Optimal. Leaf size=283 \[ \frac{a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt{c+d x^2}}-\frac{\left (70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}}+\frac{a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
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Rubi [A] time = 1.5111, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {192, 191, 5976, 6688, 12, 6715, 1619, 63, 208} \[ \frac{a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt{c+d x^2}}-\frac{\left (70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}}+\frac{a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 5976
Rule 6688
Rule 12
Rule 6715
Rule 1619
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx &=\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-a \int \frac{\frac{x}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x}{35 c^4 \sqrt{c+d x^2}}}{1-a^2 x^2} \, dx\\ &=\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-a \int \frac{x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{35 c^4 \left (1-a^2 x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx\\ &=\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{a \int \frac{x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\left (1-a^2 x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4}\\ &=\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{a \operatorname{Subst}\left (\int \frac{35 c^3+70 c^2 d x+56 c d^2 x^2+16 d^3 x^3}{\left (1-a^2 x\right ) (c+d x)^{7/2}} \, dx,x,x^2\right )}{70 c^4}\\ &=\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{a \operatorname{Subst}\left (\int \left (\frac{5 c^3 d}{\left (a^2 c+d\right ) (c+d x)^{7/2}}+\frac{c^2 d \left (11 a^2 c+6 d\right )}{\left (a^2 c+d\right )^2 (c+d x)^{5/2}}+\frac{c d \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{\left (a^2 c+d\right )^3 (c+d x)^{3/2}}+\frac{-35 a^6 c^3-70 a^4 c^2 d-56 a^2 c d^2-16 d^3}{\left (a^2 c+d\right )^3 \left (-1+a^2 x\right ) \sqrt{c+d x}}\right ) \, dx,x,x^2\right )}{70 c^4}\\ &=\frac{a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac{a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt{c+d x^2}}+\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{\left (a \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-1+a^2 x\right ) \sqrt{c+d x}} \, dx,x,x^2\right )}{70 c^4 \left (a^2 c+d\right )^3}\\ &=\frac{a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac{a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt{c+d x^2}}+\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{\left (a \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\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 )}{35 c^4 d \left (a^2 c+d\right )^3}\\ &=\frac{a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac{a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac{a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt{c+d x^2}}+\frac{x \tanh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.933125, size = 431, normalized size = 1.52 \[ \frac{2 a c \sqrt{a^2 c+d} \left (c+d x^2\right ) \left (3 \left (19 a^4 c^2+22 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^2+3 c^2 \left (a^2 c+d\right )^2+c \left (11 a^2 c+6 d\right ) \left (a^2 c+d\right ) \left (c+d x^2\right )\right )+3 \left (70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2+16 d^3\right ) \log (1-a x) \left (c+d x^2\right )^{7/2}+3 \left (70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2+16 d^3\right ) \log (a x+1) \left (c+d x^2\right )^{7/2}-3 \left (70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2+16 d^3\right ) \left (c+d x^2\right )^{7/2} \log \left (\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right )-3 \left (70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2+16 d^3\right ) \left (c+d x^2\right )^{7/2} \log \left (\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right )+6 x \left (a^2 c+d\right )^{7/2} \tanh ^{-1}(a x) \left (70 c^2 d x^2+35 c^3+56 c d^2 x^4+16 d^3 x^6\right )}{210 c^4 \left (a^2 c+d\right )^{7/2} \left (c+d x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.497, size = 0, normalized size = 0. \begin{align*} \int{{\it Artanh} \left ( ax \right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{9}{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 = 3.69069, size = 4143, normalized size = 14.64 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30643, size = 471, normalized size = 1.66 \begin{align*} \frac{1}{105} \, a{\left (\frac{3 \,{\left (35 \, a^{6} c^{3} + 70 \, a^{4} c^{2} d + 56 \, a^{2} c d^{2} + 16 \, d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} a}{\sqrt{-a^{2} c - d}}\right )}{{\left (a^{6} c^{7} + 3 \, a^{4} c^{6} d + 3 \, a^{2} c^{5} d^{2} + c^{4} d^{3}\right )} \sqrt{-a^{2} c - d} a} + \frac{57 \,{\left (d x^{2} + c\right )}^{2} a^{4} c^{2} + 11 \,{\left (d x^{2} + c\right )} a^{4} c^{3} + 3 \, a^{4} c^{4} + 66 \,{\left (d x^{2} + c\right )}^{2} a^{2} c d + 17 \,{\left (d x^{2} + c\right )} a^{2} c^{2} d + 6 \, a^{2} c^{3} d + 24 \,{\left (d x^{2} + c\right )}^{2} d^{2} + 6 \,{\left (d x^{2} + c\right )} c d^{2} + 3 \, c^{2} d^{2}}{{\left (a^{6} c^{6} + 3 \, a^{4} c^{5} d + 3 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}}}\right )} + \frac{{\left (2 \,{\left (4 \, x^{2}{\left (\frac{2 \, d^{3} x^{2}}{c^{4}} + \frac{7 \, d^{2}}{c^{3}}\right )} + \frac{35 \, d}{c^{2}}\right )} x^{2} + \frac{35}{c}\right )} x \log \left (-\frac{a x + 1}{a x - 1}\right )}{70 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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