Optimal. Leaf size=250 \[ \frac{3 b c^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac{3 b c^2 \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \cosh ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{6 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^3}-\frac{2 b c^3 x}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c^3 x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{b c}{2 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.366852, antiderivative size = 250, 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, 103, 12, 40, 39, 5754, 5721, 5461, 4182, 2279, 2391} \[ \frac{3 b c^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac{3 b c^2 \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac{a+b \cosh ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{6 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^3}-\frac{2 b c^3 x}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c^3 x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{b c}{2 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5746
Rule 103
Rule 12
Rule 40
Rule 39
Rule 5754
Rule 5721
Rule 5461
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\left (3 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{1}{x^2 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^3}\\ &=\frac{b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{3 c^2 \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)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{(b c) \int \frac{4 c^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^3}-\frac{\left (3 b c^3\right ) \int \frac{1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac{\left (3 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d}\\ &=\frac{b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b c^3 x}{4 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{3 c^2 \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)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{\left (b c^3\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac{\left (3 b c^3\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac{\left (2 b c^3\right ) \int \frac{1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^3}+\frac{\left (3 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2}\\ &=\frac{b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c^3 x}{d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 \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)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{\left (4 b c^3\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^3}\\ &=\frac{b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 \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)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}-\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 \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)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{6 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 \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)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{6 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac{\left (3 b c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}\\ &=\frac{b c}{2 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{5 b c^3 x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 c^2 \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)}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 d^3 \left (1-c^2 x^2\right )}+\frac{6 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac{3 b c^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac{3 b c^2 \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}\\ \end{align*}
Mathematica [A] time = 2.90529, size = 273, normalized size = 1.09 \[ -\frac{b c^2 \left (18 \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-18 \text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )+\frac{12 \cosh ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \cosh ^{-1}(c x)}{\left (c^2 x^2-1\right )^2}+\frac{6 \cosh ^{-1}(c x)}{c^2 x^2}+\frac{14 c x}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{c x}{\left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3}-\frac{6 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+36 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-36 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+\frac{12 a c^2}{c^2 x^2-1}-\frac{3 a c^2}{\left (c^2 x^2-1\right )^2}+18 a c^2 \log \left (1-c^2 x^2\right )-36 a c^2 \log (x)+\frac{6 a}{x^2}}{12 d^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.238, size = 641, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, a{\left (\frac{6 \, c^{4} x^{4} - 9 \, c^{2} x^{2} + 2}{c^{4} d^{3} x^{6} - 2 \, c^{2} d^{3} x^{4} + d^{3} x^{2}} + \frac{6 \, c^{2} \log \left (c x + 1\right )}{d^{3}} + \frac{6 \, c^{2} \log \left (c x - 1\right )}{d^{3}} - \frac{12 \, c^{2} \log \left (x\right )}{d^{3}}\right )} - b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{a}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx + \int \frac{b \operatorname{acosh}{\left (c x \right )}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]