Integrand size = 16, antiderivative size = 183 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \]
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Time = 0.10 (sec) , antiderivative size = 183, 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 = {5828, 759, 821, 739, 212} \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}}-\frac {b c \sqrt {c^2 x^2+1}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)} \]
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Rule 212
Rule 739
Rule 759
Rule 821
Rule 5828
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{(d+e x)^3 \sqrt {1+c^2 x^2}} \, dx}{3 e} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}-\frac {\left (b c^3\right ) \int \frac {-2 d+e x}{(d+e x)^2 \sqrt {1+c^2 x^2}} \, dx}{6 e \left (c^2 d^2+e^2\right )} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3 \left (2 c^2 d^2-e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {1+c^2 x^2}} \, dx}{6 e \left (c^2 d^2+e^2\right )^2} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}-\frac {\left (b c^3 \left (2 c^2 d^2-e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c^2 d^2+e^2-x^2} \, dx,x,\frac {e-c^2 d x}{\sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^2} \\ & = -\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}-\frac {b c \sqrt {1+c^2 x^2} \left (e^2+c^2 d (4 d+3 e x)\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {2 b \text {arcsinh}(c x)}{e (d+e x)^3}-\frac {b c^3 \left (-2 c^2 d^2+e^2\right ) \log (d+e x)}{e \left (c^2 d^2+e^2\right )^{5/2}}+\frac {b c^3 \left (-2 c^2 d^2+e^2\right ) \log \left (e-c^2 d x+\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}\right )}{e \left (c^2 d^2+e^2\right )^{5/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(168)=336\).
Time = 0.41 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.79
method | result | size |
parts | \(-\frac {a}{3 \left (e x +d \right )^{3} e}-\frac {b \,c^{3} \operatorname {arcsinh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{3} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}-\frac {b \,c^{4} d \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{5} d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {b \,c^{3} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\) | \(510\) |
derivativedivides | \(\frac {-\frac {a \,c^{4}}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{4} \operatorname {arcsinh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{4} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}-\frac {b \,c^{5} d \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\) | \(520\) |
default | \(\frac {-\frac {a \,c^{4}}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{4} \operatorname {arcsinh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{4} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}-\frac {b \,c^{5} d \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\) | \(520\) |
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Leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (167) = 334\).
Time = 0.62 (sec) , antiderivative size = 977, normalized size of antiderivative = 5.34 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=-\frac {{\left (2 \, a + 3 \, b\right )} c^{6} d^{9} + 3 \, {\left (2 \, a + b\right )} c^{4} d^{7} e^{2} + 6 \, a c^{2} d^{5} e^{4} + 2 \, a d^{3} e^{6} + 3 \, {\left (b c^{6} d^{6} e^{3} + b c^{4} d^{4} e^{5}\right )} x^{3} + 9 \, {\left (b c^{6} d^{7} e^{2} + b c^{4} d^{5} e^{4}\right )} x^{2} + {\left (2 \, b c^{5} d^{8} - b c^{3} d^{6} e^{2} + {\left (2 \, b c^{5} d^{5} e^{3} - b c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (2 \, b c^{5} d^{6} e^{2} - b c^{3} d^{4} e^{4}\right )} x^{2} + 3 \, {\left (2 \, b c^{5} d^{7} e - b c^{3} d^{5} e^{3}\right )} x\right )} \sqrt {c^{2} d^{2} + e^{2}} \log \left (-\frac {c^{3} d^{2} x - c d e - \sqrt {c^{2} d^{2} + e^{2}} {\left (c^{2} d x - e\right )} + {\left (c^{2} d^{2} - \sqrt {c^{2} d^{2} + e^{2}} c d + e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{e x + d}\right ) + 9 \, {\left (b c^{6} d^{8} e + b c^{4} d^{6} e^{3}\right )} x - 2 \, {\left ({\left (b c^{6} d^{6} e^{3} + 3 \, b c^{4} d^{4} e^{5} + 3 \, b c^{2} d^{2} e^{7} + b e^{9}\right )} x^{3} + 3 \, {\left (b c^{6} d^{7} e^{2} + 3 \, b c^{4} d^{5} e^{4} + 3 \, b c^{2} d^{3} e^{6} + b d e^{8}\right )} x^{2} + 3 \, {\left (b c^{6} d^{8} e + 3 \, b c^{4} d^{6} e^{3} + 3 \, b c^{2} d^{4} e^{5} + b d^{2} e^{7}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (b c^{6} d^{9} + 3 \, b c^{4} d^{7} e^{2} + 3 \, b c^{2} d^{5} e^{4} + b d^{3} e^{6} + {\left (b c^{6} d^{6} e^{3} + 3 \, b c^{4} d^{4} e^{5} + 3 \, b c^{2} d^{2} e^{7} + b e^{9}\right )} x^{3} + 3 \, {\left (b c^{6} d^{7} e^{2} + 3 \, b c^{4} d^{5} e^{4} + 3 \, b c^{2} d^{3} e^{6} + b d e^{8}\right )} x^{2} + 3 \, {\left (b c^{6} d^{8} e + 3 \, b c^{4} d^{6} e^{3} + 3 \, b c^{2} d^{4} e^{5} + b d^{2} e^{7}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (4 \, b c^{5} d^{8} e + 5 \, b c^{3} d^{6} e^{3} + b c d^{4} e^{5} + 3 \, {\left (b c^{5} d^{6} e^{3} + b c^{3} d^{4} e^{5}\right )} x^{2} + {\left (7 \, b c^{5} d^{7} e^{2} + 8 \, b c^{3} d^{5} e^{4} + b c d^{3} e^{6}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{6 \, {\left (c^{6} d^{12} e + 3 \, c^{4} d^{10} e^{3} + 3 \, c^{2} d^{8} e^{5} + d^{6} e^{7} + {\left (c^{6} d^{9} e^{4} + 3 \, c^{4} d^{7} e^{6} + 3 \, c^{2} d^{5} e^{8} + d^{3} e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{10} e^{3} + 3 \, c^{4} d^{8} e^{5} + 3 \, c^{2} d^{6} e^{7} + d^{4} e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{11} e^{2} + 3 \, c^{4} d^{9} e^{4} + 3 \, c^{2} d^{7} e^{6} + d^{5} e^{8}\right )} x\right )}} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x + d\right )}^{4}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x + d\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]
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