Optimal. Leaf size=293 \[ \frac {a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
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Rubi [A] time = 1.16, antiderivative size = 293, normalized size of antiderivative = 1.00, 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, 4913, 6688, 12, 6715, 1619, 63, 208} \[ \frac {a \left (19 a^4 c^2-22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c-d\right )^3 \sqrt {c+d x^2}}-\frac {\left (-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{35 c^4 \left (a^2 c-d\right )^{7/2}}+\frac {a \left (11 a^2 c-6 d\right )}{105 c^2 \left (a^2 c-d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{35 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{5/2}}+\frac {16 x \cot ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cot ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cot ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cot ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 191
Rule 192
Rule 208
Rule 1619
Rule 4913
Rule 6688
Rule 6715
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, 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 {\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 \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 \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 \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 \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 ) \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 \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 ) \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 \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 ) \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*}
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Mathematica [C] time = 1.67, size = 450, normalized size = 1.54 \[ \frac {\frac {2 a c \left (3 c^2 \left (d-a^2 c\right )^2+c \left (11 a^2 c-6 d\right ) \left (a^2 c-d\right ) \left (c+d x^2\right )+3 \left (19 a^4 c^2-22 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^2\right )}{\left (a^2 c-d\right )^3 \left (c+d x^2\right )^{5/2}}-\frac {3 \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \log \left (\frac {140 a c^4 \left (a^2 c-d\right )^{5/2} \left (\sqrt {a^2 c-d} \sqrt {c+d x^2}+a c-i d x\right )}{(a x+i) \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right )}\right )}{\left (a^2 c-d\right )^{7/2}}-\frac {3 \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right ) \log \left (\frac {140 a c^4 \left (a^2 c-d\right )^{5/2} \left (\sqrt {a^2 c-d} \sqrt {c+d x^2}+a c+i d x\right )}{(a x-i) \left (35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right )}\right )}{\left (a^2 c-d\right )^{7/2}}+\frac {6 x \cot ^{-1}(a x) \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\left (c+d x^2\right )^{7/2}}}{210 c^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.59, size = 1986, normalized size = 6.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 340, normalized size = 1.16 \[ \frac {1}{105} \, a {\left (\frac {3 \, {\left (35 \, a^{6} c^{3} - 70 \, a^{4} c^{2} d + 56 \, a^{2} c d^{2} - 16 \, d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{{\left (a^{6} c^{7} - 3 \, a^{4} c^{6} d + 3 \, a^{2} c^{5} d^{2} - c^{4} d^{3}\right )} \sqrt {-a^{2} c + d} a} + \frac {57 \, {\left (d x^{2} + c\right )}^{2} a^{4} c^{2} + 11 \, {\left (d x^{2} + c\right )} a^{4} c^{3} + 3 \, a^{4} c^{4} - 66 \, {\left (d x^{2} + c\right )}^{2} a^{2} c d - 17 \, {\left (d x^{2} + c\right )} a^{2} c^{2} d - 6 \, a^{2} c^{3} d + 24 \, {\left (d x^{2} + c\right )}^{2} d^{2} + 6 \, {\left (d x^{2} + c\right )} c d^{2} + 3 \, c^{2} d^{2}}{{\left (a^{6} c^{6} - 3 \, a^{4} c^{5} d + 3 \, a^{2} c^{4} d^{2} - c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}\right )} + \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, d^{3} x^{2}}{c^{4}} + \frac {7 \, d^{2}}{c^{3}}\right )} + \frac {35 \, d}{c^{2}}\right )} x^{2} + \frac {35}{c}\right )} x \arctan \left (\frac {1}{a x}\right )}{35 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.20, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \]
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 \[ \int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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