Optimal. Leaf size=315 \[ -\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{c^4 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 0.743373, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {5798, 5738, 30, 5748, 5761, 4180, 2279, 2391} \[ -\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{c^4 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5738
Rule 30
Rule 5748
Rule 5761
Rule 4180
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^5} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{x^4} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}-\frac{\left (b c^3 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{x^2} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^4 \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}+\frac{\left (c^4 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}+\frac{c^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b c^4 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b c^4 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}+\frac{c^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (i b c^4 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (i b c^4 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{8 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 x^4}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 x^2}+\frac{c^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.0241, size = 290, normalized size = 0.92 \[ \frac{1}{24} \left (\frac{b \sqrt{d-c^2 d x^2} \left (-3 i c^4 x^4 \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )\right )+3 c^3 x^3+3 c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-3 i c^4 x^4 \cosh ^{-1}(c x) \left (\log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )-2 c x-6 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)\right )}{x^4 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{3 a \left (c^2 x^2-2\right ) \sqrt{d-c^2 d x^2}}{x^4}+3 a c^4 \sqrt{d} \log \left (\sqrt{d} \sqrt{d-c^2 d x^2}+d\right )-3 a c^4 \sqrt{d} \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.327, size = 541, normalized size = 1.7 \begin{align*} -{\frac{a}{4\,d{x}^{4}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{a{c}^{2}}{8\,d{x}^{2}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{a{c}^{4}}{8}\sqrt{d}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{-{c}^{2}d{x}^{2}+d} \right ) } \right ) }-{\frac{a{c}^{4}}{8}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{b{\rm arccosh} \left (cx\right ){c}^{4}}{ \left ( 8\,cx+8 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{c}^{3}}{8\,x}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{3\,b{\rm arccosh} \left (cx\right ){c}^{2}}{ \left ( 8\,cx+8 \right ){x}^{2} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bc}{12\,{x}^{3}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{b{\rm arccosh} \left (cx\right )}{ \left ( 4\,cx+4 \right ){x}^{4} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{{\frac{i}{8}}b{\rm arccosh} \left (cx\right ){c}^{4}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{{\frac{i}{8}}b{\rm arccosh} \left (cx\right ){c}^{4}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\ln \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{{\frac{i}{8}}b{c}^{4}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it dilog} \left ( 1+i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{{\frac{i}{8}}b{c}^{4}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\it dilog} \left ( 1-i \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) \right ){\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{5}}, 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{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{x^{5}}\, dx \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{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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