Optimal. Leaf size=541 \[ -\frac {3 b c^2 d \sqrt {c^2 d x^2+d} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {3 b c^2 d \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {3}{2} c^2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}-\frac {3 a b c^3 d x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\frac {b c^3 d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {3 b^2 c^2 d \sqrt {c^2 d x^2+d} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {3 b^2 c^2 d \sqrt {c^2 d x^2+d} \text {Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+2 b^2 c^2 d \sqrt {c^2 d x^2+d}-\frac {b^2 c^2 d \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{\sqrt {c^2 x^2+1}}-\frac {3 b^2 c^3 d x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.65, antiderivative size = 541, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {5739, 5742, 5760, 4182, 2531, 2282, 6589, 5653, 261, 14, 5730, 446, 80, 63, 208} \[ -\frac {3 b c^2 d \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {3 b c^2 d \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {3 b^2 c^2 d \sqrt {c^2 d x^2+d} \text {PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {3 b^2 c^2 d \sqrt {c^2 d x^2+d} \text {PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {3 a b c^3 d x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\frac {b c^3 d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {3}{2} c^2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+2 b^2 c^2 d \sqrt {c^2 d x^2+d}-\frac {3 b^2 c^3 d x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}-\frac {b^2 c^2 d \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{\sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 63
Rule 80
Rule 208
Rule 261
Rule 446
Rule 2282
Rule 2531
Rule 4182
Rule 5653
Rule 5730
Rule 5739
Rule 5742
Rule 5760
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{2} \left (3 c^2 d\right ) \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx+\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (3 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {-1+c^2 x^2}{x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {3 a b c^3 d x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (3 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {-1+c^2 x}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 c^3 d \sqrt {d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-b^2 c^2 d \sqrt {d+c^2 d x^2}-\frac {3 a b c^3 d x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {3 b^2 c^3 d x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (3 b c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt {d+c^2 d x^2}-\frac {3 a b c^3 d x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {3 b^2 c^3 d x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 b c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {3 b c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 d \sqrt {d+c^2 d 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 )}{\sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt {d+c^2 d x^2}-\frac {3 a b c^3 d x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {3 b^2 c^3 d x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 b c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {3 b c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ &=2 b^2 c^2 d \sqrt {d+c^2 d x^2}-\frac {3 a b c^3 d x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {3 b^2 c^3 d x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 b c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {3 b c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {3 b^2 c^2 d \sqrt {d+c^2 d x^2} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {3 b^2 c^2 d \sqrt {d+c^2 d x^2} \text {Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 7.88, size = 771, normalized size = 1.43 \[ -\frac {3}{2} a^2 c^2 d^{3/2} \log \left (\sqrt {d} \sqrt {d \left (c^2 x^2+1\right )}+d\right )+\frac {3}{2} a^2 c^2 d^{3/2} \log (x)+\sqrt {d \left (c^2 x^2+1\right )} \left (a^2 c^2 d-\frac {a^2 d}{2 x^2}\right )+\frac {2 a b c^2 d \sqrt {d \left (c^2 x^2+1\right )} \left (\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+\text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {a b c^2 d \sqrt {d \left (c^2 x^2+1\right )} \left (4 \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-4 \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-2 \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{4 \sqrt {c^2 x^2+1}}+b^2 c^2 d \sqrt {d \left (c^2 x^2+1\right )} \left (\frac {2 \sinh ^{-1}(c x) \left (\text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {2 \left (\text {Li}_3\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_3\left (e^{-\sinh ^{-1}(c x)}\right )\right )}{\sqrt {c^2 x^2+1}}-\frac {2 c x \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+\frac {\sinh ^{-1}(c x)^2 \left (\log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt {c^2 x^2+1}}+\sinh ^{-1}(c x)^2+2\right )+\frac {b^2 c^2 d \sqrt {d \left (c^2 x^2+1\right )} \left (8 \sinh ^{-1}(c x) \text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-8 \sinh ^{-1}(c x) \text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+8 \text {Li}_3\left (-e^{-\sinh ^{-1}(c x)}\right )-8 \text {Li}_3\left (e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x)^2 \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x)^2 \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+4 \sinh ^{-1}(c x) \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-4 \sinh ^{-1}(c x) \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )+\sinh ^{-1}(c x)^2 \left (-\text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-\sinh ^{-1}(c x)^2 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )+8 \log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )}{8 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{2} d x^{2} + a^{2} d + {\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} + a b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 1131, normalized size = 2.09 \[ \text {result too large to display} \]
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 \[ -\frac {1}{2} \, {\left (3 \, c^{2} d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} - 3 \, \sqrt {c^{2} d x^{2} + d} c^{2} d + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{2}}\right )} a^{2} + \int \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{x^{3}} + \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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