Optimal. Leaf size=69 \[ -\frac {2 b \tan ^{-1}\left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x} \]
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Rubi [A] time = 0.10, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5259, 4427, 3783, 2660, 618, 204} \[ -\frac {2 b \tan ^{-1}\left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 3783
Rule 4427
Rule 5259
Rubi steps
\begin {align*} \int \frac {\csc ^{-1}(a+b x)}{x^2} \, dx &=-\left (b \operatorname {Subst}\left (\int \frac {x \cot (x) \csc (x)}{(-a+\csc (x))^2} \, dx,x,\csc ^{-1}(a+b x)\right )\right )\\ &=-\frac {\csc ^{-1}(a+b x)}{x}+b \operatorname {Subst}\left (\int \frac {1}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{a}\\ &=-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}+\frac {(2 b) \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}\\ &=-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}-\frac {(4 b) \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}\\ &=-\frac {b \csc ^{-1}(a+b x)}{a}-\frac {\csc ^{-1}(a+b x)}{x}-\frac {2 b \tan ^{-1}\left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}\\ \end {align*}
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Mathematica [C] time = 0.42, size = 115, normalized size = 1.67 \[ -\frac {\csc ^{-1}(a+b x)}{x}+\frac {b \left (-\sin ^{-1}\left (\frac {1}{a+b x}\right )+\frac {i \log \left (\frac {2 \left (-a \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} (a+b x)-\frac {i a \left (a^2+a b x-1\right )}{\sqrt {1-a^2}}\right )}{b x}\right )}{\sqrt {1-a^2}}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 278, normalized size = 4.03 \[ \left [\frac {2 \, {\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {a^{2} - 1} b x \log \left (\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - {\left (a^{3} - a\right )} \operatorname {arccsc}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}, \frac {2 \, {\left (a^{2} - 1\right )} b x \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, \sqrt {-a^{2} + 1} b x \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} \sqrt {-a^{2} + 1}}{a^{2} - 1}\right ) - {\left (a^{3} - a\right )} \operatorname {arccsc}\left (b x + a\right )}{{\left (a^{3} - a\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 96, normalized size = 1.39 \[ -b {\left (\frac {2 \, \arctan \left (\frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + a}{\sqrt {-a^{2} + 1}}\right )}{\sqrt {-a^{2} + 1} a} - \frac {\arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{a {\left (\frac {a}{b x + a} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 154, normalized size = 2.23 \[ -\frac {\mathrm {arccsc}\left (b x +a \right )}{x}-\frac {b \sqrt {-1+\left (b x +a \right )^{2}}\, \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a}+\frac {b \sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (\frac {2 \sqrt {a^{2}-1}\, \sqrt {-1+\left (b x +a \right )^{2}}+2 a \left (b x +a \right )-2}{b x}\right )}{\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) a \sqrt {a^{2}-1}} \]
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 \[ -\frac {b x \int \frac {1}{\sqrt {b x + a + 1} \sqrt {b x + a - 1} b x^{2} + \sqrt {b x + a + 1} \sqrt {b x + a - 1} a x}\,{d x} + \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )}{x} \]
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 \[ \int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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