Optimal. Leaf size=82 \[ \frac {x^6 \sqrt {1-a^2 x^2}}{a \text {ArcCos}(a x)}-\frac {5 \text {CosIntegral}(\text {ArcCos}(a x))}{64 a^7}-\frac {27 \text {CosIntegral}(3 \text {ArcCos}(a x))}{64 a^7}-\frac {25 \text {CosIntegral}(5 \text {ArcCos}(a x))}{64 a^7}-\frac {7 \text {CosIntegral}(7 \text {ArcCos}(a x))}{64 a^7} \]
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Rubi [A]
time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4728, 3383}
\begin {gather*} -\frac {5 \text {CosIntegral}(\text {ArcCos}(a x))}{64 a^7}-\frac {27 \text {CosIntegral}(3 \text {ArcCos}(a x))}{64 a^7}-\frac {25 \text {CosIntegral}(5 \text {ArcCos}(a x))}{64 a^7}-\frac {7 \text {CosIntegral}(7 \text {ArcCos}(a x))}{64 a^7}+\frac {x^6 \sqrt {1-a^2 x^2}}{a \text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 4728
Rubi steps
\begin {align*} \int \frac {x^6}{\cos ^{-1}(a x)^2} \, dx &=\frac {x^6 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}+\frac {\text {Subst}\left (\int \left (-\frac {5 \cos (x)}{64 x}-\frac {27 \cos (3 x)}{64 x}-\frac {25 \cos (5 x)}{64 x}-\frac {7 \cos (7 x)}{64 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^7}\\ &=\frac {x^6 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac {7 \text {Subst}\left (\int \frac {\cos (7 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac {27 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}\\ &=\frac {x^6 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac {5 \text {Ci}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac {27 \text {Ci}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Ci}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {7 \text {Ci}\left (7 \cos ^{-1}(a x)\right )}{64 a^7}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 86, normalized size = 1.05 \begin {gather*} -\frac {-64 a^6 x^6 \sqrt {1-a^2 x^2}+5 \text {ArcCos}(a x) \text {CosIntegral}(\text {ArcCos}(a x))+27 \text {ArcCos}(a x) \text {CosIntegral}(3 \text {ArcCos}(a x))+25 \text {ArcCos}(a x) \text {CosIntegral}(5 \text {ArcCos}(a x))+7 \text {ArcCos}(a x) \text {CosIntegral}(7 \text {ArcCos}(a x))}{64 a^7 \text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 105, normalized size = 1.28
method | result | size |
derivativedivides | \(\frac {\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {27 \cosineIntegral \left (3 \arccos \left (a x \right )\right )}{64}+\frac {5 \sin \left (5 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {25 \cosineIntegral \left (5 \arccos \left (a x \right )\right )}{64}+\frac {\sin \left (7 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {7 \cosineIntegral \left (7 \arccos \left (a x \right )\right )}{64}+\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arccos \left (a x \right )}-\frac {5 \cosineIntegral \left (\arccos \left (a x \right )\right )}{64}}{a^{7}}\) | \(105\) |
default | \(\frac {\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {27 \cosineIntegral \left (3 \arccos \left (a x \right )\right )}{64}+\frac {5 \sin \left (5 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {25 \cosineIntegral \left (5 \arccos \left (a x \right )\right )}{64}+\frac {\sin \left (7 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {7 \cosineIntegral \left (7 \arccos \left (a x \right )\right )}{64}+\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arccos \left (a x \right )}-\frac {5 \cosineIntegral \left (\arccos \left (a x \right )\right )}{64}}{a^{7}}\) | \(105\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{6}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 72, normalized size = 0.88 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} x^{6}}{a \arccos \left (a x\right )} - \frac {7 \, \operatorname {Ci}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {25 \, \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {27 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Ci}\left (\arccos \left (a x\right )\right )}{64 \, a^{7}} \end {gather*}
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 \begin {gather*} \int \frac {x^6}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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