,a+7\,b\right )}{72}+\frac {b\,c\,x^2\,\mathrm {atanh}\left (c\,x\right )}{3}}{-c^5\,x^5-3\,c^4\,x^4-2\,c^3\,x^3+2\,c^2\,x^2+3\,c\,x+1}-\frac {b\,\ln \left (c^2\,x^2-1\right )}{48\,c}+\frac {b\,\ln \left (c\,x+1\right )}{24\,c} \] Input:
int((a + b*atanh(c*x))/(c*x + 1)^4,x)
Output:
((b*c^2*x^3)/8 - (b*x)/8 - (b*atanh(c*x))/(3*c) - (12*a + 5*b)/(36*c) + (b *c^3*x^4)/24 + (c*x^2*(24*a + 7*b))/72 + (b*c*x^2*atanh(c*x))/3)/(3*c*x + 2*c^2*x^2 - 2*c^3*x^3 - 3*c^4*x^4 - c^5*x^5 + 1) - (b*log(c^2*x^2 - 1))/(4 8*c) + (b*log(c*x + 1))/(24*c)
Time = 0.18 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.95 \[ \int \frac {a+b \text {arctanh}(c x)}{(1+c x)^4} \, dx=\frac {-48 \mathit {atanh} \left (c x \right ) b -3 \,\mathrm {log}\left (c x -1\right ) b \,c^{3} x^{3}-9 \,\mathrm {log}\left (c x -1\right ) b \,c^{2} x^{2}-9 \,\mathrm {log}\left (c x -1\right ) b c x -3 \,\mathrm {log}\left (c x -1\right ) b +3 \,\mathrm {log}\left (c x +1\right ) b \,c^{3} x^{3}+9 \,\mathrm {log}\left (c x +1\right ) b \,c^{2} x^{2}+9 \,\mathrm {log}\left (c x +1\right ) b c x +3 \,\mathrm {log}\left (c x +1\right ) b -48 a +2 b \,c^{3} x^{3}-12 b c x -18 b}{144 c \left (c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1\right )} \] Input:
int((a+b*atanh(c*x))/(c*x+1)^4,x)
Output:
( - 48*atanh(c*x)*b - 3*log(c*x - 1)*b*c**3*x**3 - 9*log(c*x - 1)*b*c**2*x **2 - 9*log(c*x - 1)*b*c*x - 3*log(c*x - 1)*b + 3*log(c*x + 1)*b*c**3*x**3 + 9*log(c*x + 1)*b*c**2*x**2 + 9*log(c*x + 1)*b*c*x + 3*log(c*x + 1)*b - 48*a + 2*b*c**3*x**3 - 12*b*c*x - 18*b)/(144*c*(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1))