Optimal. Leaf size=96 \[ -\frac {2 b c \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )}{3 x^{3/2}}+\frac {1}{3} c^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2-\frac {\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{3 x^3}+b^2 c^2 \log (x)-\frac {1}{3} b^2 c^2 \log \left (1-c^2 x^3\right ) \]
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Rubi [A]
time = 0.13, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6039, 6037,
6129, 272, 36, 29, 31, 6095} \begin {gather*} \frac {1}{3} c^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2-\frac {2 b c \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )}{3 x^{3/2}}-\frac {\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{3 x^3}-\frac {1}{3} b^2 c^2 \log \left (1-c^2 x^3\right )+b^2 c^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 6037
Rule 6039
Rule 6095
Rule 6129
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{x^4} \, dx &=\int \frac {\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{x^4} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 123, normalized size = 1.28 \begin {gather*} \frac {1}{3} \left (-\frac {a^2}{x^3}-\frac {2 a b c}{x^{3/2}}-\frac {2 b \left (a+b c x^{3/2}\right ) \tanh ^{-1}\left (c x^{3/2}\right )}{x^3}+\frac {b^2 \left (-1+c^2 x^3\right ) \tanh ^{-1}\left (c x^{3/2}\right )^2}{x^3}+3 b^2 c^2 \log (x)-b (a+b) c^2 \log \left (1-c x^{3/2}\right )+(a-b) b c^2 \log \left (1+c x^{3/2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c \,x^{\frac {3}{2}}\right )\right )^{2}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs.
\(2 (80) = 160\).
time = 0.28, size = 175, normalized size = 1.82 \begin {gather*} \frac {1}{3} \, {\left ({\left (c \log \left (c x^{\frac {3}{2}} + 1\right ) - c \log \left (c x^{\frac {3}{2}} - 1\right ) - \frac {2}{x^{\frac {3}{2}}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )}{x^{3}}\right )} a b + \frac {1}{12} \, {\left ({\left (2 \, {\left (\log \left (c x^{\frac {3}{2}} - 1\right ) - 2\right )} \log \left (c x^{\frac {3}{2}} + 1\right ) - \log \left (c x^{\frac {3}{2}} + 1\right )^{2} - \log \left (c x^{\frac {3}{2}} - 1\right )^{2} - 4 \, \log \left (c x^{\frac {3}{2}} - 1\right ) + 12 \, \log \left (x\right )\right )} c^{2} + 4 \, {\left (c \log \left (c x^{\frac {3}{2}} + 1\right ) - c \log \left (c x^{\frac {3}{2}} - 1\right ) - \frac {2}{x^{\frac {3}{2}}}\right )} c \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )^{2}}{3 \, x^{3}} - \frac {a^{2}}{3 \, x^{3}} \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. 173 vs.
\(2 (80) = 160\).
time = 0.37, size = 173, normalized size = 1.80 \begin {gather*} \frac {24 \, b^{2} c^{2} x^{3} \log \left (\sqrt {x}\right ) + 4 \, {\left (a b - b^{2}\right )} c^{2} x^{3} \log \left (c x^{\frac {3}{2}} + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x^{3} \log \left (c x^{\frac {3}{2}} - 1\right ) - 8 \, a b c x^{\frac {3}{2}} + {\left (b^{2} c^{2} x^{3} - b^{2}\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )^{2} - 4 \, a^{2} - 4 \, {\left (b^{2} c x^{\frac {3}{2}} + a b\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 2.04, size = 281, normalized size = 2.93 \begin {gather*} \frac {2\,b^2\,c^2\,\ln \left (x^{3/2}\right )}{3}-\frac {a^2}{3\,x^3}-\frac {b^2\,c^2\,\ln \left (c\,x^{3/2}-1\right )}{3}-\frac {b^2\,c^2\,\ln \left (c\,x^{3/2}+1\right )}{3}+\frac {b^2\,c^2\,{\ln \left (c\,x^{3/2}+1\right )}^2}{12}+\frac {b^2\,c^2\,{\ln \left (1-c\,x^{3/2}\right )}^2}{12}-\frac {b^2\,{\ln \left (c\,x^{3/2}+1\right )}^2}{12\,x^3}-\frac {b^2\,{\ln \left (1-c\,x^{3/2}\right )}^2}{12\,x^3}-\frac {a\,b\,c^2\,\ln \left (c\,x^{3/2}-1\right )}{3}+\frac {a\,b\,c^2\,\ln \left (c\,x^{3/2}+1\right )}{3}-\frac {2\,a\,b\,c}{3\,x^{3/2}}-\frac {a\,b\,\ln \left (c\,x^{3/2}+1\right )}{3\,x^3}+\frac {a\,b\,\ln \left (1-c\,x^{3/2}\right )}{3\,x^3}-\frac {b^2\,c^2\,\ln \left (c\,x^{3/2}+1\right )\,\ln \left (1-c\,x^{3/2}\right )}{6}-\frac {b^2\,c\,\ln \left (c\,x^{3/2}+1\right )}{3\,x^{3/2}}+\frac {b^2\,c\,\ln \left (1-c\,x^{3/2}\right )}{3\,x^{3/2}}+\frac {b^2\,\ln \left (c\,x^{3/2}+1\right )\,\ln \left (1-c\,x^{3/2}\right )}{6\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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