Optimal. Leaf size=55 \[ -\frac {5 \text {Si}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac {9 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {5 \text {Si}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {\text {Si}\left (7 \cos ^{-1}(a x)\right )}{64 a^7} \]
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Rubi [A] time = 0.09, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4636, 4406, 3299} \[ -\frac {5 \text {Si}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac {9 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {5 \text {Si}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {\text {Si}\left (7 \cos ^{-1}(a x)\right )}{64 a^7} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4406
Rule 4636
Rubi steps
\begin {align*} \int \frac {x^6}{\cos ^{-1}(a x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\cos ^6(x) \sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{a^7}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {5 \sin (x)}{64 x}+\frac {9 \sin (3 x)}{64 x}+\frac {5 \sin (5 x)}{64 x}+\frac {\sin (7 x)}{64 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^7}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (7 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac {5 \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac {5 \operatorname {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac {9 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}\\ &=-\frac {5 \text {Si}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac {9 \text {Si}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {5 \text {Si}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {\text {Si}\left (7 \cos ^{-1}(a x)\right )}{64 a^7}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 40, normalized size = 0.73 \[ -\frac {5 \text {Si}\left (\cos ^{-1}(a x)\right )+9 \text {Si}\left (3 \cos ^{-1}(a x)\right )+5 \text {Si}\left (5 \cos ^{-1}(a x)\right )+\text {Si}\left (7 \cos ^{-1}(a x)\right )}{64 a^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{\arccos \left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.98, size = 47, normalized size = 0.85 \[ -\frac {\operatorname {Si}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Si}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {9 \, \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Si}\left (\arccos \left (a x\right )\right )}{64 \, a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 40, normalized size = 0.73 \[ \frac {-\frac {9 \Si \left (3 \arccos \left (a x \right )\right )}{64}-\frac {5 \Si \left (5 \arccos \left (a x \right )\right )}{64}-\frac {\Si \left (7 \arccos \left (a x \right )\right )}{64}-\frac {5 \Si \left (\arccos \left (a x \right )\right )}{64}}{a^{7}} \]
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 \[ \int \frac {x^{6}}{\arccos \left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^6}{\mathrm {acos}\left (a\,x\right )} \,d x \]
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 \[ \int \frac {x^{6}}{\operatorname {acos}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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