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 = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 12
Rule 63
Rule 191
Rule 208
Rule 444
Rule 5977
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*}
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Mathematica [A] time = 0.11, 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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fricas [B] time = 0.50, size = 354, normalized size = 5.71 \[ \left [\frac {2 \, {\left (a^{2} c + d\right )} \sqrt {d x^{2} + c} x \log \left (\frac {a x + 1}{a x - 1}\right ) + \sqrt {a^{2} c + d} {\left (d x^{2} + c\right )} \log \left (\frac {a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} + 8 \, a^{2} c d + 2 \, {\left (4 \, a^{4} c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} d x^{2} + 2 \, a^{3} c + a d\right )} \sqrt {a^{2} c + d} \sqrt {d x^{2} + c} + d^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}\right )}{4 \, {\left (a^{2} c^{3} + c^{2} d + {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )}}, \frac {{\left (a^{2} c + d\right )} \sqrt {d x^{2} + c} x \log \left (\frac {a x + 1}{a x - 1}\right ) + \sqrt {-a^{2} c - d} {\left (d x^{2} + c\right )} \arctan \left (\frac {{\left (a^{2} d x^{2} + 2 \, a^{2} c + d\right )} \sqrt {-a^{2} c - d} \sqrt {d x^{2} + c}}{2 \, {\left (a^{3} c^{2} + a c d + {\left (a^{3} c d + a d^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (a^{2} c^{3} + c^{2} d + {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )}}\right ] \]
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 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.94, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 153, normalized size = 2.47 \[ \frac {a^{2} {\left (\frac {\operatorname {arsinh}\left (-\frac {2 \, a^{2} c}{\sqrt {c d} {\left | 2 \, a^{2} x + 2 \, a \right |}} + \frac {2 \, a d x}{\sqrt {c d} {\left | 2 \, a^{2} x + 2 \, a \right |}}\right )}{a^{3} \sqrt {c + \frac {d}{a^{2}}}} - \frac {\operatorname {arsinh}\left (\frac {2 \, a^{2} c}{\sqrt {c d} {\left | 2 \, a^{2} x - 2 \, a \right |}} + \frac {2 \, a d x}{\sqrt {c d} {\left | 2 \, a^{2} x - 2 \, a \right |}}\right )}{a^{3} \sqrt {c + \frac {d}{a^{2}}}}\right )}}{2 \, c} + \frac {x \operatorname {arcoth}\left (a x\right )}{\sqrt {d x^{2} + c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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