Integrand size = 19, antiderivative size = 105 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx=\frac {b c \left (2 c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}+\frac {b c d \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{x} \]
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Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 5346, 12, 464, 270} \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx=-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{x}+\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d+9 e\right )}{9 \sqrt {c^2 x^2}}+\frac {b c d \sqrt {c^2 x^2-1}}{9 x^2 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 14
Rule 270
Rule 464
Rule 5346
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{x}-\frac {(b c x) \int \frac {-d-3 e x^2}{3 x^4 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{x}-\frac {(b c x) \int \frac {-d-3 e x^2}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{3 \sqrt {c^2 x^2}} \\ & = \frac {b c d \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{x}-\frac {\left (b c \left (-2 c^2 d-9 e\right ) x\right ) \int \frac {1}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{9 \sqrt {c^2 x^2}} \\ & = \frac {b c \left (2 c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}+\frac {b c d \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{x} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx=\frac {-3 a \left (d+3 e x^2\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+2 c^2 d x^2+9 e x^2\right )-3 b \left (d+3 e x^2\right ) \sec ^{-1}(c x)}{9 x^3} \]
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Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.03
method | result | size |
parts | \(a \left (-\frac {e}{x}-\frac {d}{3 x^{3}}\right )+b \,c^{3} \left (-\frac {\operatorname {arcsec}\left (c x \right ) e}{c^{3} x}-\frac {\operatorname {arcsec}\left (c x \right ) d}{3 x^{3} c^{3}}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{4} d \,x^{2}+9 c^{2} e \,x^{2}+c^{2} d \right )}{9 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{4}}\right )\) | \(108\) |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arcsec}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\operatorname {arcsec}\left (c x \right ) e}{c x}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{4} d \,x^{2}+9 c^{2} e \,x^{2}+c^{2} d \right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{2}}\right )\) | \(121\) |
default | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arcsec}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\operatorname {arcsec}\left (c x \right ) e}{c x}+\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{4} d \,x^{2}+9 c^{2} e \,x^{2}+c^{2} d \right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{2}}\right )\) | \(121\) |
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Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx=-\frac {9 \, a e x^{2} + 3 \, a d + 3 \, {\left (3 \, b e x^{2} + b d\right )} \operatorname {arcsec}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1} {\left ({\left (2 \, b c^{2} d + 9 \, b e\right )} x^{2} + b d\right )}}{9 \, x^{3}} \]
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Time = 2.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.43 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx=- \frac {a d}{3 x^{3}} - \frac {a e}{x} + b c e \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d \operatorname {asec}{\left (c x \right )}}{3 x^{3}} - \frac {b e \operatorname {asec}{\left (c x \right )}}{x} + \frac {b d \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} \]
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Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx={\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsec}\left (c x\right )}{x}\right )} b e - \frac {1}{9} \, b d {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {3 \, \operatorname {arcsec}\left (c x\right )}{x^{3}}\right )} - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx=\frac {1}{9} \, {\left (2 \, b c^{2} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 9 \, b e \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {9 \, b e \arccos \left (\frac {1}{c x}\right )}{c x} + \frac {b d \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{2}} - \frac {9 \, a e}{c x} - \frac {3 \, b d \arccos \left (\frac {1}{c x}\right )}{c x^{3}} - \frac {3 \, a d}{c x^{3}}\right )} c \]
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Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]
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