Optimal. Leaf size=687 \[ \frac {40}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {2}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \text {PolyLog}\left (3,-e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \text {PolyLog}\left (3,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.63, antiderivative size = 687, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 20, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5807, 5808,
5806, 5816, 4267, 2611, 2320, 6724, 5772, 267, 5784, 455, 45, 276, 5803, 12, 1265, 911, 1167, 214}
\begin {gather*} -\frac {5 b c^2 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 {5 b c^2 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 {5}{2} c^2 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {c^2 x^2+1}}-\frac {5 c^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 {5}{6} c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {2 b c^5 d^2 x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {c^2 x^2+1}}-\frac {5 a b c^3 d^2 x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\frac {b c^3 d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}+\frac {5 b^2 c^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 {5 b^2 c^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 {40}{9} b^2 c^2 d^2 \sqrt {c^2 d x^2+d}+\frac {2}{27} b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}-\frac {b^2 c^2 d^2 \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{\sqrt {c^2 x^2+1}}-\frac {5 b^2 c^3 d^2 x \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 214
Rule 267
Rule 276
Rule 455
Rule 911
Rule 1167
Rule 1265
Rule 2320
Rule 2611
Rule 4267
Rule 5772
Rule 5784
Rule 5803
Rule 5806
Rule 5807
Rule 5808
Rule 5816
Rule 6724
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^3} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{2} \left (5 c^2 d\right ) \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 (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x \sqrt {1+c^2 x^2}}+\frac {2 b c^3 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{2} \left (5 c^2 d^2\right ) \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {-3+6 c^2 x^2+c^4 x^4}{3 x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 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 {b c d^2 \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^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (5 c^2 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}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {-3+6 c^2 x^2+c^4 x^4}{x \sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 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 (5 b^2 c^4 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 {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \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^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {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^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {-3+6 c^2 x+c^4 x^2}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {c^2 x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{6 \sqrt {1+c^2 x^2}}\\ &=-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \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^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {5 c^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 (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {-8+4 x^2+x^4}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {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 (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {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 (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \text {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 )}{6 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^4 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 {55}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {5}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \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^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {5 c^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 {5 b c^2 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 {5 b c^2 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 (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (5 c^2+c^2 x^2-\frac {3}{-\frac {1}{c^2}+\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {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 (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=\frac {40}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {2}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \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^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {5 c^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 {5 b c^2 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 {5 b c^2 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 (b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {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 (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {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 (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ &=\frac {40}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {2}{27} b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \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^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 x^2}-\frac {5 c^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 {b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 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 {5 b c^2 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 {5 b^2 c^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 {5 b^2 c^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 = 7.35, size = 990, normalized size = 1.44 \begin {gather*} \sqrt {d \left (1+c^2 x^2\right )} \left (\frac {7}{3} a^2 c^2 d^2-\frac {a^2 d^2}{2 x^2}+\frac {1}{3} a^2 c^4 d^2 x^2\right )+2 a b c^2 d^2 \left (-\frac {c x \sqrt {d \left (1+c^2 x^2\right )} \left (3+c^2 x^2\right )}{9 \sqrt {1+c^2 x^2}}+\frac {1}{3} \left (1+c^2 x^2\right ) \sqrt {d \left (1+c^2 x^2\right )} \sinh ^{-1}(c x)\right )+\frac {5}{2} a^2 c^2 d^{5/2} \log (x)-\frac {5}{2} a^2 c^2 d^{5/2} \log \left (d+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right )+\frac {4 a b c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (-c x+\sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )+\text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+2 b^2 c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (2-\frac {2 c x \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\sinh ^{-1}(c x)^2+\frac {\sinh ^{-1}(c x)^2 \left (\log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\log \left (1+e^{-\sinh ^{-1}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {2 \sinh ^{-1}(c x) \left (\text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {2 \left (\text {PolyLog}\left (3,-e^{-\sinh ^{-1}(c x)}\right )-\text {PolyLog}\left (3,e^{-\sinh ^{-1}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}\right )+\frac {b^2 c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (27 \sqrt {1+c^2 x^2} \left (2+\sinh ^{-1}(c x)^2\right )+\left (2+9 \sinh ^{-1}(c x)^2\right ) \cosh \left (3 \sinh ^{-1}(c x)\right )-6 \sinh ^{-1}(c x) \left (9 c x+\sinh \left (3 \sinh ^{-1}(c x)\right )\right )\right )}{108 \sqrt {1+c^2 x^2}}+\frac {a b c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (-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 )+4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (1+e^{-\sinh ^{-1}(c x)}\right )+4 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-4 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )+2 \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{4 \sqrt {1+c^2 x^2}}+\frac {b^2 c^2 d^2 \sqrt {d \left (1+c^2 x^2\right )} \left (-4 \sinh ^{-1}(c x) \coth \left (\frac {1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x)^2 \text {csch}^2\left (\frac {1}{2} \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 (1+e^{-\sinh ^{-1}(c x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )+8 \sinh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-8 \sinh ^{-1}(c x) \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+8 \text {PolyLog}\left (3,-e^{-\sinh ^{-1}(c x)}\right )-8 \text {PolyLog}\left (3,e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x)^2 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c x)\right )+4 \sinh ^{-1}(c x) \tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )}{8 \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1403\) vs.
\(2(664)=1328\).
time = 3.79, size = 1404, normalized size = 2.04
method | result | size |
default | \(\text {Expression too large to display}\) | \(1404\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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