Optimal. Leaf size=207 \[ -\frac {b^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c^4}+\frac {b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^4}-\frac {b x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c}-\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^3}+\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b^3 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 c^4}-\frac {b^3 x \sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3} \]
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Rubi [A] time = 0.21, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5222, 4409, 4186, 3767, 8, 4184, 3719, 2190, 2279, 2391} \[ \frac {i b^3 \text {PolyLog}\left (2,-e^{2 i \sec ^{-1}(c x)}\right )}{2 c^4}+\frac {b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )}{4 c^2}-\frac {b^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c^4}-\frac {b x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{4 c}-\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^3}+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^4}+\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b^3 x \sqrt {1-\frac {1}{c^2 x^2}}}{4 c^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 3767
Rule 4184
Rule 4186
Rule 4409
Rule 5222
Rubi steps
\begin {align*} \int x^3 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^3 \sec ^4(x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^4}\\ &=\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {(3 b) \operatorname {Subst}\left (\int (a+b x)^2 \sec ^4(x) \, dx,x,\sec ^{-1}(c x)\right )}{4 c^4}\\ &=\frac {b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )}{4 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \sec ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b \operatorname {Subst}\left (\int (a+b x)^2 \sec ^2(x) \, dx,x,\sec ^{-1}(c x)\right )}{2 c^4}-\frac {b^3 \operatorname {Subst}\left (\int \sec ^2(x) \, dx,x,\sec ^{-1}(c x)\right )}{4 c^4}\\ &=\frac {b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )}{4 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \sec ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {b^2 \operatorname {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^4}+\frac {b^3 \operatorname {Subst}\left (\int 1 \, dx,x,-c \sqrt {1-\frac {1}{c^2 x^2}} x\right )}{4 c^4}\\ &=-\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^4}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \sec ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(c x)\right )}{c^4}\\ &=-\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^4}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \sec ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{c^4}+\frac {b^3 \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^4}\\ &=-\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^4}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \sec ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{c^4}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sec ^{-1}(c x)}\right )}{2 c^4}\\ &=-\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^4}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \sec ^{-1}(c x)\right )^2}{2 c^3}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \sec ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )}{c^4}+\frac {i b^3 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{2 c^4}\\ \end {align*}
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Mathematica [A] time = 0.83, size = 288, normalized size = 1.39 \[ \frac {a^3 c^4 x^4+b \sec ^{-1}(c x) \left (c x \left (3 a^2 c^3 x^3-2 a b \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 x^2+2\right )+b^2 c x\right )-4 b^2 \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )\right )-2 a^2 b c x \sqrt {1-\frac {1}{c^2 x^2}}-a^2 b c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}+a b^2 c^2 x^2-b^2 \sec ^{-1}(c x)^2 \left (-3 a c^4 x^4+b \left (2 c x \sqrt {1-\frac {1}{c^2 x^2}}+c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}-2 i\right )\right )-4 a b^2 \log \left (\frac {1}{c x}\right )+b^3 c^4 x^4 \sec ^{-1}(c x)^3-b^3 c x \sqrt {1-\frac {1}{c^2 x^2}}+2 i b^3 \text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )}{4 c^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{3} \operatorname {arcsec}\left (c x\right )^{3} + 3 \, a b^{2} x^{3} \operatorname {arcsec}\left (c x\right )^{2} + 3 \, a^{2} b x^{3} \operatorname {arcsec}\left (c x\right ) + a^{3} x^{3}, x\right ) \]
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 \[ \int {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.02, size = 447, normalized size = 2.16 \[ \frac {x^{4} a^{3}}{4}+\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{3} x^{4}}{4}-\frac {b^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \mathrm {arcsec}\left (c x \right )^{2} x^{3}}{4 c}-\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}{2 c^{3}}+\frac {i b^{3} \mathrm {arcsec}\left (c x \right )^{2}}{2 c^{4}}+\frac {b^{3} \mathrm {arcsec}\left (c x \right ) x^{2}}{4 c^{2}}-\frac {b^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}{4 c^{3}}+\frac {i b^{3} \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2 c^{4}}-\frac {b^{3} \mathrm {arcsec}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{c^{4}}-\frac {i b^{3}}{4 c^{4}}+\frac {3 a^{2} b \,x^{4} \mathrm {arcsec}\left (c x \right )}{4}-\frac {a^{2} b \,x^{3}}{4 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {a^{2} b x}{4 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {a^{2} b}{2 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {3 a \,b^{2} \mathrm {arcsec}\left (c x \right )^{2} x^{4}}{4}-\frac {a \,b^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \mathrm {arcsec}\left (c x \right ) x^{3}}{2 c}+\frac {x^{2} a \,b^{2}}{4 c^{2}}-\frac {a \,b^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \mathrm {arcsec}\left (c x \right ) x}{c^{3}}-\frac {a \,b^{2} \ln \left (\frac {1}{c x}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3}{4} \, a b^{2} x^{4} \operatorname {arcsec}\left (c x\right )^{2} + \frac {1}{4} \, a^{3} x^{4} + \frac {1}{4} \, {\left (3 \, x^{4} \operatorname {arcsec}\left (c x\right ) - \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} a^{2} b + \frac {1}{16} \, {\left (4 \, x^{4} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )^{3} - 3 \, x^{4} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \left (c^{2} x^{2}\right )^{2} - 16 \, \int \frac {3 \, {\left ({\left (4 \, x^{3} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - x^{3} \log \left (c^{2} x^{2}\right )^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + 4 \, {\left (4 \, c^{2} x^{5} \log \relax (c)^{2} - 4 \, x^{3} \log \relax (c)^{2} + 4 \, {\left (c^{2} x^{5} - x^{3}\right )} \log \relax (x)^{2} - {\left ({\left (4 \, c^{2} \log \relax (c) + c^{2}\right )} x^{5} - x^{3} {\left (4 \, \log \relax (c) + 1\right )} + 4 \, {\left (c^{2} x^{5} - x^{3}\right )} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right ) + 8 \, {\left (c^{2} x^{5} \log \relax (c) - x^{3} \log \relax (c)\right )} \log \relax (x)\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\right )}}{16 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x}\right )} b^{3} + \frac {{\left ({\left (c^{2} x^{2} + 2 \, \log \left (x^{2}\right )\right )} \sqrt {c x + 1} \sqrt {c x - 1} - 2 \, {\left (c^{4} x^{4} + c^{2} x^{2} - 2\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} a b^{2}}{4 \, \sqrt {c x + 1} \sqrt {c x - 1} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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