Integrand size = 21, antiderivative size = 170 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {16 b d^3 \sqrt {1+c^2 x^2}}{35 c}-\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{105 c}-\frac {6 b d^3 \left (1+c^2 x^2\right )^{5/2}}{175 c}-\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{49 c}+d^3 x (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x)) \]
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Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {200, 5784, 12, 1813, 1864} \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+d^3 x (a+b \text {arcsinh}(c x))-\frac {b d^3 \left (c^2 x^2+1\right )^{7/2}}{49 c}-\frac {6 b d^3 \left (c^2 x^2+1\right )^{5/2}}{175 c}-\frac {8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{105 c}-\frac {16 b d^3 \sqrt {c^2 x^2+1}}{35 c} \]
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Rule 12
Rule 200
Rule 1813
Rule 1864
Rule 5784
Rubi steps \begin{align*} \text {integral}& = d^3 x (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))-(b c) \int \frac {d^3 x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )}{35 \sqrt {1+c^2 x^2}} \, dx \\ & = d^3 x (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))-\frac {1}{35} \left (b c d^3\right ) \int \frac {x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )}{\sqrt {1+c^2 x^2}} \, dx \\ & = d^3 x (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))-\frac {1}{70} \left (b c d^3\right ) \text {Subst}\left (\int \frac {35+35 c^2 x+21 c^4 x^2+5 c^6 x^3}{\sqrt {1+c^2 x}} \, dx,x,x^2\right ) \\ & = d^3 x (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))-\frac {1}{70} \left (b c d^3\right ) \text {Subst}\left (\int \left (\frac {16}{\sqrt {1+c^2 x}}+8 \sqrt {1+c^2 x}+6 \left (1+c^2 x\right )^{3/2}+5 \left (1+c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right ) \\ & = -\frac {16 b d^3 \sqrt {1+c^2 x^2}}{35 c}-\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{105 c}-\frac {6 b d^3 \left (1+c^2 x^2\right )^{5/2}}{175 c}-\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{49 c}+d^3 x (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x)) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.70 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (105 a c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )-b \sqrt {1+c^2 x^2} \left (2161+757 c^2 x^2+351 c^4 x^4+75 c^6 x^6\right )+105 b c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{3675 c} \]
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Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.90
method | result | size |
parts | \(d^{3} a \left (\frac {1}{7} c^{6} x^{7}+\frac {3}{5} c^{4} x^{5}+x^{3} c^{2}+x \right )+\frac {d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {117 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {757 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c}\) | \(153\) |
derivativedivides | \(\frac {d^{3} a \left (\frac {1}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}+c x \right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {117 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {757 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c}\) | \(156\) |
default | \(\frac {d^{3} a \left (\frac {1}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}+c x \right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {117 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {757 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c}\) | \(156\) |
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Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {525 \, a c^{7} d^{3} x^{7} + 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} + 3675 \, a c d^{3} x + 105 \, {\left (5 \, b c^{7} d^{3} x^{7} + 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} + 35 \, b c d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (75 \, b c^{6} d^{3} x^{6} + 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} + 2161 \, b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{3675 \, c} \]
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Time = 0.58 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.30 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{6} d^{3} x^{7}}{7} + \frac {3 a c^{4} d^{3} x^{5}}{5} + a c^{2} d^{3} x^{3} + a d^{3} x + \frac {b c^{6} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {b c^{5} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {3 b c^{4} d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {117 b c^{3} d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1225} + b c^{2} d^{3} x^{3} \operatorname {asinh}{\left (c x \right )} - \frac {757 b c d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{3675} + b d^{3} x \operatorname {asinh}{\left (c x \right )} - \frac {2161 b d^{3} \sqrt {c^{2} x^{2} + 1}}{3675 c} & \text {for}\: c \neq 0 \\a d^{3} x & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (150) = 300\).
Time = 0.20 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.77 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{4} d^{3} + a c^{2} d^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{3}}{c} \]
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Exception generated. \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]
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