Optimal. Leaf size=635 \[ -\frac {2 b d^2 \sqrt {c^2 d x^2+d} \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {2 b d^2 \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}-\frac {2 a b c d^2 x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}-\frac {16 b c d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {c^2 x^2+1}}+d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+\frac {1}{5} \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 b c^5 d^2 x^5 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {c^2 x^2+1}}-\frac {22 b c^3 d^2 x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt {c^2 x^2+1}}+\frac {2 b^2 d^2 \sqrt {c^2 d x^2+d} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {2 b^2 d^2 \sqrt {c^2 d x^2+d} \text {Li}_3\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {598}{225} b^2 d^2 \sqrt {c^2 d x^2+d}+\frac {2}{125} b^2 d^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}+\frac {74}{675} b^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}-\frac {2 b^2 c d^2 x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}} \]
[Out]
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Rubi [A] time = 0.91, antiderivative size = 635, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 16, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5744, 5742, 5760, 4182, 2531, 2282, 6589, 5653, 261, 5679, 444, 43, 194, 12, 1247, 698} \[ -\frac {2 b d^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {2 b d^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 x^2+1}}+\frac {2 b^2 d^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {2 b^2 d^2 \sqrt {c^2 d x^2+d} \text {PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {2 a b c d^2 x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}-\frac {2 b c^5 d^2 x^5 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {c^2 x^2+1}}-\frac {22 b c^3 d^2 x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt {c^2 x^2+1}}-\frac {16 b c d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {c^2 x^2+1}}+d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 x^2+1}}+\frac {1}{5} \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {598}{225} b^2 d^2 \sqrt {c^2 d x^2+d}+\frac {2}{125} b^2 d^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}+\frac {74}{675} b^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}-\frac {2 b^2 c d^2 x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 194
Rule 261
Rule 444
Rule 698
Rule 1247
Rule 2282
Rule 2531
Rule 4182
Rule 5653
Rule 5679
Rule 5742
Rule 5744
Rule 5760
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx &=\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+d \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{5 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b c d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \sqrt {1+c^2 x^2}}-\frac {4 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {1+c^2 x^2}}+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1+c^2 x^2}} \, dx}{5 \sqrt {1+c^2 x^2}}\\ &=-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (d^2 \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}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b c d^2 \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 {\left (2 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx}{75 \sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (1+\frac {c^2 x^2}{3}\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 a b c d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {\left (d^2 \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 )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c d^2 \sqrt {d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {15+10 c^2 x+3 c^4 x^2}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{75 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {c^2 x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 a b c d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 d^2 \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 (2 b d^2 \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 (2 b d^2 \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^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {1+c^2 x}}+4 \sqrt {1+c^2 x}+3 \left (1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{3 \sqrt {1+c^2 x}}+\frac {1}{3} \sqrt {1+c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {598}{225} b^2 d^2 \sqrt {d+c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {74}{675} b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {2}{125} b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 d^2 \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 {2 b d^2 \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 {2 b d^2 \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 (2 b^2 d^2 \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 (2 b^2 d^2 \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 {598}{225} b^2 d^2 \sqrt {d+c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {74}{675} b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {2}{125} b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 d^2 \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 {2 b d^2 \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 {2 b d^2 \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 (2 b^2 d^2 \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 (2 b^2 d^2 \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 {598}{225} b^2 d^2 \sqrt {d+c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {74}{675} b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {2}{125} b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 d^2 \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 {2 b d^2 \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 {2 b d^2 \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 {2 b^2 d^2 \sqrt {d+c^2 d x^2} \text {Li}_3\left (-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 d^2 \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 = 4.45, size = 710, normalized size = 1.12 \[ \frac {d^2 \left (54000 a^2 \sqrt {d} \sqrt {c^2 x^2+1} \log (c x)-54000 a^2 \sqrt {d} \sqrt {c^2 x^2+1} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+d\right )+3600 a^2 \sqrt {c^2 x^2+1} \left (3 c^4 x^4+11 c^2 x^2+23\right ) \sqrt {c^2 d x^2+d}-108000 a b \sqrt {c^2 d x^2+d} \left (-\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)-\text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )+\text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )+c x-\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )+\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )-480 a b \sqrt {c^2 d x^2+d} \left (c x \left (9 c^4 x^4+5 c^2 x^2-30\right )-15 \sqrt {c^2 x^2+1} \left (3 c^4 x^4+c^2 x^2-2\right ) \sinh ^{-1}(c x)\right )-24000 a b \sqrt {c^2 d x^2+d} \left (c^3 x^3-3 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+3 c x\right )+54000 b^2 \sqrt {c^2 d x^2+d} \left (2 \sqrt {c^2 x^2+1}+\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \left (\text {Li}_2\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(c x)}\right )\right )+2 \left (\text {Li}_3\left (-e^{-\sinh ^{-1}(c x)}\right )-\text {Li}_3\left (e^{-\sinh ^{-1}(c x)}\right )\right )-2 c x \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2 \left (\log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )\right )+1000 b^2 \sqrt {c^2 d x^2+d} \left (27 \sqrt {c^2 x^2+1} \left (\sinh ^{-1}(c x)^2+2\right )-6 \sinh ^{-1}(c x) \left (9 c x+\sinh \left (3 \sinh ^{-1}(c x)\right )\right )+\left (9 \sinh ^{-1}(c x)^2+2\right ) \cosh \left (3 \sinh ^{-1}(c x)\right )\right )-b^2 \sqrt {c^2 d x^2+d} \left (6750 \sqrt {c^2 x^2+1} \left (\sinh ^{-1}(c x)^2+2\right )+480 c x \left (9 c^4 x^4+5 c^2 x^2-30\right ) \sinh ^{-1}(c x)+125 \left (9 \sinh ^{-1}(c x)^2+2\right ) \cosh \left (3 \sinh ^{-1}(c x)\right )-27 \left (25 \sinh ^{-1}(c x)^2+2\right ) \cosh \left (5 \sinh ^{-1}(c x)\right )\right )\right )}{54000 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{4} d^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} + 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{x}, x\right ) \]
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 = 0.46, size = 1321, normalized size = 2.08 \[ \text {result too large to display} \]
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 \[ -\frac {1}{15} \, {\left (15 \, d^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) - 3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} - 5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d - 15 \, \sqrt {c^{2} d x^{2} + d} d^{2}\right )} a^{2} + \int \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{x} + \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x} \,d x \]
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 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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