Integrand size = 16, antiderivative size = 115 \[ \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b e x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}+d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (6 c^2 d-e\right ) x \arctan \left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{6 c^2 \sqrt {-c^2 x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6427, 12, 396, 223, 209} \[ \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b x \arctan \left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right ) \left (6 c^2 d-e\right )}{6 c^2 \sqrt {-c^2 x^2}}+\frac {b e x^2 \sqrt {-c^2 x^2-1}}{6 c \sqrt {-c^2 x^2}} \]
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Rule 12
Rule 209
Rule 223
Rule 396
Rule 6427
Rubi steps \begin{align*} \text {integral}& = d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {3 d+e x^2}{3 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}} \\ & = d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {3 d+e x^2}{\sqrt {-1-c^2 x^2}} \, dx}{3 \sqrt {-c^2 x^2}} \\ & = \frac {b e x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}+d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c \left (6 d-\frac {e}{c^2}\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2}} \, dx}{6 \sqrt {-c^2 x^2}} \\ & = \frac {b e x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}+d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c \left (6 d-\frac {e}{c^2}\right ) x\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1-c^2 x^2}}\right )}{6 \sqrt {-c^2 x^2}} \\ & = \frac {b e x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}+d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (6 d-\frac {e}{c^2}\right ) x \arctan \left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{6 \sqrt {-c^2 x^2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.35 \[ \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=a d x+\frac {1}{3} a e x^3+\frac {b e x^2 \sqrt {\frac {1+c^2 x^2}{c^2 x^2}}}{6 c}+b d x \text {csch}^{-1}(c x)+\frac {1}{3} b e x^3 \text {csch}^{-1}(c x)+\frac {2 b d \sqrt {1+\frac {1}{c^2 x^2}} x \text {arctanh}\left (\frac {-1+\sqrt {1+c^2 x^2}}{c x}\right )}{\sqrt {1+c^2 x^2}}-\frac {b e \log \left (x \left (1+\sqrt {\frac {1+c^2 x^2}{c^2 x^2}}\right )\right )}{6 c^3} \]
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Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95
method | result | size |
parts | \(a \left (\frac {1}{3} e \,x^{3}+d x \right )+\frac {b \left (\frac {c \,\operatorname {arccsch}\left (c x \right ) e \,x^{3}}{3}+\operatorname {arccsch}\left (c x \right ) d x c +\frac {\sqrt {c^{2} x^{2}+1}\, \left (6 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+e c x \sqrt {c^{2} x^{2}+1}-e \,\operatorname {arcsinh}\left (c x \right )\right )}{6 c^{3} x \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c}\) | \(109\) |
derivativedivides | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arccsch}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccsch}\left (c x \right ) e \,c^{3} x^{3}}{3}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (6 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+e c x \sqrt {c^{2} x^{2}+1}-e \,\operatorname {arcsinh}\left (c x \right )\right )}{6 c x \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c^{2}}}{c}\) | \(126\) |
default | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arccsch}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccsch}\left (c x \right ) e \,c^{3} x^{3}}{3}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (6 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+e c x \sqrt {c^{2} x^{2}+1}-e \,\operatorname {arcsinh}\left (c x \right )\right )}{6 c x \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c^{2}}}{c}\) | \(126\) |
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (101) = 202\).
Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.13 \[ \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {2 \, a c^{3} e x^{3} + b c^{2} e x^{2} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 6 \, a c^{3} d x + 2 \, {\left (3 \, b c^{3} d + b c^{3} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (6 \, b c^{2} d - b e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 2 \, {\left (3 \, b c^{3} d + b c^{3} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x - 3 \, b c^{3} d - b c^{3} e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{6 \, c^{3}} \]
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\[ \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
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Time = 0.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.29 \[ \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{3} \, a e x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d}{2 \, c} \]
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\[ \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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