Integrand size = 19, antiderivative size = 214 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=-\frac {b \left (42 c^2 d-25 e\right ) x^2 \sqrt {-1-c^2 x^2}}{560 c^5 \sqrt {-c^2 x^2}}+\frac {b \left (42 c^2 d-25 e\right ) x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (42 c^2 d-25 e\right ) x \arctan \left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{560 c^6 \sqrt {-c^2 x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {14, 6437, 12, 470, 327, 223, 209} \[ \int x^4 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b x \arctan \left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right ) \left (42 c^2 d-25 e\right )}{560 c^6 \sqrt {-c^2 x^2}}+\frac {b e x^6 \sqrt {-c^2 x^2-1}}{42 c \sqrt {-c^2 x^2}}-\frac {b x^2 \sqrt {-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{560 c^5 \sqrt {-c^2 x^2}}+\frac {b x^4 \sqrt {-c^2 x^2-1} \left (42 c^2 d-25 e\right )}{840 c^3 \sqrt {-c^2 x^2}} \]
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Rule 12
Rule 14
Rule 209
Rule 223
Rule 327
Rule 470
Rule 6437
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^4 \left (7 d+5 e x^2\right )}{35 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}} \\ & = \frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^4 \left (7 d+5 e x^2\right )}{\sqrt {-1-c^2 x^2}} \, dx}{35 \sqrt {-c^2 x^2}} \\ & = \frac {b e x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c \left (42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {x^4}{\sqrt {-1-c^2 x^2}} \, dx}{210 \sqrt {-c^2 x^2}} \\ & = \frac {b \left (42 c^2 d-25 e\right ) x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {\left (b \left (42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2}} \, dx}{280 c \sqrt {-c^2 x^2}} \\ & = -\frac {b \left (42 c^2 d-25 e\right ) x^2 \sqrt {-1-c^2 x^2}}{560 c^5 \sqrt {-c^2 x^2}}+\frac {b \left (42 c^2 d-25 e\right ) x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b \left (42 d-\frac {25 e}{c^2}\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2}} \, dx}{560 c^3 \sqrt {-c^2 x^2}} \\ & = -\frac {b \left (42 c^2 d-25 e\right ) x^2 \sqrt {-1-c^2 x^2}}{560 c^5 \sqrt {-c^2 x^2}}+\frac {b \left (42 c^2 d-25 e\right ) x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b \left (42 d-\frac {25 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 )}{560 c^3 \sqrt {-c^2 x^2}} \\ & = -\frac {b \left (42 c^2 d-25 e\right ) x^2 \sqrt {-1-c^2 x^2}}{560 c^5 \sqrt {-c^2 x^2}}+\frac {b \left (42 c^2 d-25 e\right ) x^4 \sqrt {-1-c^2 x^2}}{840 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^6 \sqrt {-1-c^2 x^2}}{42 c \sqrt {-c^2 x^2}}+\frac {1}{5} d x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{7} e x^7 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (42 c^2 d-25 e\right ) x \arctan \left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{560 c^6 \sqrt {-c^2 x^2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.64 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {48 a c^7 x^5 \left (7 d+5 e x^2\right )+b c^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2 \left (75 e-2 c^2 \left (63 d+25 e x^2\right )+c^4 \left (84 d x^2+40 e x^4\right )\right )+48 b c^7 x^5 \left (7 d+5 e x^2\right ) \text {csch}^{-1}(c x)+3 b \left (42 c^2 d-25 e\right ) \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{1680 c^7} \]
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Time = 0.70 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93
method | result | size |
parts | \(a \left (\frac {1}{7} e \,x^{7}+\frac {1}{5} d \,x^{5}\right )+\frac {b \left (\frac {c^{5} \operatorname {arccsch}\left (c x \right ) e \,x^{7}}{7}+\frac {\operatorname {arccsch}\left (c x \right ) x^{5} c^{5} d}{5}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (84 d \,c^{5} x^{3} \sqrt {c^{2} x^{2}+1}+40 e \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-126 d \,c^{3} x \sqrt {c^{2} x^{2}+1}-50 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+126 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+75 e c x \sqrt {c^{2} x^{2}+1}-75 e \,\operatorname {arcsinh}\left (c x \right )\right )}{1680 c^{3} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x}\right )}{c^{5}}\) | \(198\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) d \,c^{7} x^{5}}{5}+\frac {\operatorname {arccsch}\left (c x \right ) e \,c^{7} x^{7}}{7}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (84 d \,c^{5} x^{3} \sqrt {c^{2} x^{2}+1}+40 e \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-126 d \,c^{3} x \sqrt {c^{2} x^{2}+1}-50 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+126 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+75 e c x \sqrt {c^{2} x^{2}+1}-75 e \,\operatorname {arcsinh}\left (c x \right )\right )}{1680 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{5}}\) | \(211\) |
default | \(\frac {\frac {a \left (\frac {1}{5} d \,c^{7} x^{5}+\frac {1}{7} e \,c^{7} x^{7}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) d \,c^{7} x^{5}}{5}+\frac {\operatorname {arccsch}\left (c x \right ) e \,c^{7} x^{7}}{7}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (84 d \,c^{5} x^{3} \sqrt {c^{2} x^{2}+1}+40 e \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-126 d \,c^{3} x \sqrt {c^{2} x^{2}+1}-50 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+126 d \,c^{2} \operatorname {arcsinh}\left (c x \right )+75 e c x \sqrt {c^{2} x^{2}+1}-75 e \,\operatorname {arcsinh}\left (c x \right )\right )}{1680 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{5}}\) | \(211\) |
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Time = 0.31 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.38 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {240 \, a c^{7} e x^{7} + 336 \, a c^{7} d x^{5} + 48 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 3 \, {\left (42 \, b c^{2} d - 25 \, b e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 48 \, {\left (7 \, b c^{7} d + 5 \, b c^{7} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 48 \, {\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5} - 7 \, b c^{7} d - 5 \, b c^{7} e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (40 \, b c^{6} e x^{6} + 2 \, {\left (42 \, b c^{6} d - 25 \, b c^{4} e\right )} x^{4} - 3 \, {\left (42 \, b c^{4} d - 25 \, b c^{2} e\right )} x^{2}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{1680 \, c^{7}} \]
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\[ \int x^4 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x^{4} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
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Time = 0.20 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.35 \[ \int x^4 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{7} \, a e x^{7} + \frac {1}{5} \, a d x^{5} + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arcsch}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b d + \frac {1}{672} \, {\left (96 \, x^{7} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, {\left (15 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{3} - 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{6}} - \frac {15 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} + \frac {15 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{6}}}{c}\right )} b e \]
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\[ \int x^4 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x^4\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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