Integrand size = 16, antiderivative size = 167 \[ \int (d+e x)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2+e^2\right ) \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c^3} \]
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Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \int (d+e x)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b d \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right ) \left (2 c^2 d^2+e^2\right )}{2 c^3}+\frac {b d e^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}+\frac {b e^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}+\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (9 c^2 d^2+e^2\right )}{6 c^3}-\frac {b d^4 \csc ^{-1}(c x)}{4 e} \]
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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)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b \int \frac {(d+e x)^4}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{4 c e} \\ & = \frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b \int \frac {\left (e+\frac {d}{x}\right )^4 x^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{4 c e} \\ & = \frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}-\frac {b \text {Subst}\left (\int \frac {(e+d x)^4}{x^4 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e} \\ & = \frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b \text {Subst}\left (\int \frac {-12 d e^3-2 e^2 \left (9 d^2+\frac {e^2}{c^2}\right ) x-12 d^3 e x^2-3 d^4 x^3}{x^3 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{12 c e} \\ & = \frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}-\frac {b \text {Subst}\left (\int \frac {4 e^2 \left (9 d^2+\frac {e^2}{c^2}\right )+12 d e \left (2 d^2+\frac {e^2}{c^2}\right ) x+6 d^4 x^2}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{24 c e} \\ & = \frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b \text {Subst}\left (\int \frac {-12 d e \left (2 d^2+\frac {e^2}{c^2}\right )-6 d^4 x}{x \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{24 c e} \\ & = \frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}-\frac {\left (b d^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}-\frac {\left (b d \left (2 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 )}{2 c^3} \\ & = \frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}-\frac {\left (b d \left (2 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 )}{4 c^3} \\ & = \frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {\left (b d \left (2 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 )}{2 c} \\ & = \frac {b e \left (9 c^2 d^2+e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \csc ^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2+e^2\right ) \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c^3} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99 \[ \int (d+e x)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {3 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (2 e^2+c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )+3 b c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \csc ^{-1}(c x)+6 b d \left (2 c^2 d^2+e^2\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{12 c^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(147)=294\).
Time = 0.63 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.40
method | result | size |
parts | \(\frac {a \left (e x +d \right )^{4}}{4 e}+\frac {b \,e^{3} \operatorname {arccsc}\left (c x \right ) x^{4}}{4}+b \,e^{2} \operatorname {arccsc}\left (c x \right ) x^{3} d +\frac {3 b e \,\operatorname {arccsc}\left (c x \right ) x^{2} d^{2}}{2}+b \,\operatorname {arccsc}\left (c x \right ) x \,d^{3}+\frac {b \,d^{4} \operatorname {arccsc}\left (c x \right )}{4 e}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right ) x}{12 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) d}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {3 b e \left (c^{2} x^{2}-1\right ) d^{2}}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right )}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(401\) |
derivativedivides | \(\frac {\frac {a \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b c \,\operatorname {arccsc}\left (c x \right ) d^{4}}{4 e}+b \,\operatorname {arccsc}\left (c x \right ) d^{3} c x +\frac {3 b c e \,\operatorname {arccsc}\left (c x \right ) d^{2} x^{2}}{2}+b c \,e^{2} \operatorname {arccsc}\left (c x \right ) d \,x^{3}+\frac {b c \,e^{3} \operatorname {arccsc}\left (c x \right ) x^{4}}{4}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b e \left (c^{2} x^{2}-1\right ) d^{2}}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) d}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right ) x}{12 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\) | \(413\) |
default | \(\frac {\frac {a \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b c \,\operatorname {arccsc}\left (c x \right ) d^{4}}{4 e}+b \,\operatorname {arccsc}\left (c x \right ) d^{3} c x +\frac {3 b c e \,\operatorname {arccsc}\left (c x \right ) d^{2} x^{2}}{2}+b c \,e^{2} \operatorname {arccsc}\left (c x \right ) d \,x^{3}+\frac {b c \,e^{3} \operatorname {arccsc}\left (c x \right ) x^{4}}{4}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b e \left (c^{2} x^{2}-1\right ) d^{2}}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) d}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right ) x}{12 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \,e^{3} \left (c^{2} x^{2}-1\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c}\) | \(413\) |
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Time = 0.39 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.74 \[ \int (d+e x)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {3 \, a c^{4} e^{3} x^{4} + 12 \, a c^{4} d e^{2} x^{3} + 18 \, a c^{4} d^{2} e x^{2} + 12 \, a c^{4} d^{3} x + 3 \, {\left (b c^{4} e^{3} x^{4} + 4 \, b c^{4} d e^{2} x^{3} + 6 \, b c^{4} d^{2} e x^{2} + 4 \, b c^{4} d^{3} x - 4 \, b c^{4} d^{3} - 6 \, b c^{4} d^{2} e - 4 \, b c^{4} d e^{2} - b c^{4} e^{3}\right )} \operatorname {arccsc}\left (c x\right ) - 6 \, {\left (4 \, b c^{4} d^{3} + 6 \, b c^{4} d^{2} e + 4 \, b c^{4} d e^{2} + b c^{4} e^{3}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (2 \, b c^{3} d^{3} + b c d e^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} e^{3} x^{2} + 6 \, b c^{2} d e^{2} x + 18 \, b c^{2} d^{2} e + 2 \, b e^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \]
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Time = 3.98 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.17 \[ \int (d+e x)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {acsc}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {acsc}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b d^{3} \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {3 b d^{2} e \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} + \frac {b d e^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{c} + \frac {b e^{3} \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} \]
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Time = 0.20 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.61 \[ \int (d+e x)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} e + \frac {1}{4} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d e^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d^{3}}{2 \, c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1130 vs. \(2 (147) = 294\).
Time = 2.48 (sec) , antiderivative size = 1130, normalized size of antiderivative = 6.77 \[ \int (d+e x)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (d+e x)^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]
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