Optimal. Leaf size=246 \[ -\frac {d n \sqrt {1-a^2 x^2}}{a}+\frac {\sqrt {1-a^2 x^2} \left (3 a^2 d+e\right ) \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}-\frac {n \sqrt {1-a^2 x^2} \left (3 a^2 d+e\right )}{3 a^3}+\frac {n \left (3 a^2 d+e\right ) \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{3 a^3}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac {e n \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )-d n x \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.23, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {4665, 444, 43, 2387, 266, 50, 63, 208, 4619, 261, 4627} \[ \frac {\sqrt {1-a^2 x^2} \left (3 a^2 d+e\right ) \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}-\frac {n \sqrt {1-a^2 x^2} \left (3 a^2 d+e\right )}{3 a^3}+\frac {n \left (3 a^2 d+e\right ) \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{3 a^3}-\frac {d n \sqrt {1-a^2 x^2}}{a}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac {e n \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )-d n x \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 50
Rule 63
Rule 208
Rule 261
Rule 266
Rule 444
Rule 2387
Rule 4619
Rule 4627
Rule 4665
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \sin ^{-1}(a x) \log \left (c x^n\right ) \, dx &=\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )-n \int \left (\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2}}{3 a^3 x}-\frac {e \left (1-a^2 x^2\right )^{3/2}}{9 a^3 x}+d \sin ^{-1}(a x)+\frac {1}{3} e x^2 \sin ^{-1}(a x)\right ) \, dx\\ &=\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )-(d n) \int \sin ^{-1}(a x) \, dx-\frac {1}{3} (e n) \int x^2 \sin ^{-1}(a x) \, dx+\frac {(e n) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x} \, dx}{9 a^3}-\frac {\left (\left (3 a^2 d+e\right ) n\right ) \int \frac {\sqrt {1-a^2 x^2}}{x} \, dx}{3 a^3}\\ &=-d n x \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-1}(a x)+\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )+(a d n) \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx+\frac {(e n) \operatorname {Subst}\left (\int \frac {\left (1-a^2 x\right )^{3/2}}{x} \, dx,x,x^2\right )}{18 a^3}+\frac {1}{9} (a e n) \int \frac {x^3}{\sqrt {1-a^2 x^2}} \, dx-\frac {\left (\left (3 a^2 d+e\right ) n\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x} \, dx,x,x^2\right )}{6 a^3}\\ &=-\frac {d n \sqrt {1-a^2 x^2}}{a}-\frac {\left (3 a^2 d+e\right ) n \sqrt {1-a^2 x^2}}{3 a^3}+\frac {e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-1}(a x)+\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {(e n) \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x} \, dx,x,x^2\right )}{18 a^3}+\frac {1}{18} (a e n) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {\left (\left (3 a^2 d+e\right ) n\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{6 a^3}\\ &=-\frac {d n \sqrt {1-a^2 x^2}}{a}+\frac {e n \sqrt {1-a^2 x^2}}{9 a^3}-\frac {\left (3 a^2 d+e\right ) n \sqrt {1-a^2 x^2}}{3 a^3}+\frac {e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-1}(a x)+\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {(e n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{18 a^3}+\frac {1}{18} (a e n) \operatorname {Subst}\left (\int \left (\frac {1}{a^2 \sqrt {1-a^2 x}}-\frac {\sqrt {1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )+\frac {\left (\left (3 a^2 d+e\right ) n\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{3 a^5}\\ &=-\frac {d n \sqrt {1-a^2 x^2}}{a}-\frac {\left (3 a^2 d+e\right ) n \sqrt {1-a^2 x^2}}{3 a^3}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-1}(a x)+\frac {\left (3 a^2 d+e\right ) n \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{3 a^3}+\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )-\frac {(e n) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{9 a^5}\\ &=-\frac {d n \sqrt {1-a^2 x^2}}{a}-\frac {\left (3 a^2 d+e\right ) n \sqrt {1-a^2 x^2}}{3 a^3}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sin ^{-1}(a x)-\frac {1}{9} e n x^3 \sin ^{-1}(a x)-\frac {e n \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{9 a^3}+\frac {\left (3 a^2 d+e\right ) n \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{3 a^3}+\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 248, normalized size = 1.01 \[ \frac {-3 a^3 x \sin ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+27 a^2 d \sqrt {1-a^2 x^2} \log \left (c x^n\right )+3 a^2 e x^2 \sqrt {1-a^2 x^2} \log \left (c x^n\right )+6 e \sqrt {1-a^2 x^2} \log \left (c x^n\right )-3 n \log (x) \left (9 a^2 d+2 e\right )-54 a^2 d n \sqrt {1-a^2 x^2}+27 a^2 d n \log \left (\sqrt {1-a^2 x^2}+1\right )-2 a^2 e n x^2 \sqrt {1-a^2 x^2}-7 e n \sqrt {1-a^2 x^2}+6 e n \log \left (\sqrt {1-a^2 x^2}+1\right )}{27 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 221, normalized size = 0.90 \[ \frac {18 \, {\left (a^{3} e x^{3} + 3 \, a^{3} d x\right )} \arcsin \left (a x\right ) \log \relax (c) + 18 \, {\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \arcsin \left (a x\right ) \log \relax (x) + 3 \, {\left (9 \, a^{2} d + 2 \, e\right )} n \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 3 \, {\left (9 \, a^{2} d + 2 \, e\right )} n \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) - 6 \, {\left (a^{3} e n x^{3} + 9 \, a^{3} d n x\right )} \arcsin \left (a x\right ) - 2 \, {\left (2 \, a^{2} e n x^{2} + {\left (54 \, a^{2} d + 7 \, e\right )} n - 3 \, {\left (a^{2} e x^{2} + 9 \, a^{2} d + 2 \, e\right )} \log \relax (c) - 3 \, {\left (a^{2} e n x^{2} + {\left (9 \, a^{2} d + 2 \, e\right )} n\right )} \log \relax (x)\right )} \sqrt {-a^{2} x^{2} + 1}}{54 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.33, size = 6894, normalized size = 28.02 \[ \text {output too large to display} \]
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 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \ln \left (c\,x^n\right )\,\mathrm {asin}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \]
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 \[ \int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {asin}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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