Integrand size = 23, antiderivative size = 214 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {298}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{15 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \]
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Time = 0.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5786, 5772, 5798, 8, 200} \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 b d^2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}-\frac {8 b d^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {16 b d^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{15 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {2}{125} b^2 c^4 d^2 x^5+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {298}{225} b^2 d^2 x \]
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Rule 8
Rule 200
Rule 5772
Rule 5786
Rule 5798
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} (4 d) \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{5} \left (2 b c d^2\right ) \int x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{15} \left (8 d^2\right ) \int (a+b \text {arcsinh}(c x))^2 \, dx+\frac {1}{25} \left (2 b^2 d^2\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac {1}{15} \left (8 b c d^2\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{25} \left (2 b^2 d^2\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{45} \left (8 b^2 d^2\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{15} \left (16 b c d^2\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = \frac {58}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{15 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{15} \left (16 b^2 d^2\right ) \int 1 \, dx \\ & = \frac {298}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{15 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.89 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (225 a^2 c x \left (15+10 c^2 x^2+3 c^4 x^4\right )-30 a b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (2235+190 c^2 x^2+27 c^4 x^4\right )-30 b \left (-15 a c x \left (15+10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )\right ) \text {arcsinh}(c x)+225 b^2 c x \left (15+10 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)^2\right )}{3375 c} \]
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Time = 0.20 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\) | \(264\) |
default | \(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\) | \(264\) |
parts | \(d^{2} a^{2} \left (\frac {1}{5} c^{4} x^{5}+\frac {2}{3} x^{3} c^{2}+x \right )+\frac {d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )}{c}+\frac {2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\) | \(264\) |
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Time = 0.29 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.30 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} x^{5} + 10 \, {\left (225 \, a^{2} + 38 \, b^{2}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} + 298 \, b^{2}\right )} c d^{2} x + 225 \, {\left (3 \, b^{2} c^{5} d^{2} x^{5} + 10 \, b^{2} c^{3} d^{2} x^{3} + 15 \, b^{2} c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{5} d^{2} x^{5} + 150 \, a b c^{3} d^{2} x^{3} + 225 \, a b c d^{2} x - {\left (9 \, b^{2} c^{4} d^{2} x^{4} + 38 \, b^{2} c^{2} d^{2} x^{2} + 149 \, b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 30 \, {\left (9 \, a b c^{4} d^{2} x^{4} + 38 \, a b c^{2} d^{2} x^{2} + 149 \, a b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{3375 \, c} \]
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Time = 0.46 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.82 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{5}}{5} + \frac {2 a^{2} c^{2} d^{2} x^{3}}{3} + a^{2} d^{2} x + \frac {2 a b c^{4} d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 a b c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {4 a b c^{2} d^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {76 a b c d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{225} + 2 a b d^{2} x \operatorname {asinh}{\left (c x \right )} - \frac {298 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} + \frac {b^{2} c^{4} d^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} c^{4} d^{2} x^{5}}{125} - \frac {2 b^{2} c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25} + \frac {2 b^{2} c^{2} d^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {76 b^{2} c^{2} d^{2} x^{3}}{675} - \frac {76 b^{2} c d^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225} + b^{2} d^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + \frac {298 b^{2} d^{2} x}{225} - \frac {298 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225 c} & \text {for}\: c \neq 0 \\a^{2} d^{2} x & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (190) = 380\).
Time = 0.23 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.14 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{5} \, b^{2} c^{4} d^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} c^{4} d^{2} x^{5} + \frac {2}{3} \, b^{2} c^{2} d^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{4} d^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{2} + \frac {2}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d^{2} - \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{2} + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]
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Exception generated. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]
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