Integrand size = 28, antiderivative size = 299 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=-\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {d+c^2 d x^2}}-\frac {2 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x}-\frac {4 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {d+c^2 d x^2}} \]
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Time = 0.32 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {5809, 5800, 5775, 3797, 2221, 2317, 2438, 5776, 270} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\frac {2 c^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x}-\frac {b c \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 d x^3}-\frac {2 c^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {c^2 d x^2+d}}-\frac {4 b c^3 \sqrt {c^2 x^2+1} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 d x^2+d}}+\frac {2 b^2 c^3 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2 \left (c^2 x^2+1\right )}{3 x \sqrt {c^2 d x^2+d}} \]
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Rule 270
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5776
Rule 5800
Rule 5809
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}-\frac {1}{3} \left (2 c^2\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x^3} \, dx}{3 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x}+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x} \, dx}{3 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x}+\frac {\left (4 c^3 \sqrt {1+c^2 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 {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {d+c^2 d x^2}}-\frac {2 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x}-\frac {\left (8 c^3 \sqrt {1+c^2 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 {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {d+c^2 d x^2}}-\frac {2 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x}-\frac {4 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1+c^2 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 {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {d+c^2 d x^2}}-\frac {2 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x}-\frac {4 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {1+c^2 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 {d+c^2 d x^2}} \\ & = -\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2 \sqrt {d+c^2 d x^2}}-\frac {2 c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 d x}-\frac {4 b c^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{3 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 c^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right )}{3 \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\frac {-a^2+a^2 c^2 x^2-b^2 c^2 x^2+2 a^2 c^4 x^4-b^2 c^4 x^4-a b c x \sqrt {1+c^2 x^2}+b^2 \left (-1+c^2 x^2+2 c^4 x^4-2 c^3 x^3 \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2-b \text {arcsinh}(c x) \left (b c x \sqrt {1+c^2 x^2}-2 a \left (-1+c^2 x^2+2 c^4 x^4\right )+4 b c^3 x^3 \sqrt {1+c^2 x^2} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )-4 a b c^3 x^3 \sqrt {1+c^2 x^2} \log (c x)+2 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{3 x^3 \sqrt {d+c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1505\) vs. \(2(281)=562\).
Time = 0.31 (sec) , antiderivative size = 1506, normalized size of antiderivative = 5.04
method | result | size |
default | \(\text {Expression too large to display}\) | \(1506\) |
parts | \(\text {Expression too large to display}\) | \(1506\) |
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{4}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4} \sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{4}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,\sqrt {d\,c^2\,x^2+d}} \,d x \]
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