Optimal. Leaf size=221 \[ \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {b^2 (d e-c f) \text {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d^2} \]
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Rubi [A]
time = 0.31, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6247, 6066,
6022, 266, 6196, 6096, 6132, 6056, 2449, 2352} \begin {gather*} \frac {\left (-\frac {\left (c^2+1\right ) f}{d}+2 c e-\frac {d e^2}{f}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {2 b (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}+\frac {a b f x}{d}-\frac {b^2 (d e-c f) \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 2352
Rule 2449
Rule 6022
Rule 6056
Rule 6066
Rule 6096
Rule 6132
Rule 6196
Rule 6247
Rubi steps
\begin {align*} \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \left (-\frac {f^2 \left (a+b \coth ^{-1}(x)\right )}{d^2}+\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x\right ) \left (a+b \coth ^{-1}(x)\right )}{d^2 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2 f}+\frac {(b f) \text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {a b f x}{d}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \left (\frac {d^2 e^2 \left (1+\frac {f \left (-2 c d e+f+c^2 f\right )}{d^2 e^2}\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2}-\frac {2 f (-d e+c f) x \left (a+b \coth ^{-1}(x)\right )}{1-x^2}\right ) \, dx,x,c+d x\right )}{d^2 f}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{d^2}-\frac {(2 b (d e-c f)) \text {Subst}\left (\int \frac {x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2}-\frac {\left (b \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{d^2 f}\\ &=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {(2 b (d e-c f)) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}+\frac {\left (2 b^2 (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^2}\\ &=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {\left (2 b^2 (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^2}\\ &=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {b^2 (d e-c f) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{d^2}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 295, normalized size = 1.33 \begin {gather*} \frac {2 a^2 c d e+2 a b c f-a^2 c^2 f+2 a^2 d^2 e x+2 a b d f x+a^2 d^2 f x^2+b^2 (-1+c+d x) (2 d e+f-c f+d f x) \coth ^{-1}(c+d x)^2+2 b \coth ^{-1}(c+d x) \left (-((c+d x) (-b f+a c f-a d (2 e+f x)))-2 b (d e-c f) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+a b f \log (1-c-d x)-a b f \log (1+c+d x)-4 a b d e \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )-2 b^2 f \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )+4 a b c f \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )+2 b^2 (d e-c f) \text {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(811\) vs.
\(2(217)=434\).
time = 0.33, size = 812, normalized size = 3.67 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.45, size = 422, normalized size = 1.91 \begin {gather*} \frac {1}{2} \, a^{2} f x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a b f + a^{2} x e + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b e}{d} + \frac {{\left (c f + f\right )} b^{2} \log \left (d x + c + 1\right )}{2 \, d^{2}} + \frac {{\left (b^{2} c f - b^{2} d e\right )} {\left (\log \left (d x + c - 1\right ) \log \left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right )\right )}}{d^{2}} + \frac {{\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} x e + 2 \, {\left (c d + d\right )} b^{2} e - {\left (c^{2} f + 2 \, c f + f\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} x e + 2 \, {\left (c d - d\right )} b^{2} e - {\left (c^{2} f - 2 \, c f + f\right )} b^{2}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (2 \, b^{2} d f x - {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} x e + 2 \, {\left (c d - d\right )} b^{2} e - {\left (c^{2} f - 2 \, c f + f\right )} b^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) - 4 \, {\left (b^{2} d f x + {\left (c f - f\right )} b^{2}\right )} \log \left (d x + c - 1\right )}{8 \, d^{2}} \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 {acoth}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, 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 )\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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