Optimal. Leaf size=197 \[ \frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {\left (53 a^2+20\right ) a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{30 b^5}-\frac {\left (58 a^2+9\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}-\frac {\left (40 a^4+40 a^2+3\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5}+\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}+\frac {1}{5} x^5 \sec ^{-1}(a+b x) \]
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Rubi [A] time = 0.23, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5258, 4426, 3782, 4056, 4048, 3770, 3767, 8} \[ \frac {\left (53 a^2+20\right ) a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{30 b^5}-\frac {\left (58 a^2+9\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}-\frac {\left (40 a^4+40 a^2+3\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5}+\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}+\frac {1}{5} x^5 \sec ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3782
Rule 4048
Rule 4056
Rule 4426
Rule 5258
Rubi steps
\begin {align*} \int x^4 \sec ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x \sec (x) (-a+\sec (x))^4 \tan (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{b^5}\\ &=\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\sec (x))^5 \, dx,x,\sec ^{-1}(a+b x)\right )}{5 b^5}\\ &=-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\sec (x))^2 \left (-4 a^3+3 \left (1+4 a^2\right ) \sec (x)-11 a \sec ^2(x)\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{20 b^5}\\ &=\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\sec (x)) \left (12 a^4-a \left (31+48 a^2\right ) \sec (x)+\left (9+58 a^2\right ) \sec ^2(x)\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{60 b^5}\\ &=\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int \left (-24 a^5+3 \left (3+40 a^2+40 a^4\right ) \sec (x)-4 a \left (20+53 a^2\right ) \sec ^2(x)\right ) \, dx,x,\sec ^{-1}(a+b x)\right )}{120 b^5}\\ &=\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)+\frac {\left (a \left (20+53 a^2\right )\right ) \operatorname {Subst}\left (\int \sec ^2(x) \, dx,x,\sec ^{-1}(a+b x)\right )}{30 b^5}-\frac {\left (3+40 a^2+40 a^4\right ) \operatorname {Subst}\left (\int \sec (x) \, dx,x,\sec ^{-1}(a+b x)\right )}{40 b^5}\\ &=\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\left (3+40 a^2+40 a^4\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5}-\frac {\left (a \left (20+53 a^2\right )\right ) \operatorname {Subst}\left (\int 1 \, dx,x,-(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{30 b^5}\\ &=\frac {a \left (20+53 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{30 b^5}+\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\left (3+40 a^2+40 a^4\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 173, normalized size = 0.88 \[ \frac {-24 a^5 \sin ^{-1}\left (\frac {1}{a+b x}\right )+\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (-9 \left (4 a^2+1\right ) b^2 x^2+2 a \left (48 a^2+31\right ) b x+a^2 \left (154 a^2+71\right )+16 a b^3 x^3-6 b^4 x^4\right )-3 \left (40 a^4+40 a^2+3\right ) \log \left ((a+b x) \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right )\right )+24 b^5 x^5 \sec ^{-1}(a+b x)}{120 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 152, normalized size = 0.77 \[ \frac {24 \, b^{5} x^{5} \operatorname {arcsec}\left (b x + a\right ) + 48 \, a^{5} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 3 \, {\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (6 \, b^{3} x^{3} - 22 \, a b^{2} x^{2} - 154 \, a^{3} + {\left (58 \, a^{2} + 9\right )} b x - 71 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{120 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 409, normalized size = 2.08 \[ -\frac {1}{960} \, b {\left (\frac {192 \, {\left (b x + a\right )}^{5} {\left (\frac {5 \, a}{b x + a} - \frac {10 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {10 \, a^{3}}{{\left (b x + a\right )}^{3}} - \frac {5 \, a^{4}}{{\left (b x + a\right )}^{4}} - 1\right )} \arccos \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{6}} - \frac {3 \, {\left (b x + a\right )}^{4} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4} + 40 \, {\left (b x + a\right )}^{3} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 240 \, {\left (b x + a\right )}^{2} a^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 960 \, {\left (b x + a\right )} a^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 24 \, {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 360 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 24 \, {\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (-{\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} {\left | b x + a \right |}\right ) - \frac {120 \, {\left (8 \, a^{3} + 3 \, a\right )} {\left (b x + a\right )}^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 24 \, {\left (10 \, a^{2} + 1\right )} {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 40 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 3}{{\left (b x + a\right )}^{4} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4}}}{b^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 509, normalized size = 2.58 \[ \frac {11 \left (-1+\left (b x +a \right )^{2}\right ) x^{2} a}{60 b^{3} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {29 \left (-1+\left (b x +a \right )^{2}\right ) x \,a^{2}}{60 b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {3 \sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{40 b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {71 \left (-1+\left (b x +a \right )^{2}\right ) a}{120 b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {77 \left (-1+\left (b x +a \right )^{2}\right ) a^{3}}{60 b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {x^{5} \mathrm {arcsec}\left (b x +a \right )}{5}-\frac {\left (-1+\left (b x +a \right )^{2}\right ) x^{3}}{20 b^{2} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{5} \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{5 b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{4} \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {3 \left (-1+\left (b x +a \right )^{2}\right ) x}{40 b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{2} \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )} \]
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 \[ \frac {1}{5} \, x^{5} \arctan \left (\sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - \int \frac {{\left (b^{2} x^{6} + a b x^{5}\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (b x + a - 1\right )\right )}}{5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} e^{\left (\log \left (b x + a + 1\right ) + \log \left (b x + a - 1\right )\right )} - 1\right )}}\,{d x} \]
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 \[ \int x^4\,\mathrm {acos}\left (\frac {1}{a+b\,x}\right ) \,d x \]
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 \[ \int x^{4} \operatorname {asec}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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