Integrand size = 14, antiderivative size = 178 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac {c^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}+\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2} \]
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Time = 0.22 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5330, 4491, 3378, 3384, 3380, 3383} \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\frac {c^3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b^2}-\frac {c^3 \sin \left (3 \sec ^{-1}(c x)\right )}{4 b \left (a+b \sec ^{-1}(c x)\right )}-\frac {c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{4 b \left (a+b \sec ^{-1}(c x)\right )} \]
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 5330
Rubi steps \begin{align*} \text {integral}& = c^3 \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right ) \\ & = c^3 \text {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 \text {Subst}\left (\int \frac {\sin (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )+\frac {1}{4} c^3 \text {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 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{4 b}+\frac {\left (3 c^3\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\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*}
Time = 0.60 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\frac {-4 b c \sqrt {1-\frac {1}{c^2 x^2}}+c^3 x^2 \left (a+b \sec ^{-1}(c x)\right ) \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )+3 c^3 x^2 \left (a+b \sec ^{-1}(c x)\right ) \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )+a c^3 x^2 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )+b c^3 x^2 \sec ^{-1}(c x) \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )+3 a c^3 x^2 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )+3 b c^3 x^2 \sec ^{-1}(c x) \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )}{4 b^2 x^2 \left (a+b \sec ^{-1}(c x)\right )} \]
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Time = 0.37 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {\sin \left (3 \,\operatorname {arcsec}\left (c x \right )\right )}{4 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b}+\frac {\frac {3 \,\operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) \sin \left (\frac {3 a}{b}\right )}{4}+\frac {3 \,\operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) \cos \left (\frac {3 a}{b}\right )}{4}}{b^{2}}-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b}+\frac {\operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{4 b^{2}}\right )\) | \(153\) |
default | \(c^{3} \left (-\frac {\sin \left (3 \,\operatorname {arcsec}\left (c x \right )\right )}{4 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b}+\frac {\frac {3 \,\operatorname {Si}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) \sin \left (\frac {3 a}{b}\right )}{4}+\frac {3 \,\operatorname {Ci}\left (\frac {3 a}{b}+3 \,\operatorname {arcsec}\left (c x \right )\right ) \cos \left (\frac {3 a}{b}\right )}{4}}{b^{2}}-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4 \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) b}+\frac {\operatorname {Si}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\frac {a}{b}+\operatorname {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{4 b^{2}}\right )\) | \(153\) |
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\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x^{4}} \,d x } \]
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\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int \frac {1}{x^{4} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (164) = 328\).
Time = 0.30 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.90 \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\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 \]
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Timed out. \[ \int \frac {1}{x^4 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int \frac {1}{x^4\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]
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