Optimal. Leaf size=259 \[ \frac{5 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{5 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}-\frac{b c^3}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{b c}{6 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{13 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d^2} \]
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Rubi [A] time = 0.307726, antiderivative size = 285, normalized size of antiderivative = 1.1, number of steps used = 19, 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} \[ \frac{5 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{5 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{2 d^2}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{d^2}-\frac{5 b c^3}{6 d^2 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^2 x^2}+\frac{b c}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{13 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d^2} \]
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^4 \left (d-c^2 d x^2\right )^2} \, dx &=-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}+\frac{1}{3} \left (5 c^2\right ) \int \frac{a+b \sin ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx+\frac{(b c) \int \frac{1}{x^3 \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\left (5 c^4\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 d^2}+\frac{\left (5 b c^3\right ) \int \frac{1}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac{b c}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d^2}+\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 d^2}-\frac{\left (5 b c^5\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac{\left (5 c^4\right ) \int \frac{a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac{5 b c^3}{6 d^2 \sqrt{1-c^2 x^2}}+\frac{b c}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^2 x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac{\left (5 c^3\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{4 d^2}+\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{6 d^2}\\ &=-\frac{5 b c^3}{6 d^2 \sqrt{1-c^2 x^2}}+\frac{b c}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^2 x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{2 d^2}-\frac{(5 b c) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{3 d^2}-\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}+\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 d^2}\\ &=-\frac{5 b c^3}{6 d^2 \sqrt{1-c^2 x^2}}+\frac{b c}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^2 x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{13 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d^2}+\frac{\left (5 i b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{\left (5 i b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac{5 b c^3}{6 d^2 \sqrt{1-c^2 x^2}}+\frac{b c}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^2 x^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left (1-c^2 x^2\right )}-\frac{5 c^2 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x \left (1-c^2 x^2\right )}+\frac{5 c^4 x \left (a+b \sin ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac{5 i c^3 \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2}-\frac{13 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 d^2}+\frac{5 i b c^3 \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{2 d^2}-\frac{5 i b c^3 \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{2 d^2}\\ \end{align*}
Mathematica [A] time = 0.929428, size = 426, normalized size = 1.64 \[ -\frac{-30 i b c^3 \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+30 i b c^3 \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+\frac{6 a c^4 x}{c^2 x^2-1}+\frac{24 a c^2}{x}+15 a c^3 \log (1-c x)-15 a c^3 \log (c x+1)+\frac{4 a}{x^3}-\frac{3 b c^3 \sqrt{1-c^2 x^2}}{c x-1}+\frac{3 b c^3 \sqrt{1-c^2 x^2}}{c x+1}+\frac{2 b c \sqrt{1-c^2 x^2}}{x^2}+26 b c^3 \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{3 b c^3 \sin ^{-1}(c x)}{c x-1}+\frac{3 b c^3 \sin ^{-1}(c x)}{c x+1}+15 i \pi b c^3 \sin ^{-1}(c x)+\frac{24 b c^2 \sin ^{-1}(c x)}{x}-30 b c^3 \sin ^{-1}(c x) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )-15 \pi b c^3 \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+30 b c^3 \sin ^{-1}(c x) \log \left (1+i e^{i \sin ^{-1}(c x)}\right )-15 \pi b c^3 \log \left (1+i e^{i \sin ^{-1}(c x)}\right )+15 \pi b c^3 \log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+15 \pi b c^3 \log \left (-\cos \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )+\frac{4 b \sin ^{-1}(c x)}{x^3}}{12 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.23, size = 426, normalized size = 1.6 \begin{align*} -{\frac{{c}^{3}a}{4\,{d}^{2} \left ( cx-1 \right ) }}-{\frac{5\,{c}^{3}a\ln \left ( cx-1 \right ) }{4\,{d}^{2}}}-{\frac{{c}^{3}a}{4\,{d}^{2} \left ( cx+1 \right ) }}+{\frac{5\,{c}^{3}a\ln \left ( cx+1 \right ) }{4\,{d}^{2}}}-{\frac{a}{3\,{d}^{2}{x}^{3}}}-2\,{\frac{{c}^{2}a}{{d}^{2}x}}-{\frac{5\,{c}^{4}b\arcsin \left ( cx \right ) x}{2\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{c}^{3}}{3\,{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{5\,{c}^{2}b\arcsin \left ( cx \right ) }{3\,{d}^{2}x \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{bc}{6\,{d}^{2}{x}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arcsin \left ( cx \right ) }{3\,{d}^{2}{x}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{13\,b{c}^{3}}{6\,{d}^{2}}\ln \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1}-1 \right ) }-{\frac{13\,b{c}^{3}}{6\,{d}^{2}}\ln \left ( 1+icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{5\,b{c}^{3}\arcsin \left ( cx \right ) }{2\,{d}^{2}}\ln \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{{\frac{5\,i}{2}}{c}^{3}b}{{d}^{2}}{\it dilog} \left ( 1+i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }-{\frac{{\frac{5\,i}{2}}{c}^{3}b}{{d}^{2}}{\it dilog} \left ( 1-i \left ( icx+\sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+{\frac{5\,b{c}^{3}\arcsin \left ( cx \right ) }{2\,{d}^{2}}\ln \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}{12} \,{\left (\frac{15 \, c^{3} \log \left (c x + 1\right )}{d^{2}} - \frac{15 \, c^{3} \log \left (c x - 1\right )}{d^{2}} - \frac{2 \,{\left (15 \, c^{4} x^{4} - 10 \, c^{2} x^{2} - 2\right )}}{c^{2} d^{2} x^{5} - d^{2} x^{3}}\right )} a + \frac{{\left (15 \,{\left (c^{5} x^{5} - c^{3} x^{3}\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} - c^{3} x^{3}\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} - 10 \, c^{2} x^{2} - 2\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) -{\left (c^{2} d^{2} x^{5} - d^{2} x^{3}\right )} \int \frac{{\left (30 \, c^{5} x^{4} - 20 \, c^{3} x^{2} - 15 \,{\left (c^{6} x^{5} - c^{4} x^{3}\right )} \log \left (c x + 1\right ) + 15 \,{\left (c^{6} x^{5} - c^{4} x^{3}\right )} \log \left (-c x + 1\right ) - 4 \, c\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{4} d^{2} x^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} x^{3}}\,{d x}\right )} b}{12 \,{\left (c^{2} d^{2} x^{5} - d^{2} x^{3}\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^{4} d^{2} x^{8} - 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, 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*} \frac{\int \frac{a}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx + \int \frac{b \operatorname{asin}{\left (c x \right )}}{c^{4} x^{8} - 2 c^{2} x^{6} + x^{4}}\, dx}{d^{2}} \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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