Integrand size = 28, antiderivative size = 398 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]
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Time = 0.34 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5807, 5785, 5783, 5776, 327, 221, 5801, 5775, 3797, 2221, 2317, 2438, 201} \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {3}{2} c^2 d x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {c^2 x^2+1}}+\frac {c d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {2 b c d \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {3 b c^3 d x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 \sqrt {c^2 x^2+1}}-\frac {b^2 c d \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {5 b^2 c d \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} b^2 c^2 d x \sqrt {c^2 d x^2+d} \]
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Rule 201
Rule 221
Rule 327
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5776
Rule 5783
Rule 5785
Rule 5801
Rule 5807
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\left (3 c^2 d\right ) \int \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx+\frac {\left (2 b c d \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}} \\ & = b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {\left (2 b c d \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 (3 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\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 \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 x (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {1}{2} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}-\frac {\left (2 c d \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 (b^2 c^2 d \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 (3 b^2 c^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{2 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {\left (4 c d \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 (3 b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \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 \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 {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \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 \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 {1}{4} b^2 c^2 d x \sqrt {d+c^2 d x^2}-\frac {5 b^2 c d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{4 \sqrt {1+c^2 x^2}}-\frac {3 b c^3 d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{2} c^2 d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {c d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{2 b \sqrt {1+c^2 x^2}}+\frac {2 b c d \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 \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 = 3.50 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {12 a^2 d \left (-2+c^2 x^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+24 a b d \sqrt {d+c^2 d x^2} \left (-2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+c x \text {arcsinh}(c x)^2+2 c x \log (c x)\right )+36 a^2 c d^{3/2} x \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-8 b^2 d \sqrt {d+c^2 d x^2} \left (\text {arcsinh}(c x) \left (3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-c x \text {arcsinh}(c x) (3+\text {arcsinh}(c x))-6 c x \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )+3 c x \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )\right )+b^2 c d x \sqrt {d+c^2 d x^2} \left (4 \text {arcsinh}(c x)^3-6 \text {arcsinh}(c x) \cosh (2 \text {arcsinh}(c x))+\left (3+6 \text {arcsinh}(c x)^2\right ) \sinh (2 \text {arcsinh}(c x))\right )-6 a b c d x \sqrt {d+c^2 d x^2} (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))}{24 x \sqrt {1+c^2 x^2}} \]
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Time = 0.31 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.19
method | result | size |
default | \(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}+\frac {3 \sqrt {c^{2} d \,x^{2}+d}\, a^{2} c^{2} d x}{2}+\frac {3 a^{2} c^{2} d^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+2 \operatorname {arcsinh}\left (c x \right )^{3} x c +c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2} x c +8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x c +8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x c -4 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2}-\operatorname {arcsinh}\left (c x \right ) c x +8 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x c +8 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x c \right ) d}{4 \sqrt {c^{2} x^{2}+1}\, x}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right ) d}{4 \sqrt {c^{2} x^{2}+1}\, x}\) | \(473\) |
parts | \(-\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+a^{2} c^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}+\frac {3 \sqrt {c^{2} d \,x^{2}+d}\, a^{2} c^{2} d x}{2}+\frac {3 a^{2} c^{2} d^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+2 \operatorname {arcsinh}\left (c x \right )^{3} x c +c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2} x c +8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x c +8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x c -4 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )^{2}-\operatorname {arcsinh}\left (c x \right ) c x +8 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x c +8 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x c \right ) d}{4 \sqrt {c^{2} x^{2}+1}\, x}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3}+6 \operatorname {arcsinh}\left (c x \right )^{2} x c -8 \,\operatorname {arcsinh}\left (c x \right ) c x +8 \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) x c -8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-c x \right ) d}{4 \sqrt {c^{2} x^{2}+1}\, x}\) | \(473\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^2} \,d x \]
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