Optimal. Leaf size=62 \[ \frac{x \coth ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{c \sqrt{a^2 c+d}} \]
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Rubi [A] time = 0.112951, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {191, 5977, 12, 444, 63, 208} \[ \frac{x \coth ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{c \sqrt{a^2 c+d}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 5977
Rule 12
Rule 444
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx &=\frac{x \coth ^{-1}(a x)}{c \sqrt{c+d x^2}}-a \int \frac{x}{c \left (1-a^2 x^2\right ) \sqrt{c+d x^2}} \, dx\\ &=\frac{x \coth ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{a \int \frac{x}{\left (1-a^2 x^2\right ) \sqrt{c+d x^2}} \, dx}{c}\\ &=\frac{x \coth ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (1-a^2 x\right ) \sqrt{c+d x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac{x \coth ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{1+\frac{a^2 c}{d}-\frac{a^2 x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{c d}\\ &=\frac{x \coth ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{c \sqrt{a^2 c+d}}\\ \end{align*}
Mathematica [A] time = 0.117746, size = 119, normalized size = 1.92 \[ \frac{\frac{-\log \left (\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right )-\log \left (\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right )+\log (1-a x)+\log (a x+1)}{\sqrt{a^2 c+d}}+\frac{2 x \coth ^{-1}(a x)}{\sqrt{c+d x^2}}}{2 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.457, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccoth} \left (ax\right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \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 = 1.74724, size = 760, normalized size = 12.26 \begin{align*} \left [\frac{2 \,{\left (a^{2} c + d\right )} \sqrt{d x^{2} + c} x \log \left (\frac{a x + 1}{a x - 1}\right ) + \sqrt{a^{2} c + d}{\left (d x^{2} + c\right )} \log \left (\frac{a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} + 8 \, a^{2} c d + 2 \,{\left (4 \, a^{4} c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \,{\left (a^{3} d x^{2} + 2 \, a^{3} c + a d\right )} \sqrt{a^{2} c + d} \sqrt{d x^{2} + c} + d^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}\right )}{4 \,{\left (a^{2} c^{3} + c^{2} d +{\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )}}, \frac{{\left (a^{2} c + d\right )} \sqrt{d x^{2} + c} x \log \left (\frac{a x + 1}{a x - 1}\right ) + \sqrt{-a^{2} c - d}{\left (d x^{2} + c\right )} \arctan \left (\frac{{\left (a^{2} d x^{2} + 2 \, a^{2} c + d\right )} \sqrt{-a^{2} c - d} \sqrt{d x^{2} + c}}{2 \,{\left (a^{3} c^{2} + a c d +{\left (a^{3} c d + a d^{2}\right )} x^{2}\right )}}\right )}{2 \,{\left (a^{2} c^{3} + c^{2} d +{\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )}}\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{\operatorname{acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac{3}{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{\operatorname{arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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