Optimal. Leaf size=195 \[ -\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^8}+\frac {2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^8}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^6}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^4}-\frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{3 c^2}+\frac {b \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^8}+\frac {11 b \tanh ^{-1}\left (c \sqrt {x}\right )}{6 c^8}-\frac {11 b \sqrt {x}}{6 c^7}-\frac {5 b x^{3/2}}{18 c^5}-\frac {b x^{5/2}}{15 c^3} \]
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Rubi [A] time = 0.60, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {43, 5980, 5916, 302, 206, 321, 5984, 5918, 2402, 2315} \[ \frac {b \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^8}-\frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{3 c^2}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^4}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^6}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^8}+\frac {2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^8}-\frac {b x^{5/2}}{15 c^3}-\frac {5 b x^{3/2}}{18 c^5}-\frac {11 b \sqrt {x}}{6 c^7}+\frac {11 b \tanh ^{-1}\left (c \sqrt {x}\right )}{6 c^8} \]
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 5916
Rule 5918
Rule 5980
Rule 5984
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^7 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \operatorname {Subst}\left (\int x^5 \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{c^2}+\frac {2 \operatorname {Subst}\left (\int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=-\frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{3 c^2}-\frac {2 \operatorname {Subst}\left (\int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{c^4}+\frac {2 \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^4}+\frac {b \operatorname {Subst}\left (\int \frac {x^6}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{3 c}\\ &=-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^4}-\frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{3 c^2}-\frac {2 \operatorname {Subst}\left (\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{c^6}+\frac {2 \operatorname {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^6}+\frac {b \operatorname {Subst}\left (\int \frac {x^4}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{2 c^3}+\frac {b \operatorname {Subst}\left (\int \left (-\frac {1}{c^6}-\frac {x^2}{c^4}-\frac {x^4}{c^2}+\frac {1}{c^6 \left (1-c^2 x^2\right )}\right ) \, dx,x,\sqrt {x}\right )}{3 c}\\ &=-\frac {b \sqrt {x}}{3 c^7}-\frac {b x^{3/2}}{9 c^5}-\frac {b x^{5/2}}{15 c^3}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^6}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^4}-\frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{3 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^8}+\frac {2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx,x,\sqrt {x}\right )}{c^7}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{3 c^7}+\frac {b \operatorname {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^5}+\frac {b \operatorname {Subst}\left (\int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 c^3}\\ &=-\frac {11 b \sqrt {x}}{6 c^7}-\frac {5 b x^{3/2}}{18 c^5}-\frac {b x^{5/2}}{15 c^3}+\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 c^8}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^6}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^4}-\frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{3 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^8}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^8}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{2 c^7}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^7}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^7}\\ &=-\frac {11 b \sqrt {x}}{6 c^7}-\frac {5 b x^{3/2}}{18 c^5}-\frac {b x^{5/2}}{15 c^3}+\frac {11 b \tanh ^{-1}\left (c \sqrt {x}\right )}{6 c^8}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^6}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^4}-\frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{3 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^8}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^8}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c \sqrt {x}}\right )}{c^8}\\ &=-\frac {11 b \sqrt {x}}{6 c^7}-\frac {5 b x^{3/2}}{18 c^5}-\frac {b x^{5/2}}{15 c^3}+\frac {11 b \tanh ^{-1}\left (c \sqrt {x}\right )}{6 c^8}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^6}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^4}-\frac {x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{3 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^8}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^8}+\frac {b \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^8}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 160, normalized size = 0.82 \[ -\frac {30 a c^6 x^3+45 a c^4 x^2+90 a c^2 x+90 a \log \left (1-c^2 x\right )+6 b c^5 x^{5/2}+25 b c^3 x^{3/2}+15 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (2 c^6 x^3+3 c^4 x^2+6 c^2 x-12 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}+1\right )-11\right )+90 b \text {Li}_2\left (-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )+165 b c \sqrt {x}-90 b \tanh ^{-1}\left (c \sqrt {x}\right )^2}{90 c^8} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b x^{3} \operatorname {artanh}\left (c \sqrt {x}\right ) + a x^{3}}{c^{2} x - 1}, x\right ) \]
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 \[ \int -\frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x^{3}}{c^{2} x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 309, normalized size = 1.58 \[ -\frac {a \,x^{3}}{3 c^{2}}-\frac {x^{2} a}{2 c^{4}}-\frac {x a}{c^{6}}-\frac {a \ln \left (c \sqrt {x}-1\right )}{c^{8}}-\frac {a \ln \left (1+c \sqrt {x}\right )}{c^{8}}-\frac {b \arctanh \left (c \sqrt {x}\right ) x^{3}}{3 c^{2}}-\frac {b \arctanh \left (c \sqrt {x}\right ) x^{2}}{2 c^{4}}-\frac {b \arctanh \left (c \sqrt {x}\right ) x}{c^{6}}-\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{c^{8}}-\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{c^{8}}-\frac {b \ln \left (c \sqrt {x}-1\right )^{2}}{4 c^{8}}+\frac {b \dilog \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{c^{8}}+\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2 c^{8}}+\frac {b \ln \left (1+c \sqrt {x}\right )^{2}}{4 c^{8}}-\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{2 c^{8}}+\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2 c^{8}}-\frac {b \,x^{\frac {5}{2}}}{15 c^{3}}-\frac {5 b \,x^{\frac {3}{2}}}{18 c^{5}}-\frac {11 b \sqrt {x}}{6 c^{7}}-\frac {11 b \ln \left (c \sqrt {x}-1\right )}{12 c^{8}}+\frac {11 b \ln \left (1+c \sqrt {x}\right )}{12 c^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 246, normalized size = 1.26 \[ -\frac {1}{6} \, a {\left (\frac {2 \, c^{4} x^{3} + 3 \, c^{2} x^{2} + 6 \, x}{c^{6}} + \frac {6 \, \log \left (c^{2} x - 1\right )}{c^{8}}\right )} - \frac {{\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b}{c^{8}} + \frac {11 \, b \log \left (c \sqrt {x} + 1\right )}{12 \, c^{8}} - \frac {11 \, b \log \left (c \sqrt {x} - 1\right )}{12 \, c^{8}} - \frac {12 \, b c^{5} x^{\frac {5}{2}} + 50 \, b c^{3} x^{\frac {3}{2}} + 45 \, b \log \left (c \sqrt {x} + 1\right )^{2} - 45 \, b \log \left (-c \sqrt {x} + 1\right )^{2} + 330 \, b c \sqrt {x} + 15 \, {\left (2 \, b c^{6} x^{3} + 3 \, b c^{4} x^{2} + 6 \, b c^{2} x\right )} \log \left (c \sqrt {x} + 1\right ) - 15 \, {\left (2 \, b c^{6} x^{3} + 3 \, b c^{4} x^{2} + 6 \, b c^{2} x + 6 \, b \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )}{180 \, c^{8}} \]
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 \[ \int -\frac {x^3\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{c^2\,x-1} \,d x \]
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 \[ - \int \frac {a x^{3}}{c^{2} x - 1}\, dx - \int \frac {b x^{3} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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