Optimal. Leaf size=192 \[ \frac {c^6 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+2 c^6 \log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{x}-\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 x^2}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}-b c^6 \text {Li}_2\left (\frac {2}{\sqrt {x} c+1}-1\right )+\frac {11}{6} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {11 b c^5}{6 \sqrt {x}}-\frac {5 b c^3}{18 x^{3/2}}-\frac {b c}{15 x^{5/2}} \]
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Rubi [A] time = 0.57, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {44, 1593, 5982, 5916, 325, 206, 5988, 5932, 2447} \[ -b c^6 \text {PolyLog}\left (2,\frac {2}{c \sqrt {x}+1}-1\right )-\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 x^2}+\frac {c^6 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}-\frac {c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{x}+2 c^6 \log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}-\frac {5 b c^3}{18 x^{3/2}}-\frac {11 b c^5}{6 \sqrt {x}}+\frac {11}{6} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b c}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 325
Rule 1593
Rule 2447
Rule 5916
Rule 5932
Rule 5982
Rule 5988
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x^4 \left (1-c^2 x\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^7-c^2 x^9} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^7 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^7} \, dx,x,\sqrt {x}\right )+\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^5 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )\\ &=-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{3} (b c) \operatorname {Subst}\left (\int \frac {1}{x^6 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )+\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^5} \, dx,x,\sqrt {x}\right )+\left (2 c^4\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b c}{15 x^{5/2}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}-\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 x^2}+\frac {1}{3} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )+\frac {1}{2} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )+\left (2 c^4\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx,x,\sqrt {x}\right )+\left (2 c^6\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b c}{15 x^{5/2}}-\frac {5 b c^3}{18 x^{3/2}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}-\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 x^2}-\frac {c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{x}+\frac {c^6 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+\frac {1}{3} \left (b c^5\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )+\frac {1}{2} \left (b c^5\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )+\left (b c^5\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )+\left (2 c^6\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b c}{15 x^{5/2}}-\frac {5 b c^3}{18 x^{3/2}}-\frac {11 b c^5}{6 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}-\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 x^2}-\frac {c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{x}+\frac {c^6 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+2 c^6 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )+\frac {1}{3} \left (b c^7\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{2} \left (b c^7\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )+\left (b c^7\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )-\left (2 b c^7\right ) \operatorname {Subst}\left (\int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b c}{15 x^{5/2}}-\frac {5 b c^3}{18 x^{3/2}}-\frac {11 b c^5}{6 \sqrt {x}}+\frac {11}{6} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}-\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 x^2}-\frac {c^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{x}+\frac {c^6 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+2 c^6 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )-b c^6 \text {Li}_2\left (-1+\frac {2}{1+c \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.81, size = 187, normalized size = 0.97 \[ -\frac {-90 a c^6 x^3 \log (x)+90 a c^4 x^2+45 a c^2 x+90 a c^6 x^3 \log \left (1-c^2 x\right )+30 a-90 b c^6 x^3 \tanh ^{-1}\left (c \sqrt {x}\right )^2+165 b c^5 x^{5/2}+25 b c^3 x^{3/2}-15 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (11 c^6 x^3+12 c^6 x^3 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )-6 c^4 x^2-3 c^2 x-2\right )+6 b c \sqrt {x}}{90 x^3}-b c^6 \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{c^{2} x^{5} - x^{4}}, x\right ) \]
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 \[ \int -\frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (c^{2} x - 1\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 381, normalized size = 1.98 \[ -\frac {c^{2} b \arctanh \left (c \sqrt {x}\right )}{2 x^{2}}-c^{6} b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )+\frac {c^{6} b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2}-\frac {c^{6} b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {c^{4} b \arctanh \left (c \sqrt {x}\right )}{x}-c^{6} b \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )+2 c^{6} b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )-c^{6} b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )+\frac {c^{6} b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2}-\frac {11 b \,c^{5}}{6 \sqrt {x}}-\frac {5 b \,c^{3}}{18 x^{\frac {3}{2}}}-\frac {b c}{15 x^{\frac {5}{2}}}-\frac {c^{4} a}{x}-\frac {c^{2} a}{2 x^{2}}-c^{6} a \ln \left (c \sqrt {x}-1\right )-c^{6} a \ln \left (1+c \sqrt {x}\right )-\frac {c^{6} b \ln \left (c \sqrt {x}-1\right )^{2}}{4}+\frac {c^{6} b \ln \left (1+c \sqrt {x}\right )^{2}}{4}-\frac {11 c^{6} b \ln \left (c \sqrt {x}-1\right )}{12}+2 c^{6} a \ln \left (c \sqrt {x}\right )+\frac {11 c^{6} b \ln \left (1+c \sqrt {x}\right )}{12}-c^{6} b \dilog \left (c \sqrt {x}\right )+c^{6} b \dilog \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )-\frac {b \arctanh \left (c \sqrt {x}\right )}{3 x^{3}}-c^{6} b \dilog \left (1+c \sqrt {x}\right )-\frac {a}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 330, normalized size = 1.72 \[ -{\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b c^{6} - {\left (\log \left (c \sqrt {x}\right ) \log \left (-c \sqrt {x} + 1\right ) + {\rm Li}_2\left (-c \sqrt {x} + 1\right )\right )} b c^{6} + {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x}\right ) + {\rm Li}_2\left (c \sqrt {x} + 1\right )\right )} b c^{6} + \frac {11}{12} \, b c^{6} \log \left (c \sqrt {x} + 1\right ) - \frac {11}{12} \, b c^{6} \log \left (c \sqrt {x} - 1\right ) - \frac {1}{6} \, {\left (6 \, c^{6} \log \left (c \sqrt {x} + 1\right ) + 6 \, c^{6} \log \left (c \sqrt {x} - 1\right ) - 6 \, c^{6} \log \relax (x) + \frac {6 \, c^{4} x^{2} + 3 \, c^{2} x + 2}{x^{3}}\right )} a - \frac {45 \, b c^{6} x^{3} \log \left (c \sqrt {x} + 1\right )^{2} - 45 \, b c^{6} x^{3} \log \left (-c \sqrt {x} + 1\right )^{2} + 330 \, b c^{5} x^{\frac {5}{2}} + 50 \, b c^{3} x^{\frac {3}{2}} + 12 \, b c \sqrt {x} + 15 \, {\left (6 \, b c^{4} x^{2} + 3 \, b c^{2} x + 2 \, b\right )} \log \left (c \sqrt {x} + 1\right ) - 15 \, {\left (6 \, b c^{6} x^{3} \log \left (c \sqrt {x} + 1\right ) + 6 \, b c^{4} x^{2} + 3 \, b c^{2} x + 2 \, b\right )} \log \left (-c \sqrt {x} + 1\right )}{180 \, x^{3}} \]
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 \[ -\int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x^4\,\left (c^2\,x-1\right )} \,d x \]
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 \[ - \int \frac {a}{c^{2} x^{5} - x^{4}}\, dx - \int \frac {b \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{5} - x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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