Optimal. Leaf size=230 \[ \frac{15 b c \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{15 b c \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac{15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{15 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3}-\frac{7 b c}{8 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d^3} \]
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Rubi [A] time = 0.244466, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {5746, 104, 21, 92, 205, 5689, 74, 5694, 4182, 2279, 2391} \[ \frac{15 b c \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{15 b c \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac{15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{15 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3}-\frac{7 b c}{8 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d^3} \]
Antiderivative was successfully verified.
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Rule 5746
Rule 104
Rule 21
Rule 92
Rule 205
Rule 5689
Rule 74
Rule 5694
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\left (5 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{1}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^3}\\ &=-\frac{b c}{3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{b \int \frac{3 c+3 c^2 x}{x (-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{3 d^3}-\frac{\left (5 b c^3\right ) \int \frac{x}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac{\left (15 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=\frac{b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac{(b c) \int \frac{1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^3}+\frac{\left (15 b c^3\right ) \int \frac{x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 d^3}+\frac{\left (15 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=\frac{b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{7 b c}{8 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{b \int \frac{c+c^2 x}{x \sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{d^3}-\frac{(15 c) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}\\ &=\frac{b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{7 b c}{8 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{15 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac{(b c) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{d^3}+\frac{(15 b c) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}-\frac{(15 b c) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}\\ &=\frac{b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{7 b c}{8 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{15 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac{(15 b c) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{(15 b c) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac{\left (b c^2\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^3}\\ &=\frac{b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{7 b c}{8 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac{5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac{15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{b c \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^3}+\frac{15 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac{15 b c \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{15 b c \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 d^3}\\ \end{align*}
Mathematica [A] time = 1.85133, size = 362, normalized size = 1.57 \[ \frac{-45 b c \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )-4 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )\right )+45 b c \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (1-e^{\cosh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )\right )-\frac{84 a c^2 x}{c^2 x^2-1}+\frac{24 a c^2 x}{\left (c^2 x^2-1\right )^2}-90 a c \log (1-c x)+90 a c \log (c x+1)-\frac{96 a}{x}+\frac{96 b c \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c \left ((c x-2) \sqrt{c x-1} \sqrt{c x+1}-3 \cosh ^{-1}(c x)\right )}{(c x-1)^2}+\frac{2 b c \left (\sqrt{c x-1} \sqrt{c x+1} (c x+2)-3 \cosh ^{-1}(c x)\right )}{(c x+1)^2}+42 b c \left (\frac{\cosh ^{-1}(c x)}{1-c x}-\frac{1}{\sqrt{\frac{c x-1}{c x+1}}}\right )+42 b c \left (\sqrt{\frac{c x-1}{c x+1}}-\frac{\cosh ^{-1}(c x)}{c x+1}\right )-\frac{96 b \cosh ^{-1}(c x)}{x}}{96 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.18, size = 392, normalized size = 1.7 \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{a}{{d}^{3}x}}-{\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{15\,b{\rm arccosh} \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{cx-1}\sqrt{cx+1}}+{\frac{25\,b{\rm arccosh} \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{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )}{{d}^{3}x \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+2\,{\frac{bc\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{{d}^{3}}}+{\frac{15\,bc}{8\,{d}^{3}}{\it dilog} \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{15\,bc}{8\,{d}^{3}}{\it dilog} \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{15\,bc{\rm arccosh} \left (cx\right )}{8\,{d}^{3}}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+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*} \text{result too large to display} \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{arcosh}\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{acosh}{\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{arcosh}\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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