Optimal. Leaf size=172 \[ -\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{4} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right ) \]
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Rubi [A]
time = 0.16, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6037, 335,
216, 648, 632, 210, 642, 212} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )-\frac {1}{4} b c^{2/3} \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )+\frac {1}{4} b c^{2/3} \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )+b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 216
Rule 335
Rule 632
Rule 642
Rule 648
Rule 6037
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x^2} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}+\frac {1}{2} (3 b c) \int \frac {1}{\sqrt {x} \left (1-c^2 x^3\right )} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}+(3 b c) \text {Subst}\left (\int \frac {1}{1-c^2 x^6} \, dx,x,\sqrt {x}\right )\\ &=-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}+(b c) \text {Subst}\left (\int \frac {1}{1-c^{2/3} x^2} \, dx,x,\sqrt {x}\right )+(b c) \text {Subst}\left (\int \frac {1-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )+(b c) \text {Subst}\left (\int \frac {1+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )\\ &=b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{4} \left (b c^{2/3}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{4} \left (b c^{2/3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{4} (3 b c) \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{4} (3 b c) \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx,x,\sqrt {x}\right )\\ &=b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{4} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{2} \left (3 b c^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} \sqrt {x}\right )-\frac {1}{2} \left (3 b c^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} \sqrt {x}\right )\\ &=-\frac {1}{2} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{4} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 205, normalized size = 1.19 \begin {gather*} -\frac {a}{x}+\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {-1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3} b c^{2/3} \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {b \tanh ^{-1}\left (c x^{3/2}\right )}{x}-\frac {1}{2} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}\right )+\frac {1}{2} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}\right )-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{4} b c^{2/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 167, normalized size = 0.97
method | result | size |
derivativedivides | \(-\frac {a}{x}-\frac {b \arctanh \left (c \,x^{\frac {3}{2}}\right )}{x}+\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\) | \(167\) |
default | \(-\frac {a}{x}-\frac {b \arctanh \left (c \,x^{\frac {3}{2}}\right )}{x}+\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\) | \(167\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 163, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, {\left ({\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {1}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {1}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c - \frac {4 \, \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )}{x}\right )} b - \frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 234, normalized size = 1.36 \begin {gather*} -\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} b x \arctan \left (\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {2}{3}} \sqrt {x} + \sqrt {3} c}{3 \, c}\right ) - 2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{3}} x \arctan \left (\frac {2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {2}{3}} \sqrt {x} - \sqrt {3} c}{3 \, c}\right ) + \left (-c^{2}\right )^{\frac {1}{3}} b x \log \left (c^{2} x - \left (-c^{2}\right )^{\frac {1}{3}} c \sqrt {x} + \left (-c^{2}\right )^{\frac {2}{3}}\right ) + b {\left (c^{2}\right )}^{\frac {1}{3}} x \log \left (c^{2} x - {\left (c^{2}\right )}^{\frac {1}{3}} c \sqrt {x} + {\left (c^{2}\right )}^{\frac {2}{3}}\right ) - 2 \, \left (-c^{2}\right )^{\frac {1}{3}} b x \log \left (c \sqrt {x} + \left (-c^{2}\right )^{\frac {1}{3}}\right ) - 2 \, b {\left (c^{2}\right )}^{\frac {1}{3}} x \log \left (c \sqrt {x} + {\left (c^{2}\right )}^{\frac {1}{3}}\right ) + 2 \, b \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right ) + 4 \, a}{4 \, x} \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 [A]
time = 0.49, size = 172, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {\log \left (x + \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {\log \left (x - \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \log \left (\sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left | \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {1}{3}}}\right )} b c - \frac {b \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right )}{2 \, x} - \frac {a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.46, size = 220, normalized size = 1.28 \begin {gather*} \frac {b\,c^{2/3}\,\ln \left (\frac {c^{1/3}\,\sqrt {x}+1}{c^{1/3}\,\sqrt {x}-1}\right )}{2}-\frac {a}{x}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (b\,x-b\,c^2\,x^4\right )}{2\,x^2-2\,c^2\,x^5}-\frac {b\,\ln \left (c\,x^{3/2}+1\right )}{2\,x}+\frac {b\,c^{2/3}\,\ln \left (\frac {\sqrt {3}+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x+1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{2}+\frac {b\,c^{2/3}\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}+c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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