Optimal. Leaf size=263 \[ \frac {a \left (a^2+1\right ) \text {Li}_2\left (-\frac {a+b x+1}{-a-b x+1}\right )}{b^4}+\frac {\left (6 a^2+1\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac {a \left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{b^4}+\frac {\left (6 a^2+1\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}+\frac {2 a \left (a^2+1\right ) \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^4}-\frac {\left (a^4+6 a^2+1\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {(a+b x)^2}{12 b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {a \tanh ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}-\frac {a x}{b^3}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.35, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {6112, 5929, 5911, 260, 5917, 321, 206, 266, 43, 6049, 5949, 5985, 5919, 2402, 2315} \[ \frac {a \left (a^2+1\right ) \text {PolyLog}\left (2,-\frac {a+b x+1}{-a-b x+1}\right )}{b^4}+\frac {\left (6 a^2+1\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac {a \left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac {\left (a^4+6 a^2+1\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {\left (6 a^2+1\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}+\frac {2 a \left (a^2+1\right ) \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^4}-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {a \tanh ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 206
Rule 260
Rule 266
Rule 321
Rule 2315
Rule 2402
Rule 5911
Rule 5917
Rule 5919
Rule 5929
Rule 5949
Rule 5985
Rule 6049
Rule 6112
Rubi steps
\begin {align*} \int x^3 \coth ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^3 \coth ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2-\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {\left (1+6 a^2\right ) \coth ^{-1}(x)}{b^4}+\frac {4 a x \coth ^{-1}(x)}{b^4}-\frac {x^2 \coth ^{-1}(x)}{b^4}+\frac {\left (1+6 a^2+a^4-4 a \left (1+a^2\right ) x\right ) \coth ^{-1}(x)}{b^4 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac {\operatorname {Subst}\left (\int x^2 \coth ^{-1}(x) \, dx,x,a+b x\right )}{2 b^4}-\frac {\operatorname {Subst}\left (\int \frac {\left (1+6 a^2+a^4-4 a \left (1+a^2\right ) x\right ) \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{2 b^4}-\frac {(2 a) \operatorname {Subst}\left (\int x \coth ^{-1}(x) \, dx,x,a+b x\right )}{b^4}+\frac {\left (1+6 a^2\right ) \operatorname {Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{2 b^4}\\ &=\frac {\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int \frac {x^3}{1-x^2} \, dx,x,a+b x\right )}{6 b^4}-\frac {\operatorname {Subst}\left (\int \left (\frac {\left (1+a^2 \left (6+a^2\right )\right ) \coth ^{-1}(x)}{1-x^2}-\frac {4 a \left (1+a^2\right ) x \coth ^{-1}(x)}{1-x^2}\right ) \, dx,x,a+b x\right )}{2 b^4}+\frac {a \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,a+b x\right )}{b^4}-\frac {\left (1+6 a^2\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x\right )}{2 b^4}\\ &=-\frac {a x}{b^3}+\frac {\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac {\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac {\operatorname {Subst}\left (\int \frac {x}{1-x} \, dx,x,(a+b x)^2\right )}{12 b^4}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,a+b x\right )}{b^4}+\frac {\left (2 a \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b^4}-\frac {\left (1+6 a^2+a^4\right ) \operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{2 b^4}\\ &=-\frac {a x}{b^3}+\frac {\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac {\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac {a \tanh ^{-1}(a+b x)}{b^4}+\frac {\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac {\operatorname {Subst}\left (\int \left (-1+\frac {1}{1-x}\right ) \, dx,x,(a+b x)^2\right )}{12 b^4}+\frac {\left (2 a \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x} \, dx,x,a+b x\right )}{b^4}\\ &=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac {\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac {a \tanh ^{-1}(a+b x)}{b^4}+\frac {2 a \left (1+a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}-\frac {\left (2 a \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{b^4}\\ &=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac {\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac {a \tanh ^{-1}(a+b x)}{b^4}+\frac {2 a \left (1+a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}+\frac {\left (2 a \left (1+a^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a-b x}\right )}{b^4}\\ &=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {\left (1+6 a^2\right ) (a+b x) \coth ^{-1}(a+b x)}{2 b^4}-\frac {a (a+b x)^2 \coth ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{6 b^4}-\frac {a \left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{b^4}-\frac {\left (1+6 a^2+a^4\right ) \coth ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \coth ^{-1}(a+b x)^2+\frac {a \tanh ^{-1}(a+b x)}{b^4}+\frac {2 a \left (1+a^2\right ) \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^4}+\frac {\log \left (1-(a+b x)^2\right )}{12 b^4}+\frac {\left (1+6 a^2\right ) \log \left (1-(a+b x)^2\right )}{4 b^4}+\frac {a \left (1+a^2\right ) \text {Li}_2\left (1-\frac {2}{1-a-b x}\right )}{b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.70, size = 203, normalized size = 0.77 \[ -\frac {12 \left (a^3+a\right ) \text {Li}_2\left (e^{-2 \coth ^{-1}(a+b x)}\right )+36 a^2 \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )+11 a^2-2 \coth ^{-1}(a+b x) \left (12 \left (a^3+a\right ) \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )+13 a^3+9 a^2 b x-3 a b^2 x^2+9 a+b^3 x^3+3 b x\right )+3 \left (a^4-4 a^3+6 a^2-4 a-b^4 x^4+1\right ) \coth ^{-1}(a+b x)^2+10 a b x+8 \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )-b^2 x^2+1}{12 b^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {arcoth}\left (b x + a\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {arcoth}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 967, normalized size = 3.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 320, normalized size = 1.22 \[ \frac {1}{4} \, x^{4} \operatorname {arcoth}\left (b x + a\right )^{2} + \frac {1}{48} \, b^{2} {\left (\frac {48 \, {\left (a^{3} + a\right )} {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )}}{b^{6}} + \frac {4 \, {\left (13 \, a^{3} + 18 \, a^{2} + 9 \, a + 4\right )} \log \left (b x + a + 1\right )}{b^{6}} + \frac {4 \, b^{2} x^{2} - 40 \, a b x + 3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 6 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + 3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, {\left (13 \, a^{3} - 18 \, a^{2} + 9 \, a - 4\right )} \log \left (b x + a - 1\right )}{b^{6}}\right )} + \frac {1}{12} \, b {\left (\frac {2 \, {\left (b^{2} x^{3} - 3 \, a b x^{2} + 3 \, {\left (3 \, a^{2} + 1\right )} x\right )}}{b^{4}} - \frac {3 \, {\left (a^{4} + 4 \, a^{3} + 6 \, a^{2} + 4 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{5}} + \frac {3 \, {\left (a^{4} - 4 \, a^{3} + 6 \, a^{2} - 4 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{5}}\right )} \operatorname {arcoth}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {acoth}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________