Optimal. Leaf size=117 \[ -\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b}-\frac {c^3 \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b} \]
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Rubi [A] time = 0.22, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5222, 4406, 3303, 3299, 3302} \[ -\frac {c^3 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b}-\frac {c^3 \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b} \]
Antiderivative was successfully verified.
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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 )} \, dx &=c^3 \operatorname {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )\\ &=c^3 \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{4 (a+b x)}+\frac {\sin (3 x)}{4 (a+b x)}\right ) \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac {1}{4} c^3 \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )+\frac {1}{4} c^3 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac {1}{4} \left (c^3 \cos \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 )+\frac {1}{4} \left (c^3 \cos \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 )-\frac {1}{4} \left (c^3 \sin \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 )-\frac {1}{4} \left (c^3 \sin \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 )\\ &=-\frac {c^3 \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{4 b}-\frac {c^3 \text {Ci}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}+\frac {c^3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{4 b}+\frac {c^3 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sec ^{-1}(c x)\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 91, normalized size = 0.78 \[ \frac {c^3 \left (\sin \left (\frac {a}{b}\right ) \left (-\text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )-\sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )+\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )\right )}{4 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x^{4} \operatorname {arcsec}\left (c x\right ) + a x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 199, normalized size = 1.70 \[ -\frac {1}{4} \, {\left (\frac {4 \, c^{2} \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b} - \frac {4 \, c^{2} \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b} - \frac {c^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b} + \frac {c^{2} \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b} + \frac {3 \, c^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (\frac {1}{c x}\right )\right )}{b} - \frac {c^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 102, normalized size = 0.87 \[ c^{3} \left (\frac {\Si \left (\frac {3 a}{b}+3 \,\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {3 a}{b}\right )}{4 b}-\frac {\Ci \left (\frac {3 a}{b}+3 \,\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}+\frac {\Si \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{4 b}-\frac {\Ci \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )}{4 b}\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )} \,d x \]
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 \[ \int \frac {1}{x^{4} \left (a + b \operatorname {asec}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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