Optimal. Leaf size=529 \[ -\frac{35 i b c^3 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3}+\frac{35 i b c^3 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3}+\frac{19 b^2 c^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{3 d^3}-\frac{19 b^2 c^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{3 d^3}+\frac{35 i b^2 c^3 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{35 i b^2 c^3 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (c^2 x^2+1\right )}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (c^2 x^2+1\right )^2}+\frac{29 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{12 d^3 \sqrt{c^2 x^2+1}}-\frac{b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (c^2 x^2+1\right )^2}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (c^2 x^2+1\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac{35 c^3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3}+\frac{38 b c^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3}+\frac{b^2 c^4 x}{12 d^3 \left (c^2 x^2+1\right )}+\frac{b^2 c^2}{6 d^3 x \left (c^2 x^2+1\right )}-\frac{b^2 c^2}{2 d^3 x}-\frac{17 b^2 c^3 \tan ^{-1}(c x)}{6 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.30742, antiderivative size = 529, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 17, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.654, Rules used = {5747, 5690, 5693, 4180, 2531, 2282, 6589, 5717, 203, 199, 5755, 5760, 4182, 2279, 2391, 290, 325} \[ -\frac{35 i b c^3 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3}+\frac{35 i b c^3 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3}+\frac{19 b^2 c^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{3 d^3}-\frac{19 b^2 c^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{3 d^3}+\frac{35 i b^2 c^3 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{35 i b^2 c^3 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (c^2 x^2+1\right )}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (c^2 x^2+1\right )^2}+\frac{29 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{12 d^3 \sqrt{c^2 x^2+1}}-\frac{b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (c^2 x^2+1\right )^2}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (c^2 x^2+1\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac{35 c^3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3}+\frac{38 b c^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3}+\frac{b^2 c^4 x}{12 d^3 \left (c^2 x^2+1\right )}+\frac{b^2 c^2}{6 d^3 x \left (c^2 x^2+1\right )}-\frac{b^2 c^2}{2 d^3 x}-\frac{17 b^2 c^3 \tan ^{-1}(c x)}{6 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5747
Rule 5690
Rule 5693
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5717
Rule 203
Rule 199
Rule 5755
Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rule 290
Rule 325
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^4 \left (d+c^2 d x^2\right )^3} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}-\frac{1}{3} \left (7 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx+\frac{(2 b c) \int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left (1+c^2 x^2\right )^{5/2}} \, dx}{3 d^3}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac{1}{3} \left (35 c^4\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^3} \, dx+\frac{\left (b^2 c^2\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )^2} \, dx}{3 d^3}-\frac{\left (5 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^{5/2}} \, dx}{3 d^3}-\frac{\left (14 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^{5/2}} \, dx}{3 d^3}\\ &=\frac{b^2 c^2}{6 d^3 x \left (1+c^2 x^2\right )}-\frac{19 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{9 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac{\left (b^2 c^2\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d^3}-\frac{\left (5 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac{\left (14 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^3}+\frac{\left (5 b^2 c^4\right ) \int \frac{1}{\left (1+c^2 x^2\right )^2} \, dx}{9 d^3}+\frac{\left (14 b^2 c^4\right ) \int \frac{1}{\left (1+c^2 x^2\right )^2} \, dx}{9 d^3}-\frac{\left (35 b c^5\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{6 d^3}+\frac{\left (35 c^4\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac{b^2 c^2}{2 d^3 x}+\frac{b^2 c^2}{6 d^3 x \left (1+c^2 x^2\right )}+\frac{19 b^2 c^4 x}{18 d^3 \left (1+c^2 x^2\right )}-\frac{b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}-\frac{19 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}-\frac{\left (5 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{3 d^3}-\frac{\left (14 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{3 d^3}+\frac{\left (5 b^2 c^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{18 d^3}-\frac{\left (b^2 c^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d^3}+\frac{\left (7 b^2 c^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{9 d^3}+\frac{\left (5 b^2 c^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{3 d^3}-\frac{\left (35 b^2 c^4\right ) \int \frac{1}{\left (1+c^2 x^2\right )^2} \, dx}{18 d^3}+\frac{\left (14 b^2 c^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{3 d^3}-\frac{\left (35 b c^5\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{4 d^3}+\frac{\left (35 c^4\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac{b^2 c^2}{2 d^3 x}+\frac{b^2 c^2}{6 d^3 x \left (1+c^2 x^2\right )}+\frac{b^2 c^4 x}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac{29 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{12 d^3 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{62 b^2 c^3 \tan ^{-1}(c x)}{9 d^3}+\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{8 d^3}-\frac{\left (5 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^3}-\frac{\left (14 b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^3}-\frac{\left (35 b^2 c^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{36 d^3}-\frac{\left (35 b^2 c^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 d^3}\\ &=-\frac{b^2 c^2}{2 d^3 x}+\frac{b^2 c^2}{6 d^3 x \left (1+c^2 x^2\right )}+\frac{b^2 c^4 x}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac{29 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{12 d^3 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{35 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{17 b^2 c^3 \tan ^{-1}(c x)}{6 d^3}+\frac{38 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^3}-\frac{\left (35 i b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 d^3}+\frac{\left (35 i b c^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 d^3}+\frac{\left (5 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^3}-\frac{\left (5 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^3}+\frac{\left (14 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^3}-\frac{\left (14 b^2 c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^3}\\ &=-\frac{b^2 c^2}{2 d^3 x}+\frac{b^2 c^2}{6 d^3 x \left (1+c^2 x^2\right )}+\frac{b^2 c^4 x}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac{29 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{12 d^3 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{35 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{17 b^2 c^3 \tan ^{-1}(c x)}{6 d^3}+\frac{38 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^3}-\frac{35 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}+\frac{35 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}+\frac{\left (35 i b^2 c^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 d^3}-\frac{\left (35 i b^2 c^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{4 d^3}+\frac{\left (5 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 d^3}-\frac{\left (5 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 d^3}+\frac{\left (14 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 d^3}-\frac{\left (14 b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 d^3}\\ &=-\frac{b^2 c^2}{2 d^3 x}+\frac{b^2 c^2}{6 d^3 x \left (1+c^2 x^2\right )}+\frac{b^2 c^4 x}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac{29 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{12 d^3 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{35 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{17 b^2 c^3 \tan ^{-1}(c x)}{6 d^3}+\frac{38 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^3}+\frac{19 b^2 c^3 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{3 d^3}-\frac{35 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}+\frac{35 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{19 b^2 c^3 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^3}+\frac{\left (35 i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{\left (35 i b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{4 d^3}\\ &=-\frac{b^2 c^2}{2 d^3 x}+\frac{b^2 c^2}{6 d^3 x \left (1+c^2 x^2\right )}+\frac{b^2 c^4 x}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac{29 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{12 d^3 \sqrt{1+c^2 x^2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac{7 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac{35 c^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{17 b^2 c^3 \tan ^{-1}(c x)}{6 d^3}+\frac{38 b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^3}+\frac{19 b^2 c^3 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{3 d^3}-\frac{35 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}+\frac{35 i b c^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{19 b^2 c^3 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^3}+\frac{35 i b^2 c^3 \text{Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac{35 i b^2 c^3 \text{Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{4 d^3}\\ \end{align*}
Mathematica [A] time = 10.1588, size = 937, normalized size = 1.77 \[ \frac{11 a^2 x c^4}{8 d^3 \left (c^2 x^2+1\right )}+\frac{a^2 x c^4}{4 d^3 \left (c^2 x^2+1\right )^2}+\frac{35 a^2 \tan ^{-1}(c x) c^3}{8 d^3}+\frac{b^2 \left (-\frac{1}{2} c x \sinh ^{-1}(c x)^2 \text{csch}^4\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\frac{8 \sinh ^{-1}(c x)^2 \sinh ^4\left (\frac{1}{2} \sinh ^{-1}(c x)\right )}{c^3 x^3}+\frac{33 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}+\frac{6 c x \sinh ^{-1}(c x)^2}{\left (c^2 x^2+1\right )^2}-2 \sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )+\frac{66 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{4 \sinh ^{-1}(c x)}{\left (c^2 x^2+1\right )^{3/2}}-136 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )+38 \sinh ^{-1}(c x)^2 \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-4 \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-152 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-105 i \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+105 i \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+152 \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )-152 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-210 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+210 i \sinh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+152 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-210 i \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )+210 i \text{PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )-38 \sinh ^{-1}(c x)^2 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )+4 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\frac{2 c x}{c^2 x^2+1}\right ) c^3}{24 d^3}+\frac{3 a^2 c^2}{d^3 x}+\frac{2 a b \left (\frac{11 \left (\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right ) c^4}{16 \left (x c^2+i c\right )}-\frac{35}{16} i \left (-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \log \left (1+i e^{\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)}{c}+\frac{2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c}\right ) c^4+\frac{35}{16} i \left (-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \log \left (1-i e^{\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)}{c}+\frac{2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c}\right ) c^4+\frac{i \left ((2 i-c x) \sqrt{c^2 x^2+1}-3 \sinh ^{-1}(c x)\right ) c^3}{48 (c x-i)^2}-\frac{11 \left (i \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1}\right ) c^3}{16 (-i c x-1)}+\frac{i \left (\sqrt{c^2 x^2+1} (c x+2 i)+3 \sinh ^{-1}(c x)\right ) c^3}{48 (c x+i)^2}+\frac{1}{6} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right ) c^3-3 \left (-\frac{\sinh ^{-1}(c x)}{x}-c \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )\right ) c^2-\frac{\sqrt{c^2 x^2+1} c}{6 x^2}-\frac{\sinh ^{-1}(c x)}{3 x^3}\right )}{d^3}-\frac{a^2}{3 d^3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.392, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{{x}^{4} \left ({c}^{2}d{x}^{2}+d \right ) ^{3}}}\, dx \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}{24} \, a^{2}{\left (\frac{105 \, c^{3} \arctan \left (c x\right )}{d^{3}} + \frac{105 \, c^{6} x^{6} + 175 \, c^{4} x^{4} + 56 \, c^{2} x^{2} - 8}{c^{4} d^{3} x^{7} + 2 \, c^{2} d^{3} x^{5} + d^{3} x^{3}}\right )} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{6} d^{3} x^{10} + 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} + d^{3} x^{4}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{6} d^{3} x^{10} + 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} + d^{3} x^{4}}\,{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^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{c^{6} d^{3} x^{10} + 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} + d^{3} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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