Optimal. Leaf size=377 \[ \frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{9/2}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{9/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}} \]
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Rubi [A] time = 0.66, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3565, 3628, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{9/2}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{9/2}}+\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3529
Rule 3534
Rule 3565
Rule 3628
Rubi steps
\begin {align*} \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx &=\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {-8 a^2 b e^2+\frac {7}{2} a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)+\frac {1}{2} b \left (5 a^2-7 b^2\right ) e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{7/2}} \, dx}{7 e^3}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {\frac {7}{2} a \left (a^2-3 b^2\right ) e^3+\frac {7}{2} b \left (3 a^2-b^2\right ) e^3 \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx}{7 e^5}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {\frac {7}{2} b \left (3 a^2-b^2\right ) e^4-\frac {7}{2} a \left (a^2-3 b^2\right ) e^4 \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{7 e^7}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {-\frac {7}{2} a \left (a^2-3 b^2\right ) e^5-\frac {7}{2} b \left (3 a^2-b^2\right ) e^5 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{7 e^9}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {4 \operatorname {Subst}\left (\int \frac {\frac {7}{2} a \left (a^2-3 b^2\right ) e^6+\frac {7}{2} b \left (3 a^2-b^2\right ) e^5 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{7 d e^9}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^4}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^4}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d e^4}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d e^4}\\ &=\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}\\ &=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}+\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.68, size = 116, normalized size = 0.31 \[ \frac {2 \tan ^4(c+d x) \sqrt {e \cot (c+d x)} \left (5 a \left (a^2-3 b^2\right ) \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\cot ^2(c+d x)\right )+b \left (7 \left (3 a^2-b^2\right ) \cot (c+d x) \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\cot ^2(c+d x)\right )+b (15 a+7 b \cot (c+d x))\right )\right )}{35 d e^5} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \cot \left (d x + c\right ) + a\right )}^{3}}{\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 829, normalized size = 2.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 342, normalized size = 0.91 \[ -\frac {e {\left (\frac {105 \, {\left (\frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )}}{e^{5}} - \frac {8 \, {\left (15 \, a^{3} e^{3} + \frac {63 \, a^{2} b e^{3}}{\tan \left (d x + c\right )} - \frac {35 \, {\left (a^{3} - 3 \, a b^{2}\right )} e^{3}}{\tan \left (d x + c\right )^{2}} - \frac {105 \, {\left (3 \, a^{2} b - b^{3}\right )} e^{3}}{\tan \left (d x + c\right )^{3}}\right )}}{e^{5} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {7}{2}}}\right )}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.26, size = 1992, normalized size = 5.28 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{3}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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