Integrand size = 21, antiderivative size = 163 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e \left (12 c^2 d+e\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{6 c^2 \sqrt {c^2 x^2}} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 5347, 12, 1279, 396, 223, 212} \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e x \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (12 c^2 d+e\right )}{6 c^2 \sqrt {c^2 x^2}}-\frac {b c d^2 \sqrt {c^2 x^2-1}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {c^2 x^2-1}}{6 c \sqrt {c^2 x^2}} \]
[In]
[Out]
Rule 12
Rule 212
Rule 223
Rule 276
Rule 396
Rule 1279
Rule 5347
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{3 x^2 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {6 d e+e^2 x^2}{\sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+-\frac {\left (b \left (-12 c^2 d e-e^2\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{6 c \sqrt {c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+-\frac {\left (b \left (-12 c^2 d e-e^2\right ) x\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{6 c \sqrt {c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {-1+c^2 x^2}}{\sqrt {c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+2 d e x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e \left (12 c^2 d+e\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{6 c^2 \sqrt {c^2 x^2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.82 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\frac {c^2 \left (b \sqrt {1-\frac {1}{c^2 x^2}} x \left (-6 c^2 d^2+e^2 x^2\right )+2 a c \left (-3 d^2+6 d e x^2+e^2 x^4\right )\right )+2 b c^3 \left (-3 d^2+6 d e x^2+e^2 x^4\right ) \csc ^{-1}(c x)+b e \left (12 c^2 d+e\right ) x \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{6 c^3 x} \]
[In]
[Out]
Time = 0.52 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.53
method | result | size |
parts | \(a \left (\frac {e^{2} x^{3}}{3}+2 d e x -\frac {d^{2}}{x}\right )+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{3}}{3}+2 b e \,\operatorname {arccsc}\left (c x \right ) x d -\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2}}{x}+\frac {b \left (c^{2} x^{2}-1\right ) e^{2}}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{c \,x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(250\) |
derivativedivides | \(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {2 b \,\operatorname {arccsc}\left (c x \right ) d e x}{c}+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{3}}{3 c}-\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2}}{c x}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) e^{2}}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\) | \(273\) |
default | \(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {2 b \,\operatorname {arccsc}\left (c x \right ) d e x}{c}+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{3}}{3 c}-\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2}}{c x}-\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \sqrt {c^{2} x^{2}-1}\, d e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) e^{2}}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )\) | \(273\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.42 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\frac {2 \, a c^{3} e^{2} x^{4} - 6 \, b c^{4} d^{2} x + 12 \, a c^{3} d e x^{2} - 6 \, a c^{3} d^{2} + 4 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (12 \, b c^{2} d e + b e^{2}\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} + {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \operatorname {arccsc}\left (c x\right ) - {\left (6 \, b c^{3} d^{2} - b c e^{2} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{6 \, c^{3} x} \]
[In]
[Out]
Time = 4.36 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.27 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=- \frac {a d^{2}}{x} + 2 a d e x + \frac {a e^{2} x^{3}}{3} - b c d^{2} \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d^{2} \operatorname {acsc}{\left (c x \right )}}{x} + 2 b d e x \operatorname {acsc}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {2 b d e \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {b e^{2} \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\frac {1}{3} \, a e^{2} x^{3} - {\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b d^{2} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e^{2} + 2 \, a d e x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d e}{c} - \frac {a d^{2}}{x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2502 vs. \(2 (145) = 290\).
Time = 2.32 (sec) , antiderivative size = 2502, normalized size of antiderivative = 15.35 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^2} \,d x \]
[In]
[Out]