Optimal. Leaf size=242 \[ \frac{15 i b c \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}-\frac{15 i b c \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}+\frac{15 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{5 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}-\frac{15 i c \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 d^3}-\frac{7 b c}{8 d^3 \sqrt{1-c^2 x^2}}-\frac{b c}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^3} \]
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Rubi [A] time = 0.24238, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {4701, 4655, 4657, 4181, 2279, 2391, 261, 266, 51, 63, 208} \[ \frac{15 i b c \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}-\frac{15 i b c \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}+\frac{15 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{5 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}-\frac{15 i c \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 d^3}-\frac{7 b c}{8 d^3 \sqrt{1-c^2 x^2}}-\frac{b c}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^3} \]
Antiderivative was successfully verified.
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Rule 4701
Rule 4655
Rule 4657
Rule 4181
Rule 2279
Rule 2391
Rule 261
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\left (5 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{1}{x \left (1-c^2 x^2\right )^{5/2}} \, dx}{d^3}\\ &=-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{2 d^3}-\frac{\left (5 b c^3\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac{\left (15 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac{b c}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^3}-\frac{\left (15 b c^3\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{8 d^3}+\frac{\left (15 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac{b c}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{7 b c}{8 d^3 \sqrt{1-c^2 x^2}}-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{(15 c) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 d^3}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d^3}\\ &=-\frac{b c}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{7 b c}{8 d^3 \sqrt{1-c^2 x^2}}-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac{15 i c \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{c d^3}-\frac{(15 b c) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 d^3}+\frac{(15 b c) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 d^3}\\ &=-\frac{b c}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{7 b c}{8 d^3 \sqrt{1-c^2 x^2}}-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac{15 i c \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^3}+\frac{(15 i b c) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 d^3}-\frac{(15 i b c) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 d^3}\\ &=-\frac{b c}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac{7 b c}{8 d^3 \sqrt{1-c^2 x^2}}-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac{15 i c \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 d^3}-\frac{b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d^3}+\frac{15 i b c \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}-\frac{15 i b c \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{8 d^3}\\ \end{align*}
Mathematica [B] time = 1.51178, size = 512, normalized size = 2.12 \[ -\frac{-30 i b c \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+30 i b c \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+\frac{14 a c^2 x}{c^2 x^2-1}-\frac{4 a c^2 x}{\left (c^2 x^2-1\right )^2}+15 a c \log (1-c x)-15 a c \log (c x+1)+\frac{16 a}{x}-\frac{b c^2 x \sqrt{1-c^2 x^2}}{3 (c x-1)^2}+\frac{b c^2 x \sqrt{1-c^2 x^2}}{3 (c x+1)^2}-\frac{7 b c \sqrt{1-c^2 x^2}}{c x-1}+\frac{7 b c \sqrt{1-c^2 x^2}}{c x+1}+\frac{2 b c \sqrt{1-c^2 x^2}}{3 (c x-1)^2}+\frac{2 b c \sqrt{1-c^2 x^2}}{3 (c x+1)^2}+16 b c \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{7 b c \sin ^{-1}(c x)}{c x-1}+\frac{7 b c \sin ^{-1}(c x)}{c x+1}-\frac{b c \sin ^{-1}(c x)}{(c x-1)^2}+\frac{b c \sin ^{-1}(c x)}{(c x+1)^2}+15 i \pi b c \sin ^{-1}(c x)+\frac{16 b \sin ^{-1}(c x)}{x}-30 b c \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-15 \pi b c \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+30 b c \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-15 \pi b c \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+15 \pi b c \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+15 \pi b c \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )}{16 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.223, size = 461, normalized size = 1.9 \begin{align*}{\frac{ca}{16\,{d}^{3} \left ( cx-1 \right ) ^{2}}}-{\frac{7\,ca}{16\,{d}^{3} \left ( cx-1 \right ) }}-{\frac{15\,ca\ln \left ( cx-1 \right ) }{16\,{d}^{3}}}-{\frac{ca}{16\,{d}^{3} \left ( cx+1 \right ) ^{2}}}-{\frac{7\,ca}{16\,{d}^{3} \left ( cx+1 \right ) }}+{\frac{15\,ca\ln \left ( cx+1 \right ) }{16\,{d}^{3}}}-{\frac{a}{{d}^{3}x}}-{\frac{15\,b\arcsin \left ( cx \right ){c}^{4}{x}^{3}}{8\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{7\,b{c}^{3}{x}^{2}}{8\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{25\,b\arcsin \left ( cx \right ){c}^{2}x}{8\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{23\,bc}{24\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b\arcsin \left ( cx \right ) }{{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) x}}+{\frac{bc}{{d}^{3}}\ln \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1}-1 \right ) }-{\frac{bc}{{d}^{3}}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{15\,bc\arcsin \left ( cx \right ) }{8\,{d}^{3}}\ln \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{15\,bc\arcsin \left ( cx \right ) }{8\,{d}^{3}}\ln \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{{\frac{15\,i}{8}}cb}{{d}^{3}}{\it dilog} \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{{\frac{15\,i}{8}}cb}{{d}^{3}}{\it dilog} \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) } \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*} -\frac{1}{16} \, a{\left (\frac{2 \,{\left (15 \, c^{4} x^{4} - 25 \, c^{2} x^{2} + 8\right )}}{c^{4} d^{3} x^{5} - 2 \, c^{2} d^{3} x^{3} + d^{3} x} - \frac{15 \, c \log \left (c x + 1\right )}{d^{3}} + \frac{15 \, c \log \left (c x - 1\right )}{d^{3}}\right )} + \frac{{\left (15 \,{\left (c^{5} x^{5} - 2 \, c^{3} x^{3} + c x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (c x + 1\right ) - 15 \,{\left (c^{5} x^{5} - 2 \, c^{3} x^{3} + c x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) \log \left (-c x + 1\right ) - 2 \,{\left (15 \, c^{4} x^{4} - 25 \, c^{2} x^{2} + 8\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (c^{4} d^{3} x^{5} - 2 \, c^{2} d^{3} x^{3} + d^{3} x\right )} \int \frac{{\left (30 \, c^{5} x^{4} - 50 \, c^{3} x^{2} - 15 \,{\left (c^{6} x^{5} - 2 \, c^{4} x^{3} + c^{2} x\right )} \log \left (c x + 1\right ) + 15 \,{\left (c^{6} x^{5} - 2 \, c^{4} x^{3} + c^{2} x\right )} \log \left (-c x + 1\right ) + 16 \, c\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{6} d^{3} x^{7} - 3 \, c^{4} d^{3} x^{5} + 3 \, c^{2} d^{3} x^{3} - d^{3} x}\,{d x}\right )} b}{16 \,{\left (c^{4} d^{3} x^{5} - 2 \, c^{2} d^{3} x^{3} + d^{3} x\right )}} \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{b \arcsin \left (c x\right ) + a}{c^{6} d^{3} x^{8} - 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} - d^{3} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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