Optimal. Leaf size=114 \[ -\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3}{a^2 \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {5}{2 a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {5 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{16 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{a^2}+\frac {9 \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{16 a^2} \]
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Rubi [A]
time = 0.35, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6179, 6175,
6113, 6181, 5556, 3379} \begin {gather*} \frac {5 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{16 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{a^2}+\frac {9 \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{16 a^2}-\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}+\frac {5}{2 a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {3}{a^2 \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 5556
Rule 6113
Rule 6175
Rule 6179
Rule 6181
Rubi steps
\begin {align*} \int \frac {x}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)^3} \, dx &=-\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\frac {1}{2} (5 a) \int \frac {x^2}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {1}{2 a^2 \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+3 \int \frac {x}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx+\frac {5 \int \frac {1}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)^2} \, dx}{2 a}-\frac {5 \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3}{a^2 \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {5}{2 a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-10 \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+15 \int \frac {x}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx+\frac {3 \text {Subst}\left (\int \frac {\cosh ^5(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3}{a^2 \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {5}{2 a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3 \text {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 x}+\frac {\sinh (4 x)}{8 x}+\frac {\sinh (6 x)}{32 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}-\frac {10 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}+\frac {15 \text {Subst}\left (\int \frac {\cosh ^5(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3}{a^2 \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {5}{2 a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3 \text {Subst}\left (\int \frac {\sinh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^2}+\frac {3 \text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^2}+\frac {15 \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^2}-\frac {10 \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}+\frac {15 \text {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 x}+\frac {\sinh (4 x)}{8 x}+\frac {\sinh (6 x)}{32 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3}{a^2 \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {5}{2 a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {15 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^2}+\frac {3 \text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2}+\frac {3 \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^2}+\frac {15 \text {Subst}\left (\int \frac {\sinh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^2}-\frac {5 \text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^2}+\frac {15 \text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^2}+\frac {75 \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^2}-\frac {5 \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}-\frac {3}{a^2 \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}+\frac {5}{2 a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {5 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{16 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{a^2}+\frac {9 \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{16 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 73, normalized size = 0.64 \begin {gather*} \frac {\frac {8 \left (a x+\left (1+5 a^2 x^2\right ) \tanh ^{-1}(a x)\right )}{\left (-1+a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}+5 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+16 \text {Shi}\left (4 \tanh ^{-1}(a x)\right )+9 \text {Shi}\left (6 \tanh ^{-1}(a x)\right )}{16 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.71, size = 121, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (4 \arctanh \left (a x \right )\right )}{16 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )}+\hyperbolicSineIntegral \left (4 \arctanh \left (a x \right )\right )-\frac {\sinh \left (6 \arctanh \left (a x \right )\right )}{64 \arctanh \left (a x \right )^{2}}-\frac {3 \cosh \left (6 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )}+\frac {9 \hyperbolicSineIntegral \left (6 \arctanh \left (a x \right )\right )}{16}-\frac {5 \sinh \left (2 \arctanh \left (a x \right )\right )}{64 \arctanh \left (a x \right )^{2}}-\frac {5 \cosh \left (2 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )}+\frac {5 \hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )}{16}}{a^{2}}\) | \(121\) |
default | \(\frac {-\frac {\sinh \left (4 \arctanh \left (a x \right )\right )}{16 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )}+\hyperbolicSineIntegral \left (4 \arctanh \left (a x \right )\right )-\frac {\sinh \left (6 \arctanh \left (a x \right )\right )}{64 \arctanh \left (a x \right )^{2}}-\frac {3 \cosh \left (6 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )}+\frac {9 \hyperbolicSineIntegral \left (6 \arctanh \left (a x \right )\right )}{16}-\frac {5 \sinh \left (2 \arctanh \left (a x \right )\right )}{64 \arctanh \left (a x \right )^{2}}-\frac {5 \cosh \left (2 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )}+\frac {5 \hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )}{16}}{a^{2}}\) | \(121\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 447 vs.
\(2 (103) = 206\).
time = 0.36, size = 447, normalized size = 3.92 \begin {gather*} \frac {{\left (9 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - 9 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) + 16 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 16 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 5 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - 5 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 64 \, a x + 32 \, {\left (5 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{32 \, {\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \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 \frac {x}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname {atanh}^{3}{\left (a 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 = -0.01 \begin {gather*} \int \frac {x}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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