Optimal. Leaf size=166 \[ \frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (2-\frac {2}{1+c+d x}\right )}{d e^3}-\frac {3 b^3 \text {PolyLog}\left (2,-1+\frac {2}{1+c+d x}\right )}{2 d e^3} \]
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Rubi [A]
time = 0.24, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6242, 12, 6037,
6129, 6135, 6079, 2497, 6095} \begin {gather*} \frac {3 b^2 \log \left (2-\frac {2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^3}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac {3 b^3 \text {Li}_2\left (\frac {2}{c+d x+1}-1\right )}{2 d e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2497
Rule 6037
Rule 6079
Rule 6095
Rule 6129
Rule 6135
Rule 6242
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{(c e+d e x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^3}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^3}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x^2 \left (1-x^2\right )} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x^2} \, dx,x,c+d x\right )}{2 d e^3}+\frac {(3 b) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x \left (1-x^2\right )} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x (1+x)} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (2-\frac {2}{1+c+d x}\right )}{d e^3}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (2-\frac {2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac {3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac {\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (2-\frac {2}{1+c+d x}\right )}{d e^3}-\frac {3 b^3 \text {Li}_2\left (-1+\frac {2}{1+c+d x}\right )}{2 d e^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.83, size = 335, normalized size = 2.02 \begin {gather*} \frac {-4 a^3-12 a^2 b c+i b^3 c^3 \pi ^3-12 a^2 b d x+2 i b^3 c^2 d \pi ^3 x+i b^3 c d^2 \pi ^3 x^2+12 b^2 (-1+c+d x) (b (c+d x)+a (1+c+d x)) \tanh ^{-1}(c+d x)^2+4 b^3 \left (-1+c^2+2 c d x+d^2 x^2\right ) \tanh ^{-1}(c+d x)^3+12 b \tanh ^{-1}(c+d x) \left (a \left (-2 b (c+d x)+a \left (-1+c^2+2 c d x+d^2 x^2\right )\right )+2 b^2 (c+d x)^2 \log \left (1-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+24 a b^2 c^2 \log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )+48 a b^2 c d x \log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )+24 a b^2 d^2 x^2 \log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )-12 b^3 (c+d x)^2 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c+d x)}\right )}{8 d e^3 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 3.13, size = 5530, normalized size = 33.31
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(5530\) |
default | \(\text {Expression too large to display}\) | \(5530\) |
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*} \frac {\int \frac {a^{3}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{3} \operatorname {atanh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a b^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a^{2} b \operatorname {atanh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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