Integrand size = 14, antiderivative size = 134 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx=\frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {3}{16} a b c^4 \sec ^{-1}(c x)+\frac {3}{32} b^2 c^4 \sec ^{-1}(c x)^2+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5330, 4490, 3391} \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx=\frac {3}{16} a b c^4 \sec ^{-1}(c x)+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac {3}{32} b^2 c^4 \sec ^{-1}(c x)^2+\frac {3 b^2 c^2}{32 x^2}+\frac {b^2}{32 x^4} \]
[In]
[Out]
Rule 3391
Rule 4490
Rule 5330
Rubi steps \begin{align*} \text {integral}& = c^4 \text {Subst}\left (\int (a+b x)^2 \cos ^3(x) \sin (x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = -\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} \left (b c^4\right ) \text {Subst}\left (\int (a+b x) \cos ^4(x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {b^2}{32 x^4}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{8} \left (3 b c^4\right ) \text {Subst}\left (\int (a+b x) \cos ^2(x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{16} \left (3 b c^4\right ) \text {Subst}\left (\int (a+b x) \, dx,x,\sec ^{-1}(c x)\right ) \\ & = \frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {3}{16} a b c^4 \sec ^{-1}(c x)+\frac {3}{32} b^2 c^4 \sec ^{-1}(c x)^2+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx=\frac {-8 a^2+b^2+4 a b c \sqrt {1-\frac {1}{c^2 x^2}} x+3 b^2 c^2 x^2+6 a b c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+2 b \left (-8 a+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (2+3 c^2 x^2\right )\right ) \sec ^{-1}(c x)+b^2 \left (-8+3 c^4 x^4\right ) \sec ^{-1}(c x)^2-6 a b c^4 x^4 \arcsin \left (\frac {1}{c x}\right )}{32 x^4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(264\) vs. \(2(116)=232\).
Time = 0.82 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.98
method | result | size |
parts | \(-\frac {a^{2}}{4 x^{4}}+b^{2} c^{4} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arcsec}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arcsec}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {a b \,\operatorname {arcsec}\left (c x \right )}{2 x^{4}}-\frac {3 a b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {3 a b c \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}+\frac {a b \left (c^{2} x^{2}-1\right )}{8 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}}\) | \(265\) |
derivativedivides | \(c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{4}}+b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arcsec}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arcsec}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {a b \,\operatorname {arcsec}\left (c x \right )}{2 c^{4} x^{4}}-\frac {3 a b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 a b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {a b \left (c^{2} x^{2}-1\right )}{8 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(274\) |
default | \(c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{4}}+b^{2} \left (-\frac {\operatorname {arcsec}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\operatorname {arcsec}\left (c x \right ) \left (3 c^{3} x^{3} \operatorname {arcsec}\left (c x \right )+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \operatorname {arcsec}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {a b \,\operatorname {arcsec}\left (c x \right )}{2 c^{4} x^{4}}-\frac {3 a b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 a b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {a b \left (c^{2} x^{2}-1\right )}{8 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(274\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx=\frac {3 \, b^{2} c^{2} x^{2} + {\left (3 \, b^{2} c^{4} x^{4} - 8 \, b^{2}\right )} \operatorname {arcsec}\left (c x\right )^{2} - 8 \, a^{2} + b^{2} + 2 \, {\left (3 \, a b c^{4} x^{4} - 8 \, a b\right )} \operatorname {arcsec}\left (c x\right ) + 2 \, {\left (3 \, a b c^{2} x^{2} + 2 \, a b + {\left (3 \, b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \operatorname {arcsec}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{32 \, x^{4}} \]
[In]
[Out]
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}}{x^{5}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx=\frac {1}{256} \, {\left (24 \, b^{2} c^{3} \arccos \left (\frac {1}{c x}\right )^{2} + 48 \, a b c^{3} \arccos \left (\frac {1}{c x}\right ) - 15 \, b^{2} c^{3} + \frac {48 \, b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )}{x} + \frac {48 \, a b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {24 \, b^{2} c}{x^{2}} + \frac {32 \, b^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arccos \left (\frac {1}{c x}\right )}{x^{3}} + \frac {32 \, a b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{3}} - \frac {64 \, b^{2} \arccos \left (\frac {1}{c x}\right )^{2}}{c x^{4}} - \frac {128 \, a b \arccos \left (\frac {1}{c x}\right )}{c x^{4}} - \frac {64 \, a^{2}}{c x^{4}} + \frac {8 \, b^{2}}{c x^{4}}\right )} c \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2}{x^5} \,d x \]
[In]
[Out]