Optimal. Leaf size=85 \[ -\frac {2 b c \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{\sqrt {x}}+c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{x}+b^2 c^2 \log (x)-b^2 c^2 \log \left (1-c^2 x\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 85, 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*} c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2-\frac {2 b c \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{\sqrt {x}}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{x}+b^2 c^2 \log (x)-b^2 c^2 \log \left (1-c^2 x\right ) \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 \sqrt {x}\right )\right )^2}{x^2} \, dx &=\int \frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 129, normalized size = 1.52 \begin {gather*} -\frac {a^2+2 a b c \sqrt {x}+2 b \left (a+b c \sqrt {x}\right ) \tanh ^{-1}\left (c \sqrt {x}\right )-b^2 \left (-1+c^2 x\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2+b (a+b) c^2 x \log \left (1-c \sqrt {x}\right )-a b c^2 x \log \left (1+c \sqrt {x}\right )+b^2 c^2 x \log \left (1+c \sqrt {x}\right )-b^2 c^2 x \log (x)}{x} \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. \(274\) vs.
\(2(77)=154\).
time = 0.11, size = 275, normalized size = 3.24
method | result | size |
derivativedivides | \(2 c^{2} \left (-\frac {a^{2}}{2 c^{2} x}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right )^{2}}{2 c^{2} x}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right )}{c \sqrt {x}}-\frac {b^{2} \ln \left (c \sqrt {x}-1\right )^{2}}{8}+\frac {b^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {b^{2} \ln \left (1+c \sqrt {x}\right )}{2}+b^{2} \ln \left (c \sqrt {x}\right )-\frac {b^{2} \ln \left (c \sqrt {x}-1\right )}{2}-\frac {b^{2} \ln \left (1+c \sqrt {x}\right )^{2}}{8}-\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{4}-\frac {a b \arctanh \left (c \sqrt {x}\right )}{c^{2} x}+\frac {a b \ln \left (1+c \sqrt {x}\right )}{2}-\frac {a b \ln \left (c \sqrt {x}-1\right )}{2}-\frac {a b}{c \sqrt {x}}\right )\) | \(275\) |
default | \(2 c^{2} \left (-\frac {a^{2}}{2 c^{2} x}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right )^{2}}{2 c^{2} x}+\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}-\frac {b^{2} \arctanh \left (c \sqrt {x}\right )}{c \sqrt {x}}-\frac {b^{2} \ln \left (c \sqrt {x}-1\right )^{2}}{8}+\frac {b^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {b^{2} \ln \left (1+c \sqrt {x}\right )}{2}+b^{2} \ln \left (c \sqrt {x}\right )-\frac {b^{2} \ln \left (c \sqrt {x}-1\right )}{2}-\frac {b^{2} \ln \left (1+c \sqrt {x}\right )^{2}}{8}-\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {b^{2} \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{4}-\frac {a b \arctanh \left (c \sqrt {x}\right )}{c^{2} x}+\frac {a b \ln \left (1+c \sqrt {x}\right )}{2}-\frac {a b \ln \left (c \sqrt {x}-1\right )}{2}-\frac {a b}{c \sqrt {x}}\right )\) | \(275\) |
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. 174 vs.
\(2 (77) = 154\).
time = 0.27, size = 174, normalized size = 2.05 \begin {gather*} {\left ({\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c \sqrt {x}\right )}{x}\right )} a b + \frac {1}{4} \, {\left ({\left (2 \, {\left (\log \left (c \sqrt {x} - 1\right ) - 2\right )} \log \left (c \sqrt {x} + 1\right ) - \log \left (c \sqrt {x} + 1\right )^{2} - \log \left (c \sqrt {x} - 1\right )^{2} - 4 \, \log \left (c \sqrt {x} - 1\right ) + 4 \, \log \left (x\right )\right )} c^{2} + 4 \, {\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c \operatorname {artanh}\left (c \sqrt {x}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2}}{x} - \frac {a^{2}}{x} \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. 157 vs.
\(2 (77) = 154\).
time = 0.37, size = 157, normalized size = 1.85 \begin {gather*} \frac {8 \, b^{2} c^{2} x \log \left (\sqrt {x}\right ) + 4 \, {\left (a b - b^{2}\right )} c^{2} x \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x \log \left (c \sqrt {x} - 1\right ) - 8 \, a b c \sqrt {x} + {\left (b^{2} c^{2} x - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} - 4 \, a^{2} - 4 \, {\left (b^{2} c \sqrt {x} + a b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 680 vs.
\(2 (78) = 156\).
time = 3.47, size = 680, normalized size = 8.00 \begin {gather*} \begin {cases} - \frac {a^{2}}{x} + \frac {2 a b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} & \text {for}\: c = - \sqrt {\frac {1}{x}} \\- \frac {a^{2}}{x} - \frac {2 a b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} & \text {for}\: c = \sqrt {\frac {1}{x}} \\- \frac {a^{2}}{x} & \text {for}\: c = 0 \\- \frac {a^{2} c^{2} x^{\frac {3}{2}}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {a^{2} \sqrt {x}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 a b c^{4} x^{\frac {5}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 a b c^{3} x^{2}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {4 a b c^{2} x^{\frac {3}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 a b c x}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 a b \sqrt {x} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} c^{4} x^{\frac {5}{2}} \log {\left (x \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{4} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \frac {1}{c} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} c^{4} x^{\frac {5}{2}} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{4} x^{\frac {5}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{3} x^{2} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {b^{2} c^{2} x^{\frac {3}{2}} \log {\left (x \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \frac {1}{c} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c x \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} \sqrt {x} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \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 = 1.79, size = 278, normalized size = 3.27 \begin {gather*} 2\,b^2\,c^2\,\ln \left (\sqrt {x}\right )-\frac {a^2}{x}-b^2\,c^2\,\ln \left (c\,\sqrt {x}-1\right )-b^2\,c^2\,\ln \left (c\,\sqrt {x}+1\right )+\frac {b^2\,c^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{4}+\frac {b^2\,c^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{4}-\frac {b^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{4\,x}-\frac {b^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{4\,x}-a\,b\,c^2\,\ln \left (c\,\sqrt {x}-1\right )+a\,b\,c^2\,\ln \left (c\,\sqrt {x}+1\right )-\frac {2\,a\,b\,c}{\sqrt {x}}-\frac {a\,b\,\ln \left (c\,\sqrt {x}+1\right )}{x}+\frac {a\,b\,\ln \left (1-c\,\sqrt {x}\right )}{x}-\frac {b^2\,c^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{2}-\frac {b^2\,c\,\ln \left (c\,\sqrt {x}+1\right )}{\sqrt {x}}+\frac {b^2\,c\,\ln \left (1-c\,\sqrt {x}\right )}{\sqrt {x}}+\frac {b^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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