Optimal. Leaf size=344 \[ -\frac{i b (e f-d g) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2}-\frac{i b (e f-d g) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^2}+\frac{(e f-d g) \log (d+e x) \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{b \sin ^{-1}(c x) (e f-d g) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2}+\frac{b \sin ^{-1}(c x) (e f-d g) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^2}+\frac{b g \sqrt{1-c^2 x^2}}{c e}-\frac{i b \sin ^{-1}(c x)^2 (e f-d g)}{2 e^2}-\frac{b \sin ^{-1}(c x) (e f-d g) \log (d+e x)}{e^2} \]
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Rubi [A] time = 0.643494, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {43, 4753, 12, 6742, 261, 216, 2404, 4741, 4519, 2190, 2279, 2391} \[ -\frac{i b (e f-d g) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2}-\frac{i b (e f-d g) \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^2}+\frac{(e f-d g) \log (d+e x) \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{b \sin ^{-1}(c x) (e f-d g) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2}+\frac{b \sin ^{-1}(c x) (e f-d g) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e^2}+\frac{b g \sqrt{1-c^2 x^2}}{c e}-\frac{i b \sin ^{-1}(c x)^2 (e f-d g)}{2 e^2}-\frac{b \sin ^{-1}(c x) (e f-d g) \log (d+e x)}{e^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 4753
Rule 12
Rule 6742
Rule 261
Rule 216
Rule 2404
Rule 4741
Rule 4519
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (a+b \sin ^{-1}(c x)\right )}{d+e x} \, dx &=\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-(b c) \int \frac{e g x+(e f-d g) \log (d+e x)}{e^2 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac{(b c) \int \frac{e g x+(e f-d g) \log (d+e x)}{\sqrt{1-c^2 x^2}} \, dx}{e^2}\\ &=\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac{(b c) \int \left (\frac{e g x}{\sqrt{1-c^2 x^2}}+\frac{(e f-d g) \log (d+e x)}{\sqrt{1-c^2 x^2}}\right ) \, dx}{e^2}\\ &=\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac{(b c g) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{e}-\frac{(b c (e f-d g)) \int \frac{\log (d+e x)}{\sqrt{1-c^2 x^2}} \, dx}{e^2}\\ &=\frac{b g \sqrt{1-c^2 x^2}}{c e}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac{b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac{(b c (e f-d g)) \int \frac{\sin ^{-1}(c x)}{c d+c e x} \, dx}{e}\\ &=\frac{b g \sqrt{1-c^2 x^2}}{c e}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac{b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac{(b c (e f-d g)) \operatorname{Subst}\left (\int \frac{x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=\frac{b g \sqrt{1-c^2 x^2}}{c e}-\frac{i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac{b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac{(b c (e f-d g)) \operatorname{Subst}\left (\int \frac{e^{i x} x}{c^2 d-c \sqrt{c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e}+\frac{(b c (e f-d g)) \operatorname{Subst}\left (\int \frac{e^{i x} x}{c^2 d+c \sqrt{c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=\frac{b g \sqrt{1-c^2 x^2}}{c e}-\frac{i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2}+\frac{b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^2}-\frac{b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac{(b (e f-d g)) \operatorname{Subst}\left (\int \log \left (1-\frac{i c e e^{i x}}{c^2 d-c \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^2}-\frac{(b (e f-d g)) \operatorname{Subst}\left (\int \log \left (1-\frac{i c e e^{i x}}{c^2 d+c \sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^2}\\ &=\frac{b g \sqrt{1-c^2 x^2}}{c e}-\frac{i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2}+\frac{b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^2}-\frac{b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac{(i b (e f-d g)) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i c e x}{c^2 d-c \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^2}+\frac{(i b (e f-d g)) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i c e x}{c^2 d+c \sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^2}\\ &=\frac{b g \sqrt{1-c^2 x^2}}{c e}-\frac{i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac{g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac{b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2}+\frac{b (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^2}-\frac{b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac{(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac{i b (e f-d g) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e^2}-\frac{i b (e f-d g) \text{Li}_2\left (\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.549131, size = 282, normalized size = 0.82 \[ \frac{-\frac{1}{2} i b (e f-d g) \left (2 \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )+2 \text{PolyLog}\left (2,\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \left (\log \left (1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )+\log \left (1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )\right )\right )\right )+(e f-d g) \log (d+e x) \left (a+b \sin ^{-1}(c x)\right )+e g x \left (a+b \sin ^{-1}(c x)\right )+\frac{b e g \sqrt{1-c^2 x^2}}{c}-b \sin ^{-1}(c x) (e f-d g) \log (d+e x)}{e^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.367, size = 1578, normalized size = 4.6 \begin{align*} \text{result 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*} a g{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + \frac{a f \log \left (e x + d\right )}{e} + \int \frac{{\left (b g x + b f\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a g x + a f +{\left (b g x + b f\right )} \arcsin \left (c x\right )}{e x + d}, x\right ) \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 \frac{\left (a + b \operatorname{asin}{\left (c x \right )}\right ) \left (f + g x\right )}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}{\left (b \arcsin \left (c x\right ) + a\right )}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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