Integrand size = 28, antiderivative size = 378 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {4 c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {4 b^2 c^3 d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}} \]
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Time = 0.47 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5807, 5805, 5775, 3797, 2221, 2317, 2438, 5783, 5802, 283, 221} \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {c^2 d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}-\frac {b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {c^2 x^2+1}}+\frac {4 c^3 d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {c^2 x^2+1}}+\frac {8 b c^3 d \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 b^2 c^3 d \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {b^2 c^3 d \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}}-\frac {b^2 c^2 d \sqrt {c^2 d x^2+d}}{3 x} \]
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Rule 221
Rule 283
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5783
Rule 5802
Rule 5805
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}{3 x^3}+\left (c^2 d\right ) \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{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^3} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {\sqrt {1+c^2 x^2}}{x^2} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 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 (c^4 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}{\sqrt {1+c^2 x^2}} \\ & = -\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {1+c^2 x^2}}-\frac {\left (2 c^3 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 )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 c^3 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^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {4 c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {1+c^2 x^2}}+\frac {\left (4 c^3 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 )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (4 c^3 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 {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {4 c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b c^3 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 )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b c^3 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 {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {4 c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^3 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 )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^3 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 {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x}+\frac {4 c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {4 b^2 c^3 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 )}{3 \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 1.32 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {-a b c d x \sqrt {d+c^2 d x^2}-a^2 d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-4 a^2 c^2 d x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-b^2 c^2 d x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+b d \sqrt {d+c^2 d x^2} \left (3 a c^3 x^3-b \left (-4 c^3 x^3+\sqrt {1+c^2 x^2}+4 c^2 x^2 \sqrt {1+c^2 x^2}\right )\right ) \text {arcsinh}(c x)^2+b^2 c^3 d x^3 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3+b d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \left (-b c x-2 a \sqrt {1+c^2 x^2} \left (1+4 c^2 x^2\right )+8 b c^3 x^3 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )+8 a b c^3 d x^3 \sqrt {d+c^2 d x^2} \log (c x)+3 a^2 c^3 d^{3/2} x^3 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-4 b^2 c^3 d x^3 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 x^3 \sqrt {1+c^2 x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1928\) vs. \(2(350)=700\).
Time = 0.37 (sec) , antiderivative size = 1929, normalized size of antiderivative = 5.10
method | result | size |
default | \(\text {Expression too large to display}\) | \(1929\) |
parts | \(\text {Expression too large to display}\) | \(1929\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^4} \,d x \]
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