Optimal. Leaf size=147 \[ -\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{d \sqrt{c^2 d^2-e^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d e} \]
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Rubi [A] time = 0.119007, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6288, 961, 266, 63, 208, 725, 204} \[ -\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tan ^{-1}\left (\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right )}{d \sqrt{c^2 d^2-e^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d e} \]
Antiderivative was successfully verified.
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Rule 6288
Rule 961
Rule 266
Rule 63
Rule 208
Rule 725
Rule 204
Rubi steps
\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^2} \, dx &=-\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x (d+e x) \sqrt{1-c^2 x^2}} \, dx}{e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \left (\frac{1}{d x \sqrt{1-c^2 x^2}}-\frac{e}{d (d+e x) \sqrt{1-c^2 x^2}}\right ) \, dx}{e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{(d+e x) \sqrt{1-c^2 x^2}} \, dx}{d}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx}{d e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{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 )}{d}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{2 d e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{d \sqrt{c^2 d^2-e^2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{c^2 d e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e (d+e x)}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tan ^{-1}\left (\frac{e+c^2 d x}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right )}{d \sqrt{c^2 d^2-e^2}}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{d e}\\ \end{align*}
Mathematica [A] time = 0.225787, size = 222, normalized size = 1.51 \[ -\frac{a}{e (d+e x)}+\frac{b \log (d+e x)}{d \sqrt{e^2-c^2 d^2}}-\frac{b \log \left (c x \sqrt{\frac{1-c x}{c x+1}} \sqrt{e^2-c^2 d^2}+\sqrt{\frac{1-c x}{c x+1}} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right )}{d \sqrt{e^2-c^2 d^2}}+\frac{b \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )}{d e}-\frac{b \text{sech}^{-1}(c x)}{e (d+e x)}-\frac{b \log (x)}{d e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.29, size = 231, normalized size = 1.6 \begin{align*} -{\frac{ac}{ \left ( cxe+cd \right ) e}}-{\frac{bc{\rm arcsech} \left (cx\right )}{ \left ( cxe+cd \right ) e}}+{\frac{xbc}{de}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{\frac{xbc}{de}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}\ln \left ( 2\,{\frac{1}{cxe+cd} \left ( \sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}\sqrt{-{c}^{2}{x}^{2}+1}e+{c}^{2}dx+e \right ) } \right ){\frac{1}{\sqrt{-{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (c^{2} \int \frac{x^{2}}{c^{2} d^{2} x^{2} +{\left (c^{2} d^{2} x^{2} - d^{2} +{\left (c^{2} d e x^{2} - d e\right )} x\right )} \sqrt{c x + 1} \sqrt{-c x + 1} - d^{2} +{\left (c^{2} d e x^{2} - d e\right )} x}\,{d x} + \frac{x \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right ) - x \log \left (c\right ) - x \log \left (x\right )}{d e x + d^{2}} - \int \frac{1}{c^{2} d^{2} x^{2} - d^{2} +{\left (c^{2} d e x^{2} - d e\right )} x}\,{d x}\right )} b - \frac{a}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8242, size = 1162, normalized size = 7.9 \begin{align*} \left [-\frac{a c^{2} d^{3} - a d e^{2} + \sqrt{-c^{2} d^{2} + e^{2}}{\left (b e^{2} x + b d e\right )} \log \left (\frac{c^{2} d e x -{\left (c^{3} d^{2} - c e^{2}\right )} x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + e^{2} - \sqrt{-c^{2} d^{2} + e^{2}}{\left (c^{2} d x + c e x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + e\right )}}{e x + d}\right ) +{\left (b c^{2} d^{3} - b d e^{2} +{\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) +{\left (b c^{2} d^{3} - b d e^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c^{2} d^{4} e - d^{2} e^{3} +{\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, -\frac{a c^{2} d^{3} - a d e^{2} - 2 \, \sqrt{c^{2} d^{2} - e^{2}}{\left (b e^{2} x + b d e\right )} \arctan \left (-\frac{\sqrt{c^{2} d^{2} - e^{2}} c d x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - \sqrt{c^{2} d^{2} - e^{2}}{\left (e x + d\right )}}{{\left (c^{2} d^{2} - e^{2}\right )} x}\right ) +{\left (b c^{2} d^{3} - b d e^{2} +{\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) +{\left (b c^{2} d^{3} - b d e^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c^{2} d^{4} e - d^{2} e^{3} +{\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asech}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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