Optimal. Leaf size=507 \[ -\frac{b c^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{b c^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]
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Rubi [A] time = 0.709962, antiderivative size = 507, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {4777, 4773, 3324, 3323, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac{b c^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{b c^2 f \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]
Antiderivative was successfully verified.
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Rule 4777
Rule 4773
Rule 3324
Rule 3323
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{(f+g x)^2 \sqrt{d-c^2 d x^2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{a+b \sin ^{-1}(c x)}{(f+g x)^2 \sqrt{1-c^2 x^2}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=\frac{\left (c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{a+b x}{(c f+g \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}+\frac{\left (c^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{a+b x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (b c g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c f+x} \, dx,x,c g x\right )}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}+\frac{\left (2 c^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (2 i c^2 f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{2 c f-2 i e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\left (2 i c^2 f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{2 c f-2 i e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}+\frac{\left (i b c^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{\left (i b c^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}+\frac{\left (b c^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{\left (b c^2 f \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ &=\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt{d-c^2 d x^2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{b c^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b c^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 0.457289, size = 295, normalized size = 0.58 \[ \frac{c \sqrt{1-c^2 x^2} \left (\frac{c f \left (-b \text{PolyLog}\left (2,-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right )+b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right )-\log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )\right )\right )}{\sqrt{c^2 f^2-g^2}}+\frac{g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c f+c g x}-b \log (f+g x)\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.408, size = 1678, normalized size = 3.3 \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*} \int \frac{b \arcsin \left (c x\right ) + a}{\sqrt{-c^{2} d x^{2} + d}{\left (g x + f\right )}^{2}}\,{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{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{c^{2} d g^{2} x^{4} + 2 \, c^{2} d f g x^{3} - 2 \, d f g x - d f^{2} +{\left (c^{2} d f^{2} - d g^{2}\right )} x^{2}}, 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{a + b \operatorname{asin}{\left (c x \right )}}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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