Optimal. Leaf size=197 \[ \frac {a \left (20+53 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{30 b^5}+\frac {11 a x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{60 b^3}-\frac {x^3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{20 b^2}-\frac {\left (9+58 a^2\right ) (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{120 b^5}+\frac {a^5 \sec ^{-1}(a+b x)}{5 b^5}+\frac {1}{5} x^5 \sec ^{-1}(a+b x)-\frac {\left (3+40 a^2+40 a^4\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{40 b^5} \]
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Rubi [A]
time = 0.18, 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 = {5366, 4511,
3867, 4141, 4133, 3855, 3852, 8} \begin {gather*} \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) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3867
Rule 4133
Rule 4141
Rule 4511
Rule 5366
Rubi steps
\begin {align*} \int x^4 \sec ^{-1}(a+b x) \, dx &=\frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 = 0.12, size = 173, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (a^2 \left (71+154 a^2\right )+2 a \left (31+48 a^2\right ) b x-9 \left (1+4 a^2\right ) b^2 x^2+16 a b^3 x^3-6 b^4 x^4\right )+24 b^5 x^5 \sec ^{-1}(a+b x)-24 a^5 \text {ArcSin}\left (\frac {1}{a+b x}\right )-3 \left (3+40 a^2+40 a^4\right ) \log \left ((a+b x) \left (1+\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{120 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 329, normalized size = 1.67
method | result | size |
derivativedivides | \(\frac {-\frac {\mathrm {arcsec}\left (b x +a \right ) a^{5}}{5}+\mathrm {arcsec}\left (b x +a \right ) a^{4} \left (b x +a \right )-2 \,\mathrm {arcsec}\left (b x +a \right ) a^{3} \left (b x +a \right )^{2}+2 \,\mathrm {arcsec}\left (b x +a \right ) a^{2} \left (b x +a \right )^{3}-\mathrm {arcsec}\left (b x +a \right ) a \left (b x +a \right )^{4}+\frac {\mathrm {arcsec}\left (b x +a \right ) \left (b x +a \right )^{5}}{5}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (24 a^{5} \arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right )+120 a^{4} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-240 a^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-40 a \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-80 a \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+9 \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{120 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{5}}\) | \(329\) |
default | \(\frac {-\frac {\mathrm {arcsec}\left (b x +a \right ) a^{5}}{5}+\mathrm {arcsec}\left (b x +a \right ) a^{4} \left (b x +a \right )-2 \,\mathrm {arcsec}\left (b x +a \right ) a^{3} \left (b x +a \right )^{2}+2 \,\mathrm {arcsec}\left (b x +a \right ) a^{2} \left (b x +a \right )^{3}-\mathrm {arcsec}\left (b x +a \right ) a \left (b x +a \right )^{4}+\frac {\mathrm {arcsec}\left (b x +a \right ) \left (b x +a \right )^{5}}{5}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (24 a^{5} \arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right )+120 a^{4} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-240 a^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-40 a \left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 \left (b x +a \right )^{3} \sqrt {\left (b x +a \right )^{2}-1}+120 a^{2} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-80 a \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+9 \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{120 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{5}}\) | \(329\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.60, size = 152, normalized size = 0.77 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \int x^{4} \operatorname {asec}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 409 vs.
\(2 (173) = 346\).
time = 0.47, size = 409, normalized size = 2.08 \begin {gather*} -\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 )} \end {gather*}
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 \begin {gather*} \int x^4\,\mathrm {acos}\left (\frac {1}{a+b\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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