Optimal. Leaf size=293 \[ -\frac{a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt{c+d x^2}}+\frac{\left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}}-\frac{a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}-\frac{a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac{16 x \tan ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{x \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
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Rubi [A] time = 1.23099, antiderivative size = 293, 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, 4912, 6688, 12, 6715, 1619, 63, 208} \[ -\frac{a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt{c+d x^2}}+\frac{\left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}}-\frac{a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}-\frac{a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac{16 x \tan ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{x \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 4912
Rule 6688
Rule 12
Rule 6715
Rule 1619
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx &=\frac{x \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tan ^{-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 \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tan ^{-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 \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tan ^{-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 \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tan ^{-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 \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tan ^{-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 \left (11 a^2 c-6 d\right ) d}{\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 \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tan ^{-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 \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tan ^{-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 \left (a^2 c-d\right )^3 d}\\ &=-\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 \tan ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \tan ^{-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 [C] time = 1.34959, size = 450, normalized size = 1.54 \[ \frac{-\frac{2 a c \left (3 \left (19 a^4 c^2-22 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^2+3 c^2 \left (d-a^2 c\right )^2+c \left (11 a^2 c-6 d\right ) \left (a^2 c-d\right ) \left (c+d x^2\right )\right )}{\left (a^2 c-d\right )^3 \left (c+d x^2\right )^{5/2}}+\frac{3 \left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \log \left (-\frac{140 a c^4 \left (a^2 c-d\right )^{5/2} \left (\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right )}{(a x+i) \left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right )}\right )}{\left (a^2 c-d\right )^{7/2}}+\frac{3 \left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \log \left (-\frac{140 a c^4 \left (a^2 c-d\right )^{5/2} \left (\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right )}{(a x-i) \left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right )}\right )}{\left (a^2 c-d\right )^{7/2}}+\frac{6 x \tan ^{-1}(a x) \left (70 c^2 d x^2+35 c^3+56 c d^2 x^4+16 d^3 x^6\right )}{\left (c+d x^2\right )^{7/2}}}{210 c^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.664, size = 0, normalized size = 0. \begin{align*} \int{\arctan \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 = 6.0733, size = 4096, normalized size = 13.98 \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 = 1.21842, size = 454, normalized size = 1.55 \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 \arctan \left (a x\right )}{35 \,{\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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