Optimal. Leaf size=56 \[ \frac{b \tan ^{-1}\left (\sqrt{c+d x-1} \sqrt{c+d x+1}\right )}{d e^2}-\frac{a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)} \]
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Rubi [A] time = 0.0516872, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {5866, 12, 5662, 92, 203} \[ \frac{b \tan ^{-1}\left (\sqrt{c+d x-1} \sqrt{c+d x+1}\right )}{d e^2}-\frac{a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5662
Rule 92
Rule 203
Rubi steps
\begin{align*} \int \frac{a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+c+d x} \sqrt{1+c+d x}\right )}{d e^2}\\ &=-\frac{a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac{b \tan ^{-1}\left (\sqrt{-1+c+d x} \sqrt{1+c+d x}\right )}{d e^2}\\ \end{align*}
Mathematica [A] time = 0.108186, size = 76, normalized size = 1.36 \[ \frac{\frac{b \sqrt{(c+d x)^2-1} \tan ^{-1}\left (\sqrt{(c+d x)^2-1}\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}-\frac{a+b \cosh ^{-1}(c+d x)}{c+d x}}{d e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 88, normalized size = 1.6 \begin{align*} -{\frac{a}{d{e}^{2} \left ( dx+c \right ) }}-{\frac{b{\rm arccosh} \left (dx+c\right )}{d{e}^{2} \left ( dx+c \right ) }}-{\frac{b}{d{e}^{2}}\sqrt{dx+c-1}\sqrt{dx+c+1}\arctan \left ({\frac{1}{\sqrt{ \left ( dx+c \right ) ^{2}-1}}} \right ){\frac{1}{\sqrt{ \left ( dx+c \right ) ^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.53022, size = 312, normalized size = 5.57 \begin{align*} \frac{b d x \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - a c + 2 \,{\left (b c d x + b c^{2}\right )} \arctan \left (-d x - c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) +{\left (b d x + b c\right )} \log \left (-d x - c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{c d^{2} e^{2} x + c^{2} d e^{2}} \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*} \frac{\int \frac{a}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{b \operatorname{acosh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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