Integrand size = 14, antiderivative size = 83 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}+\frac {b d \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c} \]
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Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5335, 1582, 1410, 1821, 858, 222, 272, 65, 214} \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}+\frac {b d \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}+\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}-\frac {b d^2 \csc ^{-1}(c x)}{2 e} \]
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 1410
Rule 1582
Rule 1821
Rule 5335
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}+\frac {b \int \frac {(d+e x)^2}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{2 c e} \\ & = \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}+\frac {b \int \frac {\left (e+\frac {d}{x}\right )^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{2 c e} \\ & = \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \text {Subst}\left (\int \frac {(e+d x)^2}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e} \\ & = \frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}+\frac {b \text {Subst}\left (\int \frac {-2 d e-d^2 x}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e} \\ & = \frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {(b d) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e} \\ & = \frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {(b d) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c} \\ & = \frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}+(b c d) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right ) \\ & = \frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}+\frac {b d \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d x+\frac {1}{2} a e x^2+\frac {b e x \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{2 c}+b d x \csc ^{-1}(c x)+\frac {1}{2} b e x^2 \csc ^{-1}(c x)+\frac {b d \sqrt {1-\frac {1}{c^2 x^2}} x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {-1+c^2 x^2}} \]
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Time = 0.48 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33
method | result | size |
parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \left (\frac {c \,\operatorname {arccsc}\left (c x \right ) x^{2} e}{2}+\operatorname {arccsc}\left (c x \right ) x c d +\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c}\) | \(110\) |
derivativedivides | \(\frac {\frac {a \left (d x \,c^{2}+\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arccsc}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c}\) | \(127\) |
default | \(\frac {\frac {a \left (d x \,c^{2}+\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arccsc}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c}\) | \(127\) |
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Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.55 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a c^{2} e x^{2} + 2 \, a c^{2} d x - 2 \, b c d \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} b e + {\left (b c^{2} e x^{2} + 2 \, b c^{2} d x - 2 \, b c^{2} d - b c^{2} e\right )} \operatorname {arccsc}\left (c x\right ) - 2 \, {\left (2 \, b c^{2} d + b c^{2} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, c^{2}} \]
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Time = 2.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.25 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d x + \frac {a e x^{2}}{2} + b d x \operatorname {acsc}{\left (c x \right )} + \frac {b e x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b d \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {b e \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} \]
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Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.11 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d}{2 \, c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (73) = 146\).
Time = 0.44 (sec) , antiderivative size = 341, normalized size of antiderivative = 4.11 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{8} \, {\left (\frac {b e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c} + \frac {4 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {4 \, a d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c} + \frac {2 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{2}} + \frac {8 \, b d \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {8 \, b d \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{2}} + \frac {2 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {2 \, a e}{c^{3}} + \frac {4 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {4 \, a d}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {2 \, b e}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {b e \arcsin \left (\frac {1}{c x}\right )}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {a e}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}\right )} c \]
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Timed out. \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\left (d+e\,x\right ) \,d x \]
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