Integrand size = 16, antiderivative size = 123 \[ \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^3} \]
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Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5335, 1582, 1489, 1821, 858, 222, 272, 65, 214} \[ \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right ) \left (6 c^2 d^2+e^2\right )}{6 c^3}+\frac {b d e x \sqrt {1-\frac {1}{c^2 x^2}}}{c}+\frac {b e^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e} \]
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1489
Rule 1582
Rule 1821
Rule 5335
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \int \frac {(d+e x)^3}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{3 c e} \\ & = \frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \int \frac {\left (e+\frac {d}{x}\right )^3 x}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{3 c e} \\ & = \frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {b \text {Subst}\left (\int \frac {(e+d x)^3}{x^3 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{3 c e} \\ & = \frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \text {Subst}\left (\int \frac {-6 d e^2-e \left (6 d^2+\frac {e^2}{c^2}\right ) x-2 d^3 x^2}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c e} \\ & = \frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {b \text {Subst}\left (\int \frac {e \left (6 d^2+\frac {e^2}{c^2}\right )+2 d^3 x}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c e} \\ & = \frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {\left (b d^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{3 c e}-\frac {\left (b \left (6 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c^3} \\ & = \frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {\left (b \left (6 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{12 c^3} \\ & = \frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (b \left (6 c^2 d^2+e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c} \\ & = \frac {b d e \sqrt {1-\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \csc ^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2+e^2\right ) \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{6 c^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99 \[ \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {c^2 x \left (b e \sqrt {1-\frac {1}{c^2 x^2}} (6 d+e x)+2 a c \left (3 d^2+3 d e x+e^2 x^2\right )\right )+2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \csc ^{-1}(c x)+b \left (6 c^2 d^2+e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{6 c^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs. \(2(109)=218\).
Time = 0.57 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.47
method | result | size |
parts | \(\frac {a \left (e x +d \right )^{3}}{3 e}+\frac {b \,e^{2} \operatorname {arccsc}\left (c x \right ) x^{3}}{3}+b e \,\operatorname {arccsc}\left (c x \right ) x^{2} d +b \,\operatorname {arccsc}\left (c x \right ) x \,d^{2}+\frac {b \,d^{3} \operatorname {arccsc}\left (c x \right )}{3 e}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{3 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(304\) |
derivativedivides | \(\frac {\frac {a \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b c \,\operatorname {arccsc}\left (c x \right ) d^{3}}{3 e}+b \,\operatorname {arccsc}\left (c x \right ) d^{2} c x +b c e \,\operatorname {arccsc}\left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \operatorname {arccsc}\left (c x \right ) x^{3}}{3}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{3 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right )}{6 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\) | \(315\) |
default | \(\frac {\frac {a \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b c \,\operatorname {arccsc}\left (c x \right ) d^{3}}{3 e}+b \,\operatorname {arccsc}\left (c x \right ) d^{2} c x +b c e \,\operatorname {arccsc}\left (c x \right ) d \,x^{2}+\frac {b c \,e^{2} \operatorname {arccsc}\left (c x \right ) x^{3}}{3}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{3 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right )}{6 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\) | \(315\) |
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Time = 0.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.70 \[ \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {2 \, a c^{3} e^{2} x^{3} + 6 \, a c^{3} d e x^{2} + 6 \, a c^{3} d^{2} x + 2 \, {\left (b c^{3} e^{2} x^{3} + 3 \, b c^{3} d e x^{2} + 3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} - 3 \, b c^{3} d e - b c^{3} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) - 4 \, {\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{2} d^{2} + b e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c e^{2} x + 6 \, b c d e\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, c^{3}} \]
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Time = 3.40 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.85 \[ \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {acsc}{\left (c x \right )} + b d e x^{2} \operatorname {acsc}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b d^{2} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {b d e \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{c} + \frac {b e^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} \]
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Time = 0.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.61 \[ \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d e + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d^{2}}{2 \, c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (109) = 218\).
Time = 2.19 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.89 \[ \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{24} \, {\left (\frac {b e^{2} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a e^{2} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c} - \frac {24 \, b d e x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {b e^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{2}} - \frac {24 \, a d e x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}}{c} + \frac {12 \, b d^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {12 \, a d^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c} + \frac {3 \, b e^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {24 \, b d e x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2}} + \frac {3 \, a e^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{3}} + \frac {24 \, b d^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {24 \, b d^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{2}} + \frac {24 \, b d e \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {24 \, a d e}{c^{3}} + \frac {4 \, b e^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {4 \, b e^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {12 \, b d^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {12 \, a d^{2}}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, b e^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, a e^{2}}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {b e^{2}}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {b e^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {a e^{2}}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}\right )} c \]
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Timed out. \[ \int (d+e x)^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \]
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