Optimal. Leaf size=19 \[ \text {Int}\left (\frac {\left (a+b \sin ^{-1}(c+d x)\right )^n}{x},x\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^n}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^n}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^n}{-\frac {c}{d}+\frac {x}{d}} \, dx,x,c+d x\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^n}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{n}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^n}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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