Optimal. Leaf size=542 \[ -\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2} \]
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Rubi [A]
time = 0.93, antiderivative size = 542, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {6438, 5959,
5957, 533, 385, 214, 5962, 5681, 2221, 2317, 2438} \begin {gather*} -\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 d^2}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (\frac {d}{x^2}+e\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^2}+\frac {b \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 d^2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 385
Rule 533
Rule 2221
Rule 2317
Rule 2438
Rule 5681
Rule 5957
Rule 5959
Rule 5962
Rule 6438
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx &=-\text {Subst}\left (\int \frac {x^3 \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \left (-\frac {e x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{d \left (e+d x^2\right )^2}+\frac {x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{d}+\frac {e \text {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{d}\\ &=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}-\frac {\text {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c d^2}\\ &=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 (-d)^{3/2}}-\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 (-d)^{3/2}}+\frac {\left (b e \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 (-d)^{3/2}}-\frac {\text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 (-d)^{3/2}}+\frac {\left (b e \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {1}{e-\left (d+\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 (-d)^{3/2}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 (-d)^{3/2}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {sech}^{-1}(c x)\right )}{2 (-d)^{3/2}}\\ &=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 d^2}\\ &=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}\\ \end {align*}
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Mathematica [F]
time = 32.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 4.67, size = 3326, normalized size = 6.14
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(3326\) |
default | \(\text {Expression too large to display}\) | \(3326\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x \left (d + e x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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