Optimal. Leaf size=222 \[ \frac{15 i b c \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac{15 i b c \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac{15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (c^2 x^2+1\right )}-\frac{5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}-\frac{15 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3}-\frac{7 b c}{8 d^3 \sqrt{c^2 x^2+1}}-\frac{b c}{12 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac{b c \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{d^3} \]
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Rubi [A] time = 0.238339, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {5747, 5690, 5693, 4180, 2279, 2391, 261, 266, 51, 63, 208} \[ \frac{15 i b c \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac{15 i b c \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac{15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (c^2 x^2+1\right )}-\frac{5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac{a+b \sinh ^{-1}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}-\frac{15 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3}-\frac{7 b c}{8 d^3 \sqrt{c^2 x^2+1}}-\frac{b c}{12 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac{b c \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{d^3} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5690
Rule 5693
Rule 4180
Rule 2279
Rule 2391
Rule 261
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left (d+c^2 d x^2\right )^3} \, dx &=-\frac{a+b \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\left (5 c^2\right ) \int \frac{a+b \sinh ^{-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 \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac{5 c^2 x \left (a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac{5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{15 c^2 x \left (a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac{5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac{(15 c) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\sinh ^{-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 \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac{5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac{15 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-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 i b c) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{8 d^3}-\frac{(15 i b c) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-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 \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac{5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac{15 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-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^{\sinh ^{-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^{\sinh ^{-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 \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac{5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac{15 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-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^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac{15 i b c \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}\\ \end{align*}
Mathematica [C] time = 1.21853, size = 298, normalized size = 1.34 \[ -\frac{\frac{2 b c \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},c^2 x^2+1\right )}{\left (c^2 x^2+1\right )^{3/2}}+\frac{15 b c \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},c^2 x^2+1\right )}{\sqrt{c^2 x^2+1}}-45 b \sqrt{-c^2} \text{PolyLog}\left (2,\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right )+45 b \sqrt{-c^2} \text{PolyLog}\left (2,\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-\frac{15 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 x^3+x}-\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{x \left (c^2 x^2+1\right )^2}+\frac{45 \left (a+b \sinh ^{-1}(c x)\right )}{x}+45 a c \tan ^{-1}(c x)+45 b c \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )+45 b \sqrt{-c^2} \sinh ^{-1}(c x) \log \left (\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right )-45 b \sqrt{-c^2} \sinh ^{-1}(c x) \log \left (\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )}{24 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.016, size = 357, normalized size = 1.6 \begin{align*} -{\frac{a}{{d}^{3}x}}-{\frac{7\,a{c}^{4}{x}^{3}}{8\,{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{9\,a{c}^{2}x}{8\,{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{15\,ca\arctan \left ( cx \right ) }{8\,{d}^{3}}}-{\frac{b{\it Arcsinh} \left ( cx \right ) }{{d}^{3}x}}-{\frac{7\,b{\it Arcsinh} \left ( cx \right ){c}^{4}{x}^{3}}{8\,{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{9\,b{\it Arcsinh} \left ( cx \right ){c}^{2}x}{8\,{d}^{3} \left ({c}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{15\,bc{\it Arcsinh} \left ( cx \right ) \arctan \left ( cx \right ) }{8\,{d}^{3}}}-{\frac{15\,bc\arctan \left ( cx \right ) }{8\,{d}^{3}}\ln \left ( 1+{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{15\,bc\arctan \left ( cx \right ) }{8\,{d}^{3}}\ln \left ( 1-{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{15\,i}{8}}cb}{{d}^{3}}{\it dilog} \left ( 1+{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{\frac{15\,i}{8}}cb}{{d}^{3}}{\it dilog} \left ( 1-{i \left ( 1+icx \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \right ) }-{\frac{15\,b{c}^{3}{x}^{2}}{8\,{d}^{3}} \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{\frac{47\,bc}{24\,{d}^{3}} \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}+{\frac{bc}{{d}^{3}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{bc}{{d}^{3}}{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \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}{8} \, a{\left (\frac{15 \, c^{4} x^{4} + 25 \, c^{2} x^{2} + 8}{c^{4} d^{3} x^{5} + 2 \, c^{2} d^{3} x^{3} + d^{3} x} + \frac{15 \, c \arctan \left (c x\right )}{d^{3}}\right )} + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{6} d^{3} x^{8} + 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} + d^{3} x^{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{b \operatorname{arsinh}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx + \int \frac{b \operatorname{asinh}{\left (c x \right )}}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx}{d^{3}} \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 \operatorname{arsinh}\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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