Optimal. Leaf size=178 \[ \frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.27, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5222, 4406, 3297, 3303, 3299, 3302} \[ \frac {c^3 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rule 5222
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx &=c^3 \operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )\\ &=c^3 \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{4 (a+b x)^2}+\frac {\sin (3 x)}{4 (a+b x)^2}\right ) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac {1}{4} c^3 \operatorname {Subst}\left (\int \frac {\sin (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )+\frac {1}{4} c^3 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac {c^3 \operatorname {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}\\ &=-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac {\left (c^3 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac {\left (3 c^3 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac {\left (c^3 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac {\left (3 c^3 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}\\ &=-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 223, normalized size = 1.25 \[ \frac {c^3 x^2 \cos \left (\frac {a}{b}\right ) \left (a+b \sec ^{-1}(c x)\right ) \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )+3 c^3 x^2 \cos \left (\frac {3 a}{b}\right ) \left (a+b \sec ^{-1}(c x)\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )+a c^3 x^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )+b c^3 x^2 \sin \left (\frac {a}{b}\right ) \sec ^{-1}(c x) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )+3 a c^3 x^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )+3 b c^3 x^2 \sin \left (\frac {3 a}{b}\right ) \sec ^{-1}(c x) \text {Si}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )-4 b c \sqrt {1-\frac {1}{c^2 x^2}}}{4 b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x^{4} \operatorname {arcsec}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {arcsec}\left (c x\right ) + a^{2} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 694, normalized size = 3.90 \[ \frac {1}{4} \, {\left (\frac {12 \, b c^{2} \arccos \left (\frac {1}{c x}\right ) \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {12 \, b c^{2} \arccos \left (\frac {1}{c x}\right ) \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {12 \, a c^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {12 \, a c^{2} \cos \left (\frac {a}{b}\right )^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} - \frac {9 \, b c^{2} \arccos \left (\frac {1}{c x}\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {b c^{2} \arccos \left (\frac {1}{c x}\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} - \frac {3 \, b c^{2} \arccos \left (\frac {1}{c x}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {b c^{2} \arccos \left (\frac {1}{c x}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} - \frac {9 \, a c^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {a c^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} - \frac {3 \, a c^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {a c^{2} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} - \frac {4 \, b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{{\left (b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}\right )} x^{2}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 153, normalized size = 0.86 \[ c^{3} \left (-\frac {\sin \left (3 \,\mathrm {arcsec}\left (c x \right )\right )}{4 \left (a +b \,\mathrm {arcsec}\left (c x \right )\right ) b}+\frac {\frac {3 \Si \left (\frac {3 a}{b}+3 \,\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {3 a}{b}\right )}{4}+\frac {3 \Ci \left (\frac {3 a}{b}+3 \,\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {3 a}{b}\right )}{4}}{b^{2}}-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4 \left (a +b \,\mathrm {arcsec}\left (c x \right )\right ) b}+\frac {\Si \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )+\Ci \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{4 b^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {4 \, \sqrt {c x + 1} \sqrt {c x - 1} {\left (b \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) + a\right )} + 4 \, {\left (4 \, b^{3} x^{3} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} + b^{3} x^{3} \log \left (c^{2} x^{2}\right )^{2} + 8 \, b^{3} x^{3} \log \relax (c) \log \relax (x) + 4 \, b^{3} x^{3} \log \relax (x)^{2} + 8 \, a b^{2} x^{3} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) + 4 \, {\left (b^{3} \log \relax (c)^{2} + a^{2} b\right )} x^{3} - 4 \, {\left (b^{3} x^{3} \log \relax (c) + b^{3} x^{3} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right )\right )} \int \frac {{\left (2 \, a c^{2} x^{2} + {\left (2 \, b c^{2} x^{2} - 3 \, b\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) - 3 \, a\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{4 \, {\left (b^{3} c^{2} \log \relax (c)^{2} + a^{2} b c^{2}\right )} x^{6} - 4 \, {\left (b^{3} \log \relax (c)^{2} + a^{2} b\right )} x^{4} + 4 \, {\left (b^{3} c^{2} x^{6} - b^{3} x^{4}\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} + {\left (b^{3} c^{2} x^{6} - b^{3} x^{4}\right )} \log \left (c^{2} x^{2}\right )^{2} + 4 \, {\left (b^{3} c^{2} x^{6} - b^{3} x^{4}\right )} \log \relax (x)^{2} + 8 \, {\left (a b^{2} c^{2} x^{6} - a b^{2} x^{4}\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) - 4 \, {\left (b^{3} c^{2} x^{6} \log \relax (c) - b^{3} x^{4} \log \relax (c) + {\left (b^{3} c^{2} x^{6} - b^{3} x^{4}\right )} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right ) + 8 \, {\left (b^{3} c^{2} x^{6} \log \relax (c) - b^{3} x^{4} \log \relax (c)\right )} \log \relax (x)}\,{d x}}{4 \, b^{3} x^{3} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} + b^{3} x^{3} \log \left (c^{2} x^{2}\right )^{2} + 8 \, b^{3} x^{3} \log \relax (c) \log \relax (x) + 4 \, b^{3} x^{3} \log \relax (x)^{2} + 8 \, a b^{2} x^{3} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) + 4 \, {\left (b^{3} \log \relax (c)^{2} + a^{2} b\right )} x^{3} - 4 \, {\left (b^{3} x^{3} \log \relax (c) + b^{3} x^{3} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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