Integrand size = 12, antiderivative size = 165 \[ \int e^{\text {arcsinh}(a+b x)} x^3 \, dx=\frac {e^{-3 \text {arcsinh}(a+b x)}}{48 b^4}+\frac {3 a e^{-2 \text {arcsinh}(a+b x)}}{16 b^4}-\frac {\left (1-6 a^2\right ) e^{-\text {arcsinh}(a+b x)}}{8 b^4}+\frac {a \left (3-4 a^2\right ) e^{2 \text {arcsinh}(a+b x)}}{16 b^4}-\frac {\left (1-6 a^2\right ) e^{3 \text {arcsinh}(a+b x)}}{24 b^4}-\frac {3 a e^{4 \text {arcsinh}(a+b x)}}{32 b^4}+\frac {e^{5 \text {arcsinh}(a+b x)}}{80 b^4}+\frac {a \left (3-4 a^2\right ) \text {arcsinh}(a+b x)}{8 b^4} \]
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Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5873, 2320, 12, 1642} \[ \int e^{\text {arcsinh}(a+b x)} x^3 \, dx=\frac {\left (3-4 a^2\right ) a e^{2 \text {arcsinh}(a+b x)}}{16 b^4}+\frac {\left (3-4 a^2\right ) a \text {arcsinh}(a+b x)}{8 b^4}-\frac {\left (1-6 a^2\right ) e^{-\text {arcsinh}(a+b x)}}{8 b^4}-\frac {\left (1-6 a^2\right ) e^{3 \text {arcsinh}(a+b x)}}{24 b^4}+\frac {3 a e^{-2 \text {arcsinh}(a+b x)}}{16 b^4}-\frac {3 a e^{4 \text {arcsinh}(a+b x)}}{32 b^4}+\frac {e^{-3 \text {arcsinh}(a+b x)}}{48 b^4}+\frac {e^{5 \text {arcsinh}(a+b x)}}{80 b^4} \]
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Rule 12
Rule 1642
Rule 2320
Rule 5873
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x \cosh (x) \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^3 \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\left (-1-x^2\right ) \left (1+2 a x-x^2\right )^3}{16 b^3 x^4} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\left (-1-x^2\right ) \left (1+2 a x-x^2\right )^3}{x^4} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{16 b^4} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{x^4}-\frac {6 a}{x^3}-\frac {2 \left (-1+6 a^2\right )}{x^2}+\frac {2 a \left (3-4 a^2\right )}{x}+2 a \left (3-4 a^2\right ) x+2 \left (-1+6 a^2\right ) x^2-6 a x^3+x^4\right ) \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{16 b^4} \\ & = \frac {e^{-3 \text {arcsinh}(a+b x)}}{48 b^4}+\frac {3 a e^{-2 \text {arcsinh}(a+b x)}}{16 b^4}-\frac {\left (1-6 a^2\right ) e^{-\text {arcsinh}(a+b x)}}{8 b^4}+\frac {a \left (3-4 a^2\right ) e^{2 \text {arcsinh}(a+b x)}}{16 b^4}-\frac {\left (1-6 a^2\right ) e^{3 \text {arcsinh}(a+b x)}}{24 b^4}-\frac {3 a e^{4 \text {arcsinh}(a+b x)}}{32 b^4}+\frac {e^{5 \text {arcsinh}(a+b x)}}{80 b^4}+\frac {a \left (3-4 a^2\right ) \text {arcsinh}(a+b x)}{8 b^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.72 \[ \int e^{\text {arcsinh}(a+b x)} x^3 \, dx=\frac {30 a b^4 x^4+24 b^5 x^5-\sqrt {1+a^2+2 a b x+b^2 x^2} \left (16-83 a^2+6 a^4+a \left (29-6 a^2\right ) b x+2 \left (-4+3 a^2\right ) b^2 x^2-6 a b^3 x^3-24 b^4 x^4\right )+15 a \left (3-4 a^2\right ) \text {arcsinh}(a+b x)}{120 b^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(205)=410\).
