Integrand size = 28, antiderivative size = 530 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {31}{64} b^2 c^2 d^2 x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 c^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 \sqrt {1+c^2 x^2}}-\frac {15 b c^3 d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+b c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{8 b \sqrt {1+c^2 x^2}}+\frac {2 b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]
[Out]
Time = 0.49 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {5807, 5786, 5785, 5783, 5776, 327, 221, 5798, 201, 5801, 5775, 3797, 2221, 2317, 2438} \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {15}{8} c^2 d^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2+\frac {5 c d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{8 b \sqrt {c^2 x^2+1}}+\frac {c d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {1}{8} b c d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+b c d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {2 b c d^2 \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}+\frac {5}{4} c^2 d x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}-\frac {15 b c^3 d^2 x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{8 \sqrt {c^2 x^2+1}}-\frac {b^2 c d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {89 b^2 c d^2 \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{64 \sqrt {c^2 x^2+1}}+\frac {31}{64} b^2 c^2 d^2 x \sqrt {c^2 d x^2+d}+\frac {1}{32} b^2 c^2 d^2 x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \]
[In]
[Out]
Rule 201
Rule 221
Rule 327
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rule 5801
Rule 5807
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\left (5 c^2 d\right ) \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx+\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{x} \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{2} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {1}{4} \left (15 c^2 d^2\right ) \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx+\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x)) \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {1}{8} b^2 c^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+b c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (15 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{8 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (15 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{4 \sqrt {1+c^2 x^2}} \\ & = -\frac {11}{16} b^2 c^2 d^2 x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 c^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {15 b c^3 d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+b c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{8 b \sqrt {1+c^2 x^2}}-\frac {\left (2 c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (15 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{32 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}+\frac {\left (15 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}} \\ & = \frac {31}{64} b^2 c^2 d^2 x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 c^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {11 b^2 c d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{16 \sqrt {1+c^2 x^2}}-\frac {15 b c^3 d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+b c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{8 b \sqrt {1+c^2 x^2}}+\frac {\left (4 c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (15 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (15 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}} \\ & = \frac {31}{64} b^2 c^2 d^2 x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 c^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 \sqrt {1+c^2 x^2}}-\frac {15 b c^3 d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+b c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{8 b \sqrt {1+c^2 x^2}}+\frac {2 b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}} \\ & = \frac {31}{64} b^2 c^2 d^2 x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 c^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 \sqrt {1+c^2 x^2}}-\frac {15 b c^3 d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+b c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{8 b \sqrt {1+c^2 x^2}}+\frac {2 b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right )}{\sqrt {1+c^2 x^2}} \\ & = \frac {31}{64} b^2 c^2 d^2 x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 c^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}-\frac {89 b^2 c d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 \sqrt {1+c^2 x^2}}-\frac {15 b c^3 d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+b c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{8} c^2 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}+\frac {5}{4} c^2 d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {5 c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{8 b \sqrt {1+c^2 x^2}}+\frac {2 b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right )}{\sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 2.79 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.04 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {d^2 \left (-256 a^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+288 a^2 c^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+64 a^2 c^4 x^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+160 b^2 c x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3-128 a b c x \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-4 a b c x \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))+512 a b c x \sqrt {d+c^2 d x^2} \log (c x)+480 a^2 c \sqrt {d} x \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-256 b^2 c x \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )+64 b^2 c x \sqrt {d+c^2 d x^2} \sinh (2 \text {arcsinh}(c x))+b^2 c x \sqrt {d+c^2 d x^2} \sinh (4 \text {arcsinh}(c x))-4 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \left (128 a \sqrt {1+c^2 x^2}+32 b c x \cosh (2 \text {arcsinh}(c x))+b c x \cosh (4 \text {arcsinh}(c x))-128 b c x \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-64 a c x \sinh (2 \text {arcsinh}(c x))-4 a c x \sinh (4 \text {arcsinh}(c x))\right )+8 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2 \left (60 a c x+32 b c x-32 b \sqrt {1+c^2 x^2}+16 b c x \sinh (2 \text {arcsinh}(c x))+b c x \sinh (4 \text {arcsinh}(c x))\right )\right )}{256 x \sqrt {1+c^2 x^2}} \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.11
method | result | size |
default | \(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}+\frac {5 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2} c^{2} d x}{4}+\frac {15 a^{2} d^{2} \sqrt {c^{2} d \,x^{2}+d}\, c^{2} x}{8}+\frac {15 a^{2} c^{2} d^{3} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}+2 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+72 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-72 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+40 \operatorname {arcsinh}\left (c x \right )^{3} x c +33 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-64 \operatorname {arcsinh}\left (c x \right )^{2} x c +128 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x c +128 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x c -64 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2}-33 \,\operatorname {arcsinh}\left (c x \right ) c x +128 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x c +128 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x c \right ) d^{2}}{64 \sqrt {c^{2} x^{2}+1}\, x}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-8 c^{5} x^{5}+144 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-72 c^{3} x^{3}+120 \operatorname {arcsinh}\left (c x \right )^{2} x c -128 \,\operatorname {arcsinh}\left (c x \right ) c x +128 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-33 c x \right ) d^{2}}{64 \sqrt {c^{2} x^{2}+1}\, x}\) | \(589\) |
parts | \(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}+\frac {5 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a^{2} c^{2} d x}{4}+\frac {15 a^{2} d^{2} \sqrt {c^{2} d \,x^{2}+d}\, c^{2} x}{8}+\frac {15 a^{2} c^{2} d^{3} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}+2 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+72 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-72 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+40 \operatorname {arcsinh}\left (c x \right )^{3} x c +33 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-64 \operatorname {arcsinh}\left (c x \right )^{2} x c +128 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x c +128 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x c -64 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2}-33 \,\operatorname {arcsinh}\left (c x \right ) c x +128 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x c +128 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x c \right ) d^{2}}{64 \sqrt {c^{2} x^{2}+1}\, x}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-8 c^{5} x^{5}+144 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-72 c^{3} x^{3}+120 \operatorname {arcsinh}\left (c x \right )^{2} x c -128 \,\operatorname {arcsinh}\left (c x \right ) c x +128 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-33 c x \right ) d^{2}}{64 \sqrt {c^{2} x^{2}+1}\, x}\) | \(589\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^2} \,d x \]
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