Integrand size = 16, antiderivative size = 293 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}} \]
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Time = 0.84 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {198, 197, 5033, 6820, 12, 6847, 1633, 65, 214} \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
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Rule 12
Rule 65
Rule 197
Rule 198
Rule 214
Rule 1633
Rule 5033
Rule 6820
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-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 \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-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 \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-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 \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {a \text {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 \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {a \text {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 \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-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 ) \text {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 \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-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 ) \text {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 \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cot ^{-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 ) \text {arctanh}\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*}
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.54 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {\frac {2 a c \left (3 c^2 \left (-a^2 c+d\right )^2+c \left (11 a^2 c-6 d\right ) \left (a^2 c-d\right ) \left (c+d x^2\right )+3 \left (19 a^4 c^2-22 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^2\right )}{\left (a^2 c-d\right )^3 \left (c+d x^2\right )^{5/2}}+\frac {6 x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right ) \cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}}-\frac {3 \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \log \left (\frac {140 a c^4 \left (a^2 c-d\right )^{5/2} \left (a c-i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) (i+a x)}\right )}{\left (a^2 c-d\right )^{7/2}}-\frac {3 \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \log \left (\frac {140 a c^4 \left (a^2 c-d\right )^{5/2} \left (a c+i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) (-i+a x)}\right )}{\left (a^2 c-d\right )^{7/2}}}{210 c^4} \]
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\[\int \frac {\operatorname {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 972 vs. \(2 (257) = 514\).
Time = 0.61 (sec) , antiderivative size = 1986, normalized size of antiderivative = 6.78 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {9}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.16 \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {1}{105} \, a {\left (\frac {3 \, {\left (35 \, a^{6} c^{3} - 70 \, a^{4} c^{2} d + 56 \, a^{2} c d^{2} - 16 \, d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{{\left (a^{6} c^{7} - 3 \, a^{4} c^{6} d + 3 \, a^{2} c^{5} d^{2} - c^{4} d^{3}\right )} \sqrt {-a^{2} c + d} a} + \frac {57 \, {\left (d x^{2} + c\right )}^{2} a^{4} c^{2} + 11 \, {\left (d x^{2} + c\right )} a^{4} c^{3} + 3 \, a^{4} c^{4} - 66 \, {\left (d x^{2} + c\right )}^{2} a^{2} c d - 17 \, {\left (d x^{2} + c\right )} a^{2} c^{2} d - 6 \, a^{2} c^{3} d + 24 \, {\left (d x^{2} + c\right )}^{2} d^{2} + 6 \, {\left (d x^{2} + c\right )} c d^{2} + 3 \, c^{2} d^{2}}{{\left (a^{6} c^{6} - 3 \, a^{4} c^{5} d + 3 \, a^{2} c^{4} d^{2} - c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}\right )} + \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, d^{3} x^{2}}{c^{4}} + \frac {7 \, d^{2}}{c^{3}}\right )} + \frac {35 \, d}{c^{2}}\right )} x^{2} + \frac {35}{c}\right )} x \arctan \left (\frac {1}{a x}\right )}{35 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \]
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