Optimal. Leaf size=123 \[ \frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}+\frac {\left (1-2 a^2\right ) b^2 \tan ^{-1}\left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}-\frac {\csc ^{-1}(a+b x)}{2 x^2} \]
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Rubi [A] time = 0.19, antiderivative size = 123, 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 = {5259, 4427, 3785, 3919, 3831, 2660, 618, 204} \[ \frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}+\frac {\left (1-2 a^2\right ) b^2 \tan ^{-1}\left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}-\frac {\csc ^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 3785
Rule 3831
Rule 3919
Rule 4427
Rule 5259
Rubi steps
\begin {align*} \int \frac {\csc ^{-1}(a+b x)}{x^3} \, dx &=-\left (b^2 \operatorname {Subst}\left (\int \frac {x \cot (x) \csc (x)}{(-a+\csc (x))^3} \, dx,x,\csc ^{-1}(a+b x)\right )\right )\\ &=-\frac {\csc ^{-1}(a+b x)}{2 x^2}+\frac {1}{2} b^2 \operatorname {Subst}\left (\int \frac {1}{(-a+\csc (x))^2} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}-\frac {\csc ^{-1}(a+b x)}{2 x^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1-a^2-a \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a \left (1-a^2\right )}\\ &=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {\csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )}\\ &=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{2 a^2 \left (1-a^2\right )}\\ &=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}-\frac {\left (\left (1-2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x+x^2} \, dx,x,\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )\right )}{a^2 \left (1-a^2\right )}\\ &=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}+\frac {\left (2 \left (1-2 a^2\right ) b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (1-a^2\right )-x^2} \, dx,x,-2 a+2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )\right )}{a^2 \left (1-a^2\right )}\\ &=-\frac {b (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 a \left (1-a^2\right ) x}+\frac {b^2 \csc ^{-1}(a+b x)}{2 a^2}-\frac {\csc ^{-1}(a+b x)}{2 x^2}+\frac {\left (1-2 a^2\right ) b^2 \tan ^{-1}\left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.77, size = 199, normalized size = 1.62 \[ \frac {\frac {b x (a+b x) \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}}{a \left (a^2-1\right )}+\frac {i \left (2 a^2-1\right ) b^2 x^2 \log \left (\frac {4 (a-1) a^2 (a+1) \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)+\frac {i \left (a^2+a b x-1\right )}{\sqrt {1-a^2}}\right )}{\left (2 a^2-1\right ) b^2 x}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac {b^2 x^2 \sin ^{-1}\left (\frac {1}{a+b x}\right )}{a^2}-\csc ^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 428, normalized size = 3.48 \[ \left [\frac {{\left (2 \, a^{2} - 1\right )} \sqrt {a^{2} - 1} b^{2} x^{2} \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (a^{3} - a\right )} b^{2} x^{2} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{3} - a\right )} b x - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \operatorname {arccsc}\left (b x + a\right )}{2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}, \frac {2 \, {\left (2 \, a^{2} - 1\right )} \sqrt {-a^{2} + 1} b^{2} x^{2} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) - 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b^{2} x^{2} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (a^{3} - a\right )} b^{2} x^{2} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{3} - a\right )} b x - {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} \operatorname {arccsc}\left (b x + a\right )}{2 \, {\left (a^{6} - 2 \, a^{4} + a^{2}\right )} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 217, normalized size = 1.76 \[ \frac {1}{2} \, b {\left (\frac {2 \, {\left (2 \, a^{2} b - b\right )} \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{{\left (a^{4} - a^{2}\right )} \sqrt {-a^{2} + 1}} + \frac {2 \, {\left ({\left (b x + a\right )} a b {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + b\right )}}{{\left ({\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 2 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1\right )} {\left (a^{3} - a\right )}} - \frac {{\left (\frac {2 \, a b}{b x + a} - b\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a^{2} {\left (\frac {a}{b x + a} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 453, normalized size = 3.68 \[ -\frac {\mathrm {arccsc}\left (b x +a \right )}{2 x^{2}}+\frac {b^{2} \sqrt {-1+\left (b x +a \right )^{2}}\, \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{2 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}-1\right )}-\frac {b^{2} \sqrt {-1+\left (b x +a \right )^{2}}\, a^{2} \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}-1\right )^{\frac {5}{2}}}-\frac {b^{2} \sqrt {-1+\left (b x +a \right )^{2}}\, \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{2 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )}+\frac {b \left (-1+\left (b x +a \right )^{2}\right )}{2 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \left (a^{2}-1\right ) x}+\frac {3 b^{2} \sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{2 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \left (a^{2}-1\right )^{\frac {5}{2}}}-\frac {b^{2} \sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{2 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a^{2} \left (a^{2}-1\right )^{\frac {5}{2}}} \]
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 \[ -\frac {x^{2} \int \frac {{\left (b^{2} x + a b\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (b x + a - 1\right )\right )}}{b^{2} x^{4} + 2 \, a b x^{3} + {\left (b^{2} x^{4} + 2 \, a b x^{3} + {\left (a^{2} - 1\right )} x^{2}\right )} {\left (b x + a + 1\right )} {\left (b x + a - 1\right )} + {\left (a^{2} - 1\right )} x^{2}}\,{d x} + \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )}{2 \, x^{2}} \]
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 \[ \int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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