Optimal. Leaf size=145 \[ -\frac {a^2 \csc ^{-1}(a+b x)^2}{2 b^2}+\frac {2 i a \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac {2 i a \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac {\log (a+b x)}{b^2}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^2}-\frac {4 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5259, 4427, 4190, 4183, 2279, 2391, 4184, 3475} \[ \frac {2 i a \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac {2 i a \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac {a^2 \csc ^{-1}(a+b x)^2}{2 b^2}+\frac {\log (a+b x)}{b^2}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^2}-\frac {4 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2391
Rule 3475
Rule 4183
Rule 4184
Rule 4190
Rule 4427
Rule 5259
Rubi steps
\begin {align*} \int x \csc ^{-1}(a+b x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \cot (x) \csc (x) (-a+\csc (x)) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {1}{2} x^2 \csc ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int x (-a+\csc (x))^2 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {1}{2} x^2 \csc ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int \left (a^2 x-2 a x \csc (x)+x \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {a^2 \csc ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}+\frac {(2 a) \operatorname {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^2}-\frac {a^2 \csc ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)^2-\frac {4 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac {\operatorname {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}-\frac {(2 a) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}+\frac {(2 a) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^2}-\frac {a^2 \csc ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)^2-\frac {4 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac {\log (a+b x)}{b^2}+\frac {(2 i a) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac {(2 i a) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}\\ &=\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^2}-\frac {a^2 \csc ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)^2-\frac {4 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac {\log (a+b x)}{b^2}+\frac {2 i a \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac {2 i a \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.82, size = 213, normalized size = 1.47 \[ \frac {2 a \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \csc ^{-1}(a+b x)+2 b x \sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \csc ^{-1}(a+b x)-a^2 \csc ^{-1}(a+b x)^2+b^2 x^2 \csc ^{-1}(a+b x)^2+4 i a \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )-4 i a \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )-2 \log \left (\frac {1}{a+b x}\right )+4 a \csc ^{-1}(a+b x) \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-4 a \csc ^{-1}(a+b x) \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )}{2 b^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {arccsc}\left (b x + a\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {arccsc}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.34, size = 250, normalized size = 1.72 \[ -\frac {a^{2} \mathrm {arccsc}\left (b x +a \right )^{2}}{2 b^{2}}+\frac {2 a \,\mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{2}}-\frac {2 a \,\mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{2}}+\frac {2 i a \dilog \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{2}}-\frac {2 i a \dilog \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{2}}+\frac {x^{2} \mathrm {arccsc}\left (b x +a \right )^{2}}{2}+\frac {\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) x}{b}+\frac {\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) a}{b^{2}}-\frac {\ln \left (\frac {1}{b x +a}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )^{2} - \frac {1}{8} \, x^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} + \int \frac {2 \, \sqrt {b x + a + 1} \sqrt {b x + a - 1} b x^{2} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - 2 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} - 1\right )} b x^{2} + {\left (a^{3} - a\right )} x\right )} \log \left (b x + a\right )^{2} + {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + {\left (a^{2} - 1\right )} b x^{2} + 2 \, {\left (b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} - 1\right )} b x^{2} + {\left (a^{3} - a\right )} x\right )} \log \left (b x + a\right )\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} - 1\right )} b x - a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {acsc}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________