Optimal. Leaf size=181 \[ -\frac {b^3 \sec ^{-1}(a+b x)}{3 a^3}-\frac {\left (2-5 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a^2 \left (1-a^2\right )^2 x}+\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}+\frac {\left (6 a^4-5 a^2+2\right ) b^3 \tan ^{-1}\left (\frac {\sqrt {a+1} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}}-\frac {\sec ^{-1}(a+b x)}{3 x^3} \]
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Rubi [A] time = 0.29, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5258, 4426, 3785, 4060, 3919, 3831, 2659, 205} \[ -\frac {\left (2-5 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \sec ^{-1}(a+b x)}{3 a^3}+\frac {\left (6 a^4-5 a^2+2\right ) b^3 \tan ^{-1}\left (\frac {\sqrt {a+1} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}}+\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}-\frac {\sec ^{-1}(a+b x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 3785
Rule 3831
Rule 3919
Rule 4060
Rule 4426
Rule 5258
Rubi steps
\begin {align*} \int \frac {\sec ^{-1}(a+b x)}{x^4} \, dx &=b^3 \operatorname {Subst}\left (\int \frac {x \sec (x) \tan (x)}{(-a+\sec (x))^4} \, dx,x,\sec ^{-1}(a+b x)\right )\\ &=-\frac {\sec ^{-1}(a+b x)}{3 x^3}+\frac {1}{3} b^3 \operatorname {Subst}\left (\int \frac {1}{(-a+\sec (x))^3} \, dx,x,\sec ^{-1}(a+b x)\right )\\ &=\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}-\frac {\sec ^{-1}(a+b x)}{3 x^3}-\frac {b^3 \operatorname {Subst}\left (\int \frac {2 \left (1-a^2\right )-2 a \sec (x)-\sec ^2(x)}{(-a+\sec (x))^2} \, dx,x,\sec ^{-1}(a+b x)\right )}{6 a \left (1-a^2\right )}\\ &=\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a^2 \left (1-a^2\right )^2 x}-\frac {\sec ^{-1}(a+b x)}{3 x^3}+\frac {b^3 \operatorname {Subst}\left (\int \frac {2 \left (1-a^2\right )^2-a \left (1-4 a^2\right ) \sec (x)}{-a+\sec (x)} \, dx,x,\sec ^{-1}(a+b x)\right )}{6 a^2 \left (1-a^2\right )^2}\\ &=\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \sec ^{-1}(a+b x)}{3 a^3}-\frac {\sec ^{-1}(a+b x)}{3 x^3}+\frac {\left (\left (2-5 a^2+6 a^4\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {\sec (x)}{-a+\sec (x)} \, dx,x,\sec ^{-1}(a+b x)\right )}{6 a^3 \left (1-a^2\right )^2}\\ &=\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \sec ^{-1}(a+b x)}{3 a^3}-\frac {\sec ^{-1}(a+b x)}{3 x^3}+\frac {\left (\left (2-5 a^2+6 a^4\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a \cos (x)} \, dx,x,\sec ^{-1}(a+b x)\right )}{6 a^3 \left (1-a^2\right )^2}\\ &=\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \sec ^{-1}(a+b x)}{3 a^3}-\frac {\sec ^{-1}(a+b x)}{3 x^3}+\frac {\left (\left (2-5 a^2+6 a^4\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a+(1+a) x^2} \, dx,x,\tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )\right )}{3 a^3 \left (1-a^2\right )^2}\\ &=\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a \left (1-a^2\right ) x^2}-\frac {\left (2-5 a^2\right ) b^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 a^2 \left (1-a^2\right )^2 x}-\frac {b^3 \sec ^{-1}(a+b x)}{3 a^3}-\frac {\sec ^{-1}(a+b x)}{3 x^3}+\frac {\left (2-5 a^2+6 a^4\right ) b^3 \tan ^{-1}\left (\frac {\sqrt {1+a} \tan \left (\frac {1}{2} \sec ^{-1}(a+b x)\right )}{\sqrt {1-a}}\right )}{3 a^3 \left (1-a^2\right )^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.43, size = 241, normalized size = 1.33 \[ \frac {1}{6} \left (\frac {2 b^3 \sin ^{-1}\left (\frac {1}{a+b x}\right )}{a^3}-\frac {b \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (a^4-4 a^3 b x-a^2 \left (5 b^2 x^2+1\right )+a b x+2 b^2 x^2\right )}{a^2 \left (a^2-1\right )^2 x^2}-\frac {i \left (6 a^4-5 a^2+2\right ) b^3 \log \left (\frac {12 a^3 \left (a^2-1\right )^2 \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac {i \left (a^2+a b x-1\right )}{\sqrt {1-a^2}}\right )}{\left (6 a^4-5 a^2+2\right ) b^3 x}\right )}{a^3 \left (1-a^2\right )^{5/2}}-\frac {2 \sec ^{-1}(a+b x)}{x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.93, size = 548, normalized size = 3.03 \[ \left [\frac {{\left (6 \, a^{4} - 5 \, a^{2} + 2\right )} \sqrt {a^{2} - 1} b^{3} x^{3} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 4 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b^{3} x^{3} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{3} x^{3} - 2 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} \operatorname {arcsec}\left (b x + a\right ) + {\left ({\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{2} x^{2} - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{6 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} x^{3}}, \frac {2 \, {\left (6 \, a^{4} - 5 \, a^{2} + 2\right )} \sqrt {-a^{2} + 1} b^{3} x^{3} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) - 4 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b^{3} x^{3} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{3} x^{3} - 2 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} \operatorname {arcsec}\left (b x + a\right ) + {\left ({\left (5 \, a^{5} - 7 \, a^{3} + 2 \, a\right )} b^{2} x^{2} - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} b x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{6 \, {\left (a^{9} - 3 \, a^{7} + 3 \, a^{5} - a^{3}\right )} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 451, normalized size = 2.49 \[ \frac {1}{3} \, b {\left (\frac {{\left (6 \, a^{4} b^{2} - 5 \, a^{2} b^{2} + 2 \, b^{2}\right )} \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{7} - 2 \, a^{5} + a^{3}\right )} \sqrt {-a^{2} + 1}} + \frac {4 \, {\left (b x + a\right )}^{3} a^{3} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 10 \, {\left (b x + a\right )}^{2} a^{4} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} - {\left (b x + a\right )}^{3} a b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + {\left (b x + a\right )}^{2} a^{2} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 16 \, {\left (b x + a\right )} a^{3} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} - 2 \, {\left (b x + a\right )}^{2} b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} - 7 \, {\left (b x + a\right )} a b^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 5 \, a^{2} b^{2} - 2 \, b^{2}}{{\left (a^{6} - 2 \, a^{4} + a^{2}\right )} {\left ({\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 2 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1\right )}^{2}} - \frac {{\left (\frac {3 \, a b^{2}}{b x + a} - \frac {3 \, a^{2} b^{2}}{{\left (b x + a\right )}^{2}} - b^{2}\right )} \arccos \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a^{3} {\left (\frac {a}{b x + a} - 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 760, normalized size = 4.20 \[ -\frac {\mathrm {arcsec}\left (b x +a \right )}{3 x^{3}}+\frac {b^{3} \sqrt {-1+\left (b x +a \right )^{2}}\, a \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{3 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}-1\right )^{2}}-\frac {b^{3} \sqrt {-1+\left (b x +a \right )^{2}}\, a^{3} \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}-1\right )^{\frac {7}{2}}}-\frac {2 b^{3} \sqrt {-1+\left (b x +a \right )^{2}}\, \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{3 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \left (a^{2}-1\right )^{2}}+\frac {5 b^{2} \left (-1+\left (b x +a \right )^{2}\right )}{6 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}-1\right )^{2} x}+\frac {11 b^{3} \sqrt {-1+\left (b x +a \right )^{2}}\, a \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{6 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}-1\right )^{\frac {7}{2}}}-\frac {b \left (-1+\left (b x +a \right )^{2}\right ) a}{6 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}-1\right )^{2} x^{2}}+\frac {b^{3} \sqrt {-1+\left (b x +a \right )^{2}}\, \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{3 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{3} \left (a^{2}-1\right )^{2}}-\frac {b^{2} \left (-1+\left (b x +a \right )^{2}\right )}{3 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )^{2} x}-\frac {7 b^{3} \sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{6 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \left (a^{2}-1\right )^{\frac {7}{2}}}+\frac {b \left (-1+\left (b x +a \right )^{2}\right )}{6 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \left (a^{2}-1\right )^{2} x^{2}}+\frac {b^{3} \sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{3 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{3} \left (a^{2}-1\right )^{\frac {7}{2}}} \]
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 \[ \frac {x^{3} \int \frac {{\left (b^{2} x + a b\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (b x + a - 1\right )\right )}}{b^{2} x^{5} + 2 \, a b x^{4} + {\left (a^{2} - 1\right )} x^{3} + {\left (b^{2} x^{5} + 2 \, a b x^{4} + {\left (a^{2} - 1\right )} x^{3}\right )} {\left (b x + a + 1\right )} {\left (b x + a - 1\right )}}\,{d x} - \arctan \left (\sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )}{3 \, x^{3}} \]
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 \[ \int \frac {\mathrm {acos}\left (\frac {1}{a+b\,x}\right )}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {asec}{\left (a + b x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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