Optimal. Leaf size=117 \[ \frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}+\frac {2}{3} b^2 c^4 \log (x)-\frac {b^2 c^2}{12 x^2}-\frac {1}{3} b^2 c^4 \log \left (1-c^2 x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5916, 5982, 266, 44, 36, 29, 31, 5948} \[ \frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac {b^2 c^2}{12 x^2}-\frac {1}{3} b^2 c^4 \log \left (1-c^2 x^2\right )+\frac {2}{3} b^2 c^4 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 44
Rule 266
Rule 5916
Rule 5948
Rule 5982
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^5} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} (b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} (b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^4} \, dx+\frac {1}{2} \left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{6} \left (b^2 c^2\right ) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx+\frac {1}{2} \left (b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac {1}{2} \left (b c^5\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{2} \left (b^2 c^4\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{12} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {c^2}{x}-\frac {c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2}{12 x^2}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{6} b^2 c^4 \log (x)-\frac {1}{12} b^2 c^4 \log \left (1-c^2 x^2\right )+\frac {1}{4} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (b^2 c^6\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2}{12 x^2}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac {b c^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+\frac {1}{4} c^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac {2}{3} b^2 c^4 \log (x)-\frac {1}{3} b^2 c^4 \log \left (1-c^2 x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 164, normalized size = 1.40 \[ -\frac {3 a^2+3 a b c^4 x^4 \log (1-c x)-3 a b c^4 x^4 \log (c x+1)+6 a b c^3 x^3+2 b \tanh ^{-1}(c x) \left (3 a+3 b c^3 x^3+b c x\right )+2 a b c x-8 b^2 c^4 x^4 \log (x)+4 b^2 c^4 x^4 \log (1-c x)+4 b^2 c^4 x^4 \log (c x+1)-3 b^2 \left (c^4 x^4-1\right ) \tanh ^{-1}(c x)^2+b^2 c^2 x^2}{12 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 173, normalized size = 1.48 \[ \frac {32 \, b^{2} c^{4} x^{4} \log \relax (x) + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{4} x^{4} \log \left (c x + 1\right ) - 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{4} x^{4} \log \left (c x - 1\right ) - 24 \, a b c^{3} x^{3} - 4 \, b^{2} c^{2} x^{2} - 8 \, a b c x + 3 \, {\left (b^{2} c^{4} x^{4} - b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 12 \, a^{2} - 4 \, {\left (3 \, b^{2} c^{3} x^{3} + b^{2} c x + 3 \, a b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{48 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.15, size = 612, normalized size = 5.23 \[ \frac {1}{6} \, {\left (4 \, b^{2} c^{3} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 4 \, b^{2} c^{3} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {3 \, {\left (\frac {{\left (c x + 1\right )}^{3} b^{2} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {{\left (c x + 1\right )} b^{2} c^{3}}{c x - 1}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {2 \, {\left (\frac {6 \, {\left (c x + 1\right )}^{3} a b c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )} a b c^{3}}{c x - 1} + \frac {3 \, {\left (c x + 1\right )}^{3} b^{2} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2} b^{2} c^{3}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )} b^{2} c^{3}}{c x - 1} + 2 \, b^{2} c^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {2 \, {\left (\frac {6 \, {\left (c x + 1\right )}^{3} a^{2} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )} a^{2} c^{3}}{c x - 1} + \frac {6 \, {\left (c x + 1\right )}^{3} a b c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {12 \, {\left (c x + 1\right )}^{2} a b c^{3}}{{\left (c x - 1\right )}^{2}} + \frac {10 \, {\left (c x + 1\right )} a b c^{3}}{c x - 1} + 4 \, a b c^{3} + \frac {{\left (c x + 1\right )}^{3} b^{2} c^{3}}{{\left (c x - 1\right )}^{3}} + \frac {2 \, {\left (c x + 1\right )}^{2} b^{2} c^{3}}{{\left (c x - 1\right )}^{2}} + \frac {{\left (c x + 1\right )} b^{2} c^{3}}{c x - 1}\right )}}{\frac {{\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {6 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {4 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 290, normalized size = 2.48 \[ -\frac {a^{2}}{4 x^{4}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{4 x^{4}}-\frac {c \,b^{2} \arctanh \left (c x \right )}{6 x^{3}}-\frac {c^{3} b^{2} \arctanh \left (c x \right )}{2 x}-\frac {c^{4} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{4}+\frac {c^{4} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{4}-\frac {c^{4} b^{2} \ln \left (c x -1\right )^{2}}{16}+\frac {c^{4} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{8}-\frac {c^{4} b^{2} \ln \left (c x +1\right )^{2}}{16}-\frac {c^{4} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{8}+\frac {c^{4} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{8}-\frac {b^{2} c^{2}}{12 x^{2}}+\frac {2 c^{4} b^{2} \ln \left (c x \right )}{3}-\frac {c^{4} b^{2} \ln \left (c x -1\right )}{3}-\frac {c^{4} b^{2} \ln \left (c x +1\right )}{3}-\frac {a b \arctanh \left (c x \right )}{2 x^{4}}-\frac {a b c}{6 x^{3}}-\frac {c^{3} a b}{2 x}-\frac {c^{4} a b \ln \left (c x -1\right )}{4}+\frac {c^{4} a b \ln \left (c x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 224, normalized size = 1.91 \[ \frac {1}{12} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} a b + \frac {1}{48} \, {\left ({\left (32 \, c^{2} \log \relax (x) - \frac {3 \, c^{2} x^{2} \log \left (c x + 1\right )^{2} + 3 \, c^{2} x^{2} \log \left (c x - 1\right )^{2} + 16 \, c^{2} x^{2} \log \left (c x - 1\right ) - 2 \, {\left (3 \, c^{2} x^{2} \log \left (c x - 1\right ) - 8 \, c^{2} x^{2}\right )} \log \left (c x + 1\right ) + 4}{x^{2}}\right )} c^{2} + 4 \, {\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c \operatorname {artanh}\left (c x\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x\right )^{2}}{4 \, x^{4}} - \frac {a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.90, size = 303, normalized size = 2.59 \[ \frac {b^2\,c^4\,{\ln \left (c\,x+1\right )}^2}{16}-\frac {a^2}{4\,x^4}+\frac {b^2\,c^4\,{\ln \left (1-c\,x\right )}^2}{16}-\frac {b^2\,{\ln \left (c\,x+1\right )}^2}{16\,x^4}-\frac {b^2\,{\ln \left (1-c\,x\right )}^2}{16\,x^4}-\frac {b^2\,c^2}{12\,x^2}+\frac {2\,b^2\,c^4\,\ln \relax (x)}{3}-\frac {b^2\,c^4\,\ln \left (c\,x-1\right )}{3}-\frac {b^2\,c^4\,\ln \left (c\,x+1\right )}{3}-\frac {a\,b\,\ln \left (c\,x+1\right )}{4\,x^4}+\frac {a\,b\,\ln \left (1-c\,x\right )}{4\,x^4}+\frac {b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{8\,x^4}-\frac {a\,b\,c}{6\,x^3}-\frac {b^2\,c\,\ln \left (c\,x+1\right )}{12\,x^3}+\frac {b^2\,c\,\ln \left (1-c\,x\right )}{12\,x^3}-\frac {a\,b\,c^3}{2\,x}-\frac {b^2\,c^3\,\ln \left (c\,x+1\right )}{4\,x}+\frac {b^2\,c^3\,\ln \left (1-c\,x\right )}{4\,x}-\frac {a\,b\,c^4\,\ln \left (c\,x-1\right )}{4}+\frac {a\,b\,c^4\,\ln \left (c\,x+1\right )}{4}-\frac {b^2\,c^4\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.00, size = 184, normalized size = 1.57 \[ \begin {cases} - \frac {a^{2}}{4 x^{4}} + \frac {a b c^{4} \operatorname {atanh}{\left (c x \right )}}{2} - \frac {a b c^{3}}{2 x} - \frac {a b c}{6 x^{3}} - \frac {a b \operatorname {atanh}{\left (c x \right )}}{2 x^{4}} + \frac {2 b^{2} c^{4} \log {\relax (x )}}{3} - \frac {2 b^{2} c^{4} \log {\left (x - \frac {1}{c} \right )}}{3} + \frac {b^{2} c^{4} \operatorname {atanh}^{2}{\left (c x \right )}}{4} - \frac {2 b^{2} c^{4} \operatorname {atanh}{\left (c x \right )}}{3} - \frac {b^{2} c^{3} \operatorname {atanh}{\left (c x \right )}}{2 x} - \frac {b^{2} c^{2}}{12 x^{2}} - \frac {b^{2} c \operatorname {atanh}{\left (c x \right )}}{6 x^{3}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{4 x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________