Optimal. Leaf size=116 \[ \frac {a^3 \csc ^{-1}(a+b x)}{3 b^3}+\frac {\left (6 a^2+1\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{6 b^3}-\frac {5 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^3}+\frac {x (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^2}+\frac {1}{3} x^3 \csc ^{-1}(a+b x) \]
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Rubi [A] time = 0.09, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5259, 4427, 3782, 3770, 3767, 8} \[ \frac {a^3 \csc ^{-1}(a+b x)}{3 b^3}+\frac {\left (6 a^2+1\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{6 b^3}-\frac {5 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^3}+\frac {x (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^2}+\frac {1}{3} x^3 \csc ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3782
Rule 4427
Rule 5259
Rubi steps
\begin {align*} \int x^2 \csc ^{-1}(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x \cot (x) \csc (x) (-a+\csc (x))^2 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {1}{3} x^3 \csc ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\csc (x))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac {x (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^2}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int \left (-2 a^3+\left (1+6 a^2\right ) \csc (x)-5 a \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{6 b^3}\\ &=\frac {x (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^2}+\frac {a^3 \csc ^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)+\frac {(5 a) \operatorname {Subst}\left (\int \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{6 b^3}-\frac {\left (1+6 a^2\right ) \operatorname {Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{6 b^3}\\ &=\frac {x (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^2}+\frac {a^3 \csc ^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)+\frac {\left (1+6 a^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{6 b^3}-\frac {(5 a) \operatorname {Subst}\left (\int 1 \, dx,x,(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{6 b^3}\\ &=-\frac {5 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^3}+\frac {x (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{6 b^2}+\frac {a^3 \csc ^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)+\frac {\left (1+6 a^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{6 b^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 129, normalized size = 1.11 \[ \frac {2 a^3 \sin ^{-1}\left (\frac {1}{a+b x}\right )+\left (-5 a^2-4 a b x+b^2 x^2\right ) \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+\left (6 a^2+1\right ) \log \left ((a+b x) \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right )\right )+2 b^3 x^3 \csc ^{-1}(a+b x)}{6 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 117, normalized size = 1.01 \[ \frac {2 \, b^{3} x^{3} \operatorname {arccsc}\left (b x + a\right ) - 4 \, a^{3} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (6 \, a^{2} + 1\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (b x - 5 \, a\right )}}{6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 203, normalized size = 1.75 \[ -\frac {1}{24} \, b {\left (\frac {8 \, {\left (b x + a\right )}^{3} {\left (\frac {3 \, a}{b x + a} - \frac {3 \, a^{2}}{{\left (b x + a\right )}^{2}} - 1\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{4}} + \frac {{\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 12 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 4 \, {\left (6 \, a^{2} + 1\right )} \log \left (-{\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} {\left | b x + a \right |}\right ) - \frac {12 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1}{{\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2}}}{b^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 272, normalized size = 2.34 \[ \frac {x^{3} \mathrm {arccsc}\left (b x +a \right )}{3}+\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{3} \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{3 b^{3} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {\left (-1+\left (b x +a \right )^{2}\right ) x}{6 b^{2} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{2} \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{b^{3} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {5 \left (-1+\left (b x +a \right )^{2}\right ) a}{6 b^{3} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{6 b^{3} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )} \]
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 \[ \frac {1}{3} \, x^{3} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) + \int \frac {{\left (b^{2} x^{4} + a b x^{3}\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (b x + a - 1\right )\right )}}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} e^{\left (\log \left (b x + a + 1\right ) + \log \left (b x + a - 1\right )\right )} - 1\right )}}\,{d x} \]
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 \[ \int x^2\,\mathrm {asin}\left (\frac {1}{a+b\,x}\right ) \,d x \]
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 \[ \int x^{2} \operatorname {acsc}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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