Optimal. Leaf size=208 \[ \frac {a \left (7 a^2 c-4 d\right )}{15 c^2 \left (a^2 c-d\right )^2 \sqrt {c+d x^2}}+\frac {a}{15 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{3/2}}-\frac {\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{15 c^3 \left (a^2 c-d\right )^{5/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
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Rubi [A] time = 0.93, antiderivative size = 208, 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, 897, 1261, 208} \[ -\frac {\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{15 c^3 \left (a^2 c-d\right )^{5/2}}+\frac {a \left (7 a^2 c-4 d\right )}{15 c^2 \left (a^2 c-d\right )^2 \sqrt {c+d x^2}}+\frac {a}{15 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 192
Rule 208
Rule 897
Rule 1261
Rule 4913
Rule 6688
Rule 6715
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx &=\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+a \int \frac {\frac {x}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {c+d x^2}}}{1+a^2 x^2} \, dx\\ &=\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{15 c^3 \left (1+a^2 x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx\\ &=\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\left (1+a^2 x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3}\\ &=\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {a \operatorname {Subst}\left (\int \frac {15 c^2+20 c d x+8 d^2 x^2}{\left (1+a^2 x\right ) (c+d x)^{5/2}} \, dx,x,x^2\right )}{30 c^3}\\ &=\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {a \operatorname {Subst}\left (\int \frac {3 c^2+4 c x^2+8 x^4}{x^4 \left (\frac {-a^2 c+d}{d}+\frac {a^2 x^2}{d}\right )} \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 d}\\ &=\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {a \operatorname {Subst}\left (\int \left (\frac {3 c^2 d}{\left (-a^2 c+d\right ) x^4}-\frac {c \left (7 a^2 c-4 d\right ) d}{\left (-a^2 c+d\right )^2 x^2}+\frac {d \left (15 a^4 c^2-20 a^2 c d+8 d^2\right )}{\left (-a^2 c+d\right )^2 \left (-a^2 c+d+a^2 x^2\right )}\right ) \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 d}\\ &=\frac {a}{15 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c-4 d\right )}{15 c^2 \left (a^2 c-d\right )^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {\left (a \left (15 a^4 c^2-20 a^2 c d+8 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a^2 c+d+a^2 x^2} \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 \left (a^2 c-d\right )^2}\\ &=\frac {a}{15 c \left (a^2 c-d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c-4 d\right )}{15 c^2 \left (a^2 c-d\right )^2 \sqrt {c+d x^2}}+\frac {x \cot ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{15 c^3 \left (a^2 c-d\right )^{5/2}}\\ \end {align*}
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Mathematica [C] time = 1.14, size = 345, normalized size = 1.66 \[ -\frac {-\frac {2 a c \left (a^2 c \left (8 c+7 d x^2\right )-d \left (5 c+4 d x^2\right )\right )}{\left (d-a^2 c\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \log \left (\frac {60 a c^3 \left (a^2 c-d\right )^{3/2} \left (\sqrt {a^2 c-d} \sqrt {c+d x^2}+a c-i d x\right )}{(a x+i) \left (15 a^4 c^2-20 a^2 c d+8 d^2\right )}\right )}{\left (a^2 c-d\right )^{5/2}}+\frac {\left (15 a^4 c^2-20 a^2 c d+8 d^2\right ) \log \left (\frac {60 a c^3 \left (a^2 c-d\right )^{3/2} \left (\sqrt {a^2 c-d} \sqrt {c+d x^2}+a c+i d x\right )}{(a x-i) \left (15 a^4 c^2-20 a^2 c d+8 d^2\right )}\right )}{\left (a^2 c-d\right )^{5/2}}-\frac {2 x \cot ^{-1}(a x) \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\left (c+d x^2\right )^{5/2}}}{30 c^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.41, size = 1278, normalized size = 6.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 208, normalized size = 1.00 \[ \frac {1}{15} \, a {\left (\frac {{\left (15 \, a^{4} c^{2} - 20 \, a^{2} c d + 8 \, d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{{\left (a^{4} c^{5} - 2 \, a^{2} c^{4} d + c^{3} d^{2}\right )} \sqrt {-a^{2} c + d} a} + \frac {7 \, {\left (d x^{2} + c\right )} a^{2} c + a^{2} c^{2} - 4 \, {\left (d x^{2} + c\right )} d - c d}{{\left (a^{4} c^{4} - 2 \, a^{2} c^{3} d + c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\right )} + \frac {{\left (4 \, x^{2} {\left (\frac {2 \, d^{2} x^{2}}{c^{3}} + \frac {5 \, d}{c^{2}}\right )} + \frac {15}{c}\right )} x \arctan \left (\frac {1}{a x}\right )}{15 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.18, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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