Optimal. Leaf size=134 \[ \frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {3}{16} a b c^4 \sec ^{-1}(c x)+\frac {3}{32} b^2 c^4 \sec ^{-1}(c x)^2+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{8 x^3}+\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \sec ^{-1}(c x)\right )^2}{4 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5330, 4490,
3391} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3391
Rule 4490
Rule 5330
Rubi steps
\begin {align*} \int \frac {\left (a+b \sec ^{-1}(c x)\right )^2}{x^5} \, dx &=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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 148, normalized size = 1.10 \begin {gather*} \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 \text {ArcSin}\left (\frac {1}{c x}\right )}{32 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(273\) vs.
\(2(116)=232\).
time = 0.24, size = 274, normalized size = 2.04
method | result | size |
derivativedivides | \(c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{4}}+b^{2} \left (-\frac {\mathrm {arcsec}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\mathrm {arcsec}\left (c x \right ) \left (3 \,\mathrm {arcsec}\left (c x \right ) c^{3} x^{3}+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \mathrm {arcsec}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {a b \,\mathrm {arcsec}\left (c x \right )}{2 c^{4} x^{4}}-\frac {3 a b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 a b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {a b \left (c^{2} x^{2}-1\right )}{8 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(274\) |
default | \(c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{4}}+b^{2} \left (-\frac {\mathrm {arcsec}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\mathrm {arcsec}\left (c x \right ) \left (3 \,\mathrm {arcsec}\left (c x \right ) c^{3} x^{3}+3 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\right )}{16 c^{3} x^{3}}-\frac {3 \mathrm {arcsec}\left (c x \right )^{2}}{32}+\frac {\left (3 c^{2} x^{2}+2\right )^{2}}{128 c^{4} x^{4}}\right )-\frac {a b \,\mathrm {arcsec}\left (c x \right )}{2 c^{4} x^{4}}-\frac {3 a b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 a b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}+\frac {a b \left (c^{2} x^{2}-1\right )}{8 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(274\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.25, size = 120, normalized size = 0.90 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 215, normalized size = 1.60 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________