Optimal. Leaf size=128 \[ -\frac{\left (3 a^2 c+2 d\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{3 c^2 \left (a^2 c+d\right )^{3/2}}+\frac{a}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.339828, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {192, 191, 5977, 6688, 12, 571, 78, 63, 208} \[ -\frac{\left (3 a^2 c+2 d\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{3 c^2 \left (a^2 c+d\right )^{3/2}}+\frac{a}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 5977
Rule 6688
Rule 12
Rule 571
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-a \int \frac{\frac{x}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x}{3 c^2 \sqrt{c+d x^2}}}{1-a^2 x^2} \, dx\\ &=\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-a \int \frac{x \left (3 c+2 d x^2\right )}{3 c^2 \left (1-a^2 x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\\ &=\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{a \int \frac{x \left (3 c+2 d x^2\right )}{\left (1-a^2 x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{a \operatorname{Subst}\left (\int \frac{3 c+2 d x}{\left (1-a^2 x\right ) (c+d x)^{3/2}} \, dx,x,x^2\right )}{6 c^2}\\ &=\frac{a}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{\left (a \left (3 a^2 c+2 d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-a^2 x\right ) \sqrt{c+d x}} \, dx,x,x^2\right )}{6 c^2 \left (a^2 c+d\right )}\\ &=\frac{a}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{\left (a \left (3 a^2 c+2 d\right )\right ) \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 )}{3 c^2 d \left (a^2 c+d\right )}\\ &=\frac{a}{3 c \left (a^2 c+d\right ) \sqrt{c+d x^2}}+\frac{x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac{2 x \coth ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}-\frac{\left (3 a^2 c+2 d\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{3 c^2 \left (a^2 c+d\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.321201, size = 226, normalized size = 1.77 \[ \frac{\frac{2 a c}{\left (a^2 c+d\right ) \sqrt{c+d x^2}}-\frac{\left (3 a^2 c+2 d\right ) \log \left (\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right )}{\left (a^2 c+d\right )^{3/2}}-\frac{\left (3 a^2 c+2 d\right ) \log \left (\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right )}{\left (a^2 c+d\right )^{3/2}}+\frac{\left (3 a^2 c+2 d\right ) \log (1-a x)}{\left (a^2 c+d\right )^{3/2}}+\frac{\left (3 a^2 c+2 d\right ) \log (a x+1)}{\left (a^2 c+d\right )^{3/2}}+\frac{2 x \coth ^{-1}(a x) \left (3 c+2 d x^2\right )}{\left (c+d x^2\right )^{3/2}}}{6 c^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.456, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccoth} \left (ax\right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{5}{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 = 1.98491, size = 1496, normalized size = 11.69 \begin{align*} \left [\frac{{\left (3 \, a^{2} c^{3} +{\left (3 \, a^{2} c d^{2} + 2 \, d^{3}\right )} x^{4} + 2 \, c^{2} d + 2 \,{\left (3 \, a^{2} c^{2} d + 2 \, c d^{2}\right )} x^{2}\right )} \sqrt{a^{2} c + d} \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 ) + 2 \,{\left (2 \, a^{3} c^{3} + 2 \, a c^{2} d + 2 \,{\left (a^{3} c^{2} d + a c d^{2}\right )} x^{2} +{\left (2 \,{\left (a^{4} c^{2} d + 2 \, a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \,{\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} x\right )} \log \left (\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{d x^{2} + c}}{12 \,{\left (a^{4} c^{6} + 2 \, a^{2} c^{5} d + c^{4} d^{2} +{\left (a^{4} c^{4} d^{2} + 2 \, a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{4} c^{5} d + 2 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, \frac{{\left (3 \, a^{2} c^{3} +{\left (3 \, a^{2} c d^{2} + 2 \, d^{3}\right )} x^{4} + 2 \, c^{2} d + 2 \,{\left (3 \, a^{2} c^{2} d + 2 \, c d^{2}\right )} x^{2}\right )} \sqrt{-a^{2} c - d} \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 ) +{\left (2 \, a^{3} c^{3} + 2 \, a c^{2} d + 2 \,{\left (a^{3} c^{2} d + a c d^{2}\right )} x^{2} +{\left (2 \,{\left (a^{4} c^{2} d + 2 \, a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \,{\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} x\right )} \log \left (\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{d x^{2} + c}}{6 \,{\left (a^{4} c^{6} + 2 \, a^{2} c^{5} d + c^{4} d^{2} +{\left (a^{4} c^{4} d^{2} + 2 \, a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \,{\left (a^{4} c^{5} d + 2 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\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{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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