Time = 0.75 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.81
method | result | size |
default | \(\frac {x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{5 b^{2}}-\frac {7 a \left (\frac {x \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 b^{2}}-\frac {5 a \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{b}\right )}{4 b}-\frac {\left (a^{2}+1\right ) \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{5 b}-\frac {2 \left (a^{2}+1\right ) \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{b}\right )}{5 b^{2}}+\frac {b \,x^{5}}{5}+\frac {a \,x^{4}}{4}\) | \(464\) |
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Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.84 \[ \int e^{\text {arcsinh}(a+b x)} x^3 \, dx=\frac {24 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} + 15 \, {\left (4 \, a^{3} - 3 \, a\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (24 \, b^{4} x^{4} + 6 \, a b^{3} x^{3} - 2 \, {\left (3 \, a^{2} - 4\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (6 \, a^{3} - 29 \, a\right )} b x + 83 \, a^{2} - 16\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{120 \, b^{4}} \]
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Time = 0.90 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.76 \[ \int e^{\text {arcsinh}(a+b x)} x^3 \, dx=\frac {a x^{4}}{4} + \frac {b x^{5}}{5} + \begin {cases} \frac {\left (- \frac {a \left (- \frac {3 a \left (- \frac {5 a \left (\frac {1}{5} - \frac {3 a^{2}}{20}\right )}{3 b} - \frac {a \left (3 a^{2} + 3\right )}{20 b}\right )}{2 b} - \frac {\left (\frac {1}{5} - \frac {3 a^{2}}{20}\right ) \left (2 a^{2} + 2\right )}{3 b^{2}}\right )}{b} - \frac {\left (a^{2} + 1\right ) \left (- \frac {5 a \left (\frac {1}{5} - \frac {3 a^{2}}{20}\right )}{3 b} - \frac {a \left (3 a^{2} + 3\right )}{20 b}\right )}{2 b^{2}}\right ) \log {\left (2 a b + 2 b^{2} x + 2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \sqrt {b^{2}} \right )}}{\sqrt {b^{2}}} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \left (\frac {a x^{3}}{20 b} + \frac {x^{4}}{5} + \frac {x^{2} \cdot \left (\frac {1}{5} - \frac {3 a^{2}}{20}\right )}{3 b^{2}} + \frac {x \left (- \frac {5 a \left (\frac {1}{5} - \frac {3 a^{2}}{20}\right )}{3 b} - \frac {a \left (3 a^{2} + 3\right )}{20 b}\right )}{2 b^{2}} + \frac {- \frac {3 a \left (- \frac {5 a \left (\frac {1}{5} - \frac {3 a^{2}}{20}\right )}{3 b} - \frac {a \left (3 a^{2} + 3\right )}{20 b}\right )}{2 b} - \frac {\left (\frac {1}{5} - \frac {3 a^{2}}{20}\right ) \left (2 a^{2} + 2\right )}{3 b^{2}}}{b^{2}}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {\frac {\left (- 3 a^{2} - 3\right ) \left (a^{2} + 2 a b x + 1\right )^{\frac {7}{2}}}{7} + \frac {\left (a^{2} + 2 a b x + 1\right )^{\frac {9}{2}}}{9} + \frac {\left (a^{2} + 2 a b x + 1\right )^{\frac {5}{2}} \cdot \left (3 a^{4} + 6 a^{2} + 3\right )}{5} + \frac {\left (a^{2} + 2 a b x + 1\right )^{\frac {3}{2}} \left (- a^{6} - 3 a^{4} - 3 a^{2} - 1\right )}{3}}{8 a^{4} b^{4}} & \text {for}\: a b \neq 0 \\\frac {x^{4} \sqrt {a^{2} + 1}}{4} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (205) = 410\).
Time = 0.30 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.98 \[ \int e^{\text {arcsinh}(a+b x)} x^3 \, dx=\frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{2}} - \frac {{\left (a^{2} + 1\right )} a^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{5 \, b^{4}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a x}{20 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a x}{5 \, b^{3}} + \frac {{\left (a^{2} + 1\right )}^{2} a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{5 \, b^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{12 \, b^{4}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a^{2}}{5 \, b^{4}} + \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} a^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{40 \, b^{6}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} {\left (a^{2} + 1\right )}}{15 \, b^{4}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{40 \, b^{5}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} {\left (a^{2} + 1\right )} a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{40 \, b^{6}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{40 \, b^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.05 \[ \int e^{\text {arcsinh}(a+b x)} x^3 \, dx=\frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} + \frac {1}{120} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left ({\left (2 \, {\left (3 \, {\left (4 \, x + \frac {a}{b}\right )} x - \frac {3 \, a^{2} b^{5} - 4 \, b^{5}}{b^{7}}\right )} x + \frac {6 \, a^{3} b^{4} - 29 \, a b^{4}}{b^{7}}\right )} x - \frac {6 \, a^{4} b^{3} - 83 \, a^{2} b^{3} + 16 \, b^{3}}{b^{7}}\right )} + \frac {{\left (4 \, a^{3} - 3 \, a\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{3} {\left | b \right |}} \]
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Timed out. \[ \int e^{\text {arcsinh}(a+b x)} x^3 \, dx=\int x^3\,\left (a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x\right ) \,d x \]
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