Integrand size = 14, antiderivative size = 240 \[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{8 b^3}+\frac {9 c^3 \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^3}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3} \]
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Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6420, 5556, 3378, 3384, 3379, 3382} \[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}+\frac {9 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 6420
Rubi steps \begin{align*} \text {integral}& = -\left (c^3 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{(a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right ) \\ & = -\left (c^3 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)^3}+\frac {\sinh (3 x)}{4 (a+b x)^3}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )\right ) \\ & = -\left (\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\sinh (x)}{(a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right )\right )-\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\sinh (3 x)}{(a+b x)^3} \, dx,x,\text {sech}^{-1}(c x)\right ) \\ & = \frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^3 \text {Subst}\left (\int \frac {\cosh (x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b}-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b} \\ & = \frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^3 \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b^2}-\frac {\left (9 c^3\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b^2} \\ & = \frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {\left (c^3 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b^2}-\frac {\left (9 c^3 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b^2}+\frac {\left (c^3 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b^2}+\frac {\left (9 c^3 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{8 b^2} \\ & = \frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 b x \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {c^2}{8 b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {3 c^3 \cosh \left (3 \text {sech}^{-1}(c x)\right )}{8 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{8 b^3}+\frac {9 c^3 \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^3}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{8 b \left (a+b \text {sech}^{-1}(c x)\right )^2}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{8 b^3}-\frac {9 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{8 b^3} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\frac {\frac {4 b^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2}+\frac {4 b \left (3-2 c^2 x^2\right )}{x^3 \left (a+b \text {sech}^{-1}(c x)\right )}-8 c^3 \left (\text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+9 c^3 \left (\text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )+\text {Chi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )-\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )\right )}{8 b^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(627\) vs. \(2(222)=444\).
Time = 1.00 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.62
method | result | size |
derivativedivides | \(c^{3} \left (\frac {\left (\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}-3 c^{2} x^{2}+4\right ) \left (3 b \,\operatorname {arcsech}\left (c x \right )+3 a -b \right )}{16 c^{3} x^{3} b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {9 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (\frac {3 a}{b}+3 \,\operatorname {arcsech}\left (c x \right )\right )}{16 b^{3}}-\frac {\left (\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}-1\right ) \left (b \,\operatorname {arcsech}\left (c x \right )+a -b \right )}{16 c x \,b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{16 b^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{16 b c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{16 b^{2} c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+3 c^{2} x^{2}-4}{16 b \,c^{3} x^{3} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}-\frac {3 \left (\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+3 c^{2} x^{2}-4\right )}{16 b^{2} c^{3} x^{3} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {9 \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{16 b^{3}}\right )\) | \(628\) |
default | \(c^{3} \left (\frac {\left (\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}-3 c^{2} x^{2}+4\right ) \left (3 b \,\operatorname {arcsech}\left (c x \right )+3 a -b \right )}{16 c^{3} x^{3} b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {9 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (\frac {3 a}{b}+3 \,\operatorname {arcsech}\left (c x \right )\right )}{16 b^{3}}-\frac {\left (\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}-1\right ) \left (b \,\operatorname {arcsech}\left (c x \right )+a -b \right )}{16 c x \,b^{2} \left (b^{2} \operatorname {arcsech}\left (c x \right )^{2}+2 a b \,\operatorname {arcsech}\left (c x \right )+a^{2}\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\frac {a}{b}+\operatorname {arcsech}\left (c x \right )\right )}{16 b^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{16 b c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+1}{16 b^{2} c x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsech}\left (c x \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+3 c^{2} x^{2}-4}{16 b \,c^{3} x^{3} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )^{2}}-\frac {3 \left (\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}+3 c^{2} x^{2}-4\right )}{16 b^{2} c^{3} x^{3} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}+\frac {9 \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{16 b^{3}}\right )\) | \(628\) |
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\[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{4}} \,d x } \]
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\[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}\, dx \]
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\[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{4}} \,d x } \]
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\[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^3} \, dx=\int \frac {1}{x^4\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \]
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