Optimal. Leaf size=155 \[ -\frac {a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac {\left (17 a^2+2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^4}-\frac {\left (2 a^2+1\right ) a \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{2 b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{3 b^4}+\frac {x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^2}+\frac {1}{4} x^4 \csc ^{-1}(a+b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5259, 4427, 3782, 4048, 3770, 3767, 8} \[ \frac {\left (17 a^2+2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^4}-\frac {a^4 \csc ^{-1}(a+b x)}{4 b^4}-\frac {\left (2 a^2+1\right ) a \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{2 b^4}+\frac {x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{3 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rule 3770
Rule 3782
Rule 4048
Rule 4427
Rule 5259
Rubi steps
\begin {align*} \int x^3 \csc ^{-1}(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x \cot (x) \csc (x) (-a+\csc (x))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}\\ &=\frac {1}{4} x^4 \csc ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\csc (x))^4 \, dx,x,\csc ^{-1}(a+b x)\right )}{4 b^4}\\ &=\frac {x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^2}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\csc (x)) \left (-3 a^3+\left (2+9 a^2\right ) \csc (x)-8 a \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{12 b^4}\\ &=\frac {x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{3 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int \left (6 a^4-12 a \left (1+2 a^2\right ) \csc (x)+2 \left (2+17 a^2\right ) \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{24 b^4}\\ &=\frac {x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)+\frac {\left (a \left (1+2 a^2\right )\right ) \operatorname {Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^4}-\frac {\left (2+17 a^2\right ) \operatorname {Subst}\left (\int \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{12 b^4}\\ &=\frac {x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)-\frac {a \left (1+2 a^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{2 b^4}+\frac {\left (2+17 a^2\right ) \operatorname {Subst}\left (\int 1 \, dx,x,(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )}{12 b^4}\\ &=\frac {\left (2+17 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^4}+\frac {x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)-\frac {a \left (1+2 a^2\right ) \tanh ^{-1}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{2 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.29, size = 149, normalized size = 0.96 \[ \frac {-3 a^4 \sin ^{-1}\left (\frac {1}{a+b x}\right )-6 \left (2 a^2+1\right ) a \log \left ((a+b x) \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right )\right )+\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (13 a^3+9 a^2 b x-3 a b^2 x^2+2 a+b^3 x^3+2 b x\right )+3 b^4 x^4 \csc ^{-1}(a+b x)}{12 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 129, normalized size = 0.83 \[ \frac {3 \, b^{4} x^{4} \operatorname {arccsc}\left (b x + a\right ) + 6 \, a^{4} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 6 \, {\left (2 \, a^{3} + a\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (b^{2} x^{2} - 4 \, a b x + 13 \, a^{2} + 2\right )}}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 300, normalized size = 1.94 \[ -\frac {1}{96} \, b {\left (\frac {24 \, {\left (b x + a\right )}^{4} {\left (\frac {4 \, a}{b x + a} - \frac {6 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {4 \, a^{3}}{{\left (b x + a\right )}^{3}} - 1\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{5}} - \frac {{\left (b x + a\right )}^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 12 \, {\left (b x + a\right )}^{2} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 72 \, {\left (b x + a\right )} a^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 9 \, {\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 48 \, {\left (2 \, a^{3} + a\right )} \log \left (-{\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} {\left | b x + a \right |}\right ) - \frac {9 \, {\left (8 \, a^{2} + 1\right )} {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 12 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1}{{\left (b x + a\right )}^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3}}}{b^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 360, normalized size = 2.32 \[ \frac {x^{4} \mathrm {arccsc}\left (b x +a \right )}{4}+\frac {\left (-1+\left (b x +a \right )^{2}\right ) x^{2}}{12 b^{2} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{4} \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{4 b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {\left (-1+\left (b x +a \right )^{2}\right ) x a}{3 b^{3} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{3} \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {13 \left (-1+\left (b x +a \right )^{2}\right ) a^{2}}{12 b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{2 b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {-1+\left (b x +a \right )^{2}}{6 b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{4} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) + \int \frac {{\left (b^{2} x^{5} + a b x^{4}\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (b x + a - 1\right )\right )}}{4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} e^{\left (\log \left (b x + a + 1\right ) + \log \left (b x + a - 1\right )\right )} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\mathrm {asin}\left (\frac {1}{a+b\,x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {acsc}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________