Optimal. Leaf size=546 \[ \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d^3} \]
[Out]
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Rubi [A]
time = 0.75, antiderivative size = 546, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used =
{6246, 6065, 6021, 6131, 6055, 2449, 2352, 6037, 6127, 266, 6095, 6195, 6205, 6745}
\begin {gather*} -\frac {b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_2\left (1-\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^3}-\frac {6 b^2 f (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d^3}+\frac {a b^2 f^2 x}{d^2}-\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (c+d x) (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}+\frac {b^3 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_3\left (1-\frac {2}{-c-d x+1}\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6065
Rule 6095
Rule 6127
Rule 6131
Rule 6195
Rule 6205
Rule 6246
Rule 6745
Rubi steps
\begin {align*} \int (e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \tanh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \text {Subst}\left (\int \left (-\frac {3 f^2 (d e-c f) \left (a+b \tanh ^{-1}(x)\right )^2}{d^3}-\frac {f^3 x \left (a+b \tanh ^{-1}(x)\right )^2}{d^3}+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \tanh ^{-1}(x)\right )^2}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \text {Subst}\left (\int \frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^3 f}+\frac {\left (b f^2\right ) \text {Subst}\left (\int x \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}+\frac {(3 b f (d e-c f)) \text {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}\\ &=\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {b \text {Subst}\left (\int \left (\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2}+\frac {f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2}\right ) \, dx,x,c+d x\right )}{d^3 f}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b (d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^3 f}-\frac {\left (b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \tanh ^{-1}(x) \, dx,x,c+d x\right )}{d^3}+\frac {\left (6 b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{1-x} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{d^3}+\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right ) \log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {\left (b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \tanh ^{-1}(c+d x)}{d^3}-\frac {b f^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 b f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \tanh ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 f^2 \log \left (1-(c+d x)^2\right )}{2 d^3}-\frac {3 b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \tanh ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text {Li}_3\left (1-\frac {2}{1-c-d x}\right )}{2 d^3}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1646\) vs. \(2(546)=1092\).
time = 8.07, size = 1646, normalized size = 3.01 \begin {gather*} a^2 \left (a e^2+\frac {b f (3 d e-2 c f)}{d^2}\right ) x+\frac {a^2 f (2 a d e+b f) x^2}{2 d}+\frac {1}{3} a^3 f^2 x^3+a^2 b x \left (3 e^2+3 e f x+f^2 x^2\right ) \tanh ^{-1}(c+d x)-\frac {a^2 b (-1+c) \left (3 d^2 e^2-3 (-1+c) d e f+(-1+c)^2 f^2\right ) \log (1-c-d x)}{2 d^3}+\frac {a^2 b (1+c) \left (3 d^2 e^2-3 (1+c) d e f+(1+c)^2 f^2\right ) \log (1+c+d x)}{2 d^3}+\frac {3 a b^2 e^2 \left (\tanh ^{-1}(c+d x) \left ((-1+c+d x) \tanh ^{-1}(c+d x)-2 \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{d}-\frac {3 a b^2 e f \left (\left (1-2 c+c^2-d^2 x^2\right ) \tanh ^{-1}(c+d x)^2-2 \tanh ^{-1}(c+d x) \left (c+d x+2 c \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+2 \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+2 c \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{d^2}+\frac {b^3 e^2 \left (2 \tanh ^{-1}(c+d x)^2 \left ((-1+c+d x) \tanh ^{-1}(c+d x)-3 \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+6 \tanh ^{-1}(c+d x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )+3 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{2 d}-\frac {b^3 e f \left (\tanh ^{-1}(c+d x) \left (\left (1-2 c+c^2-d^2 x^2\right ) \tanh ^{-1}(c+d x)^2+6 \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )-3 \tanh ^{-1}(c+d x) \left (-1+c+d x+2 c \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )\right )+\left (-3+6 c \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )+3 c \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )}{d^2}-\frac {a b^2 f^2 \left (1-(c+d x)^2\right )^{3/2} \left (-\frac {c+d x}{\sqrt {1-(c+d x)^2}}+\frac {6 c (c+d x) \tanh ^{-1}(c+d x)}{\sqrt {1-(c+d x)^2}}+\frac {3 (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt {1-(c+d x)^2}}-\frac {3 c^2 (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt {1-(c+d x)^2}}+\tanh ^{-1}(c+d x)^2 \cosh \left (3 \tanh ^{-1}(c+d x)\right )+3 c^2 \tanh ^{-1}(c+d x)^2 \cosh \left (3 \tanh ^{-1}(c+d x)\right )+2 \tanh ^{-1}(c+d x) \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )+6 c^2 \tanh ^{-1}(c+d x) \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )-6 c \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\frac {3 \left (1-4 c+3 c^2\right ) \tanh ^{-1}(c+d x)^2+2 \tanh ^{-1}(c+d x) \left (2+\left (3+9 c^2\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )-18 c \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )}{\sqrt {1-(c+d x)^2}}-\frac {4 \left (1+3 c^2\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )}{\left (1-(c+d x)^2\right )^{3/2}}-\sinh \left (3 \tanh ^{-1}(c+d x)\right )+6 c \tanh ^{-1}(c+d x) \sinh \left (3 \tanh ^{-1}(c+d x)\right )-\tanh ^{-1}(c+d x)^2 \sinh \left (3 \tanh ^{-1}(c+d x)\right )-3 c^2 \tanh ^{-1}(c+d x)^2 \sinh \left (3 \tanh ^{-1}(c+d x)\right )\right )}{4 d^3}+\frac {b^3 f^2 \left (\left (-3 c+\left (1+3 c^2\right ) \tanh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )-\frac {1}{12} \left (1-(c+d x)^2\right )^{3/2} \left (-\frac {3 (c+d x) \tanh ^{-1}(c+d x)}{\sqrt {1-(c+d x)^2}}+\frac {9 c (c+d x) \tanh ^{-1}(c+d x)^2}{\sqrt {1-(c+d x)^2}}+\frac {3 (c+d x) \tanh ^{-1}(c+d x)^3}{\sqrt {1-(c+d x)^2}}-\frac {3 c^2 (c+d x) \tanh ^{-1}(c+d x)^3}{\sqrt {1-(c+d x)^2}}-9 c \tanh ^{-1}(c+d x)^2 \cosh \left (3 \tanh ^{-1}(c+d x)\right )+\tanh ^{-1}(c+d x)^3 \cosh \left (3 \tanh ^{-1}(c+d x)\right )+3 c^2 \tanh ^{-1}(c+d x)^3 \cosh \left (3 \tanh ^{-1}(c+d x)\right )-18 c \tanh ^{-1}(c+d x) \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )+3 \tanh ^{-1}(c+d x)^2 \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )+9 c^2 \tanh ^{-1}(c+d x)^2 \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )+3 \cosh \left (3 \tanh ^{-1}(c+d x)\right ) \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\frac {3 \left (\left (1-4 c+3 c^2\right ) \tanh ^{-1}(c+d x)^3-18 c \tanh ^{-1}(c+d x) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )+\tanh ^{-1}(c+d x)^2 \left (2-9 c+\left (3+9 c^2\right ) \log \left (1+e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+3 \log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )\right )}{\sqrt {1-(c+d x)^2}}-\frac {6 \left (1+3 c^2\right ) \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c+d x)}\right )}{\left (1-(c+d x)^2\right )^{3/2}}-3 \tanh ^{-1}(c+d x) \sinh \left (3 \tanh ^{-1}(c+d x)\right )+9 c \tanh ^{-1}(c+d x)^2 \sinh \left (3 \tanh ^{-1}(c+d x)\right )-\tanh ^{-1}(c+d x)^3 \sinh \left (3 \tanh ^{-1}(c+d x)\right )-3 c^2 \tanh ^{-1}(c+d x)^3 \sinh \left (3 \tanh ^{-1}(c+d x)\right )\right )\right )}{d^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 50.18, size = 12235, normalized size = 22.41
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(12235\) |
default | \(\text {Expression too large to display}\) | \(12235\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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