Optimal. Leaf size=246 \[ \frac{\sqrt{1-a^2 x^2} \left (3 a^2 d+e\right ) \log \left (c x^n\right )}{3 a^3}-\frac{e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}-\frac{n \sqrt{1-a^2 x^2} \left (3 a^2 d+e\right )}{3 a^3}+\frac{n \left (3 a^2 d+e\right ) \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{3 a^3}-\frac{d n \sqrt{1-a^2 x^2}}{a}+\frac{2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-\frac{e n \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{9 a^3}+d x \sin ^{-1}(a x) \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x) \log \left (c x^n\right )-d n x \sin ^{-1}(a x)-\frac{1}{9} e n x^3 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.233246, antiderivative size = 246, normalized size of antiderivative = 1., 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 4665
Rule 444
Rule 43
Rule 2387
Rule 266
Rule 50
Rule 63
Rule 208
Rule 4619
Rule 261
Rule 4627
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*}
Mathematica [A] time = 0.166231, 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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Maple [C] time = 2.663, size = 10458, normalized size = 42.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80221, size = 537, normalized size = 2.18 \begin{align*} \frac{18 \,{\left (a^{3} e x^{3} + 3 \, a^{3} d x\right )} \arcsin \left (a x\right ) \log \left (c\right ) + 18 \,{\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \arcsin \left (a x\right ) \log \left (x\right ) + 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 \left (c\right ) - 3 \,{\left (a^{2} e n x^{2} +{\left (9 \, a^{2} d + 2 \, e\right )} n\right )} \log \left (x\right )\right )} \sqrt{-a^{2} x^{2} + 1}}{54 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x^{2}\right ) \log{\left (c x^{n} \right )} \operatorname{asin}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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