Integrand size = 22, antiderivative size = 135 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {5}{32} b^2 d x^2+\frac {1}{32} b^2 c^2 d x^4-\frac {3 b d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {3 d (a+b \text {arcsinh}(c x))^2}{32 c^2}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{4 c^2} \]
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Time = 0.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5798, 5786, 5785, 5783, 30, 14} \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {b d x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {3 b d x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{16 c}+\frac {d \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{4 c^2}-\frac {3 d (a+b \text {arcsinh}(c x))^2}{32 c^2}+\frac {1}{32} b^2 c^2 d x^4+\frac {5}{32} b^2 d x^2 \]
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Rule 14
Rule 30
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rubi steps \begin{align*} \text {integral}& = \frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{4 c^2}-\frac {(b d) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{2 c} \\ & = -\frac {b d x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{4 c^2}+\frac {1}{8} \left (b^2 d\right ) \int x \left (1+c^2 x^2\right ) \, dx-\frac {(3 b d) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{8 c} \\ & = -\frac {3 b d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{4 c^2}+\frac {1}{8} \left (b^2 d\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{16} \left (3 b^2 d\right ) \int x \, dx-\frac {(3 b d) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c} \\ & = \frac {5}{32} b^2 d x^2+\frac {1}{32} b^2 c^2 d x^4-\frac {3 b d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c}-\frac {b d x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {3 d (a+b \text {arcsinh}(c x))^2}{32 c^2}+\frac {d \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{4 c^2} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.15 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \left (c x \left (8 a^2 c x \left (2+c^2 x^2\right )+b^2 c x \left (5+c^2 x^2\right )-2 a b \sqrt {1+c^2 x^2} \left (5+2 c^2 x^2\right )\right )+2 b \left (-b c x \sqrt {1+c^2 x^2} \left (5+2 c^2 x^2\right )+a \left (5+16 c^2 x^2+8 c^4 x^4\right )\right ) \text {arcsinh}(c x)+b^2 \left (5+16 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)^2\right )}{32 c^2} \]
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Time = 0.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {\frac {d \,a^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}+d \,b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}-\frac {\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}}{16}-\frac {3 \operatorname {arcsinh}\left (c x \right )^{2}}{32}+\frac {\left (c^{2} x^{2}+1\right )^{2}}{32}+\frac {3 c^{2} x^{2}}{32}+\frac {3}{32}\right )+2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right )}{32}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{32}\right )}{c^{2}}\) | \(182\) |
default | \(\frac {\frac {d \,a^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}+d \,b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}-\frac {\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}}{16}-\frac {3 \operatorname {arcsinh}\left (c x \right )^{2}}{32}+\frac {\left (c^{2} x^{2}+1\right )^{2}}{32}+\frac {3 c^{2} x^{2}}{32}+\frac {3}{32}\right )+2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right )}{32}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{32}\right )}{c^{2}}\) | \(182\) |
parts | \(\frac {d \,a^{2} \left (c^{2} x^{2}+1\right )^{2}}{4 c^{2}}+\frac {d \,b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}-\frac {\operatorname {arcsinh}\left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}}{16}-\frac {3 \operatorname {arcsinh}\left (c x \right )^{2}}{32}+\frac {\left (c^{2} x^{2}+1\right )^{2}}{32}+\frac {3 c^{2} x^{2}}{32}+\frac {3}{32}\right )}{c^{2}}+\frac {2 d a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {5 \,\operatorname {arcsinh}\left (c x \right )}{32}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{16}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{32}\right )}{c^{2}}\) | \(187\) |
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Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.51 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (8 \, a^{2} + b^{2}\right )} c^{4} d x^{4} + {\left (16 \, a^{2} + 5 \, b^{2}\right )} c^{2} d x^{2} + {\left (8 \, b^{2} c^{4} d x^{4} + 16 \, b^{2} c^{2} d x^{2} + 5 \, b^{2} d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (8 \, a b c^{4} d x^{4} + 16 \, a b c^{2} d x^{2} + 5 \, a b d - {\left (2 \, b^{2} c^{3} d x^{3} + 5 \, b^{2} c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (2 \, a b c^{3} d x^{3} + 5 \, a b c d x\right )} \sqrt {c^{2} x^{2} + 1}}{32 \, c^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (129) = 258\).
Time = 0.43 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.99 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} d x^{4}}{4} + \frac {a^{2} d x^{2}}{2} + \frac {a b c^{2} d x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {a b c d x^{3} \sqrt {c^{2} x^{2} + 1}}{8} + a b d x^{2} \operatorname {asinh}{\left (c x \right )} - \frac {5 a b d x \sqrt {c^{2} x^{2} + 1}}{16 c} + \frac {5 a b d \operatorname {asinh}{\left (c x \right )}}{16 c^{2}} + \frac {b^{2} c^{2} d x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {b^{2} c^{2} d x^{4}}{32} - \frac {b^{2} c d x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {b^{2} d x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {5 b^{2} d x^{2}}{32} - \frac {5 b^{2} d x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c} + \frac {5 b^{2} d \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{2}}{2} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (119) = 238\).
Time = 0.22 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.57 \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{4} \, b^{2} c^{2} d x^{4} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} c^{2} d x^{4} + \frac {1}{2} \, b^{2} d x^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b c^{2} d + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{2} d + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} a b d + \frac {1}{4} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} d \]
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Exception generated. \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right ) \,d x \]
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