Optimal. Leaf size=687 \[ \frac {5 b c^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5 b c^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {c^2 d x^2+d}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {26 b c^2 \sqrt {c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {5 c^2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {13 i b^2 c^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {13 i b^2 c^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {5 b^2 c^2 \sqrt {c^2 x^2+1} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {c^2 d x^2+d}}+\frac {5 b^2 c^2 \sqrt {c^2 x^2+1} \text {Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2 c^2}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{d^2 \sqrt {c^2 d x^2+d}} \]
[Out]
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Rubi [A] time = 1.26, antiderivative size = 687, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 18, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5747, 5755, 5764, 5760, 4182, 2531, 2282, 6589, 5693, 4180, 2279, 2391, 5690, 261, 266, 51, 63, 208} \[ \frac {5 b c^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5 b c^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {13 i b^2 c^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {13 i b^2 c^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {5 b^2 c^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {c^2 d x^2+d}}+\frac {5 b^2 c^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {c^2 d x^2+d}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {26 b c^2 \sqrt {c^2 x^2+1} \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {5 c^2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^2 \sqrt {c^2 d x^2+d}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (c^2 d x^2+d\right )^{3/2}}+\frac {b^2 c^2}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{d^2 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 208
Rule 261
Rule 266
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4180
Rule 4182
Rule 5690
Rule 5693
Rule 5747
Rule 5755
Rule 5760
Rule 5764
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^3 \left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {1}{2} \left (5 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (5 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx}{2 d}+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt {d+c^2 d x^2}} \, dx}{2 d^2}+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 b^2 c^4 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 c^4 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt {1+c^2 x^2}} \, dx}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 b c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {26 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {26 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 b c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (3 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 i b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {26 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {5 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {26 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {5 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (5 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (5 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {2 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{6 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2 \left (d+c^2 d x^2\right )^{3/2}}-\frac {5 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^2 \sqrt {d+c^2 d x^2}}+\frac {26 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 c^2 \sqrt {1+c^2 x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {13 i b^2 c^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {5 b c^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^2 \sqrt {1+c^2 x^2} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b^2 c^2 \sqrt {1+c^2 x^2} \text {Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 8.02, size = 983, normalized size = 1.43 \[ -\frac {5 a^2 \log (x) c^2}{2 d^{5/2}}+\frac {5 a^2 \log \left (d+\sqrt {d \left (c^2 x^2+1\right )} \sqrt {d}\right ) c^2}{2 d^{5/2}}+\frac {a b \left (-3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\frac {8 \sinh ^{-1}(c x)}{c^2 x^2+1}-48 \sinh ^{-1}(c x)+104 \sqrt {c^2 x^2+1} \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-6 \sqrt {c^2 x^2+1} \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-60 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+60 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )-60 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )+60 \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+6 \sqrt {c^2 x^2+1} \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+\frac {4 c x}{\sqrt {c^2 x^2+1}}\right ) c^2}{12 d^2 \sqrt {d \left (c^2 x^2+1\right )}}+\frac {b^2 \left (-3 \sqrt {c^2 x^2+1} \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)^2-3 \sqrt {c^2 x^2+1} \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)^2-60 \sqrt {c^2 x^2+1} \log \left (1-e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)^2+60 \sqrt {c^2 x^2+1} \log \left (1+e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)^2-\frac {8 \sinh ^{-1}(c x)^2}{c^2 x^2+1}-48 \sinh ^{-1}(c x)^2-12 \sqrt {c^2 x^2+1} \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)-104 i \sqrt {c^2 x^2+1} \log \left (1-i e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+104 i \sqrt {c^2 x^2+1} \log \left (1+i e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)-120 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+120 \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right ) \sinh ^{-1}(c x)+12 \sqrt {c^2 x^2+1} \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\frac {8 c x \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+24 \sqrt {c^2 x^2+1} \log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-104 i \sqrt {c^2 x^2+1} \text {Li}_2\left (-i e^{-\sinh ^{-1}(c x)}\right )+104 i \sqrt {c^2 x^2+1} \text {Li}_2\left (i e^{-\sinh ^{-1}(c x)}\right )-120 \sqrt {c^2 x^2+1} \text {Li}_3\left (-e^{-\sinh ^{-1}(c x)}\right )+120 \sqrt {c^2 x^2+1} \text {Li}_3\left (e^{-\sinh ^{-1}(c x)}\right )+8\right ) c^2}{24 d^2 \sqrt {d \left (c^2 x^2+1\right )}}+\sqrt {d \left (c^2 x^2+1\right )} \left (-\frac {2 c^2 a^2}{d^3 \left (c^2 x^2+1\right )}-\frac {a^2}{2 d^3 x^2}-\frac {c^2 a^2}{3 d^3 \left (c^2 x^2+1\right )^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{9} + 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} + d^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{2}}{x^{3} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, a^{2} {\left (\frac {15 \, c^{2} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right )}{d^{\frac {5}{2}}} - \frac {15 \, c^{2}}{\sqrt {c^{2} d x^{2} + d} d^{2}} - \frac {5 \, c^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {3}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{2}}\right )} + \int \frac {b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} + \frac {2 \, a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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