Optimal. Leaf size=165 \[ \frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{9/2}}+\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3646, 3709,
3610, 3613, 214} \begin {gather*} \frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{9/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{7 d e (e \cot (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 3610
Rule 3613
Rule 3646
Rule 3709
Rubi steps
\begin {align*} \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx &=\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {-8 a^3 e^2-7 a^3 e^2 \cot (c+d x)-a^3 e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{7/2}} \, dx}{7 e^3}\\ &=\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {-7 a^3 e^3+7 a^3 e^3 \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx}{7 e^5}\\ &=\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {7 a^3 e^4+7 a^3 e^4 \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{7 e^7}\\ &=\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int \frac {7 a^3 e^5-7 a^3 e^5 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{7 e^9}\\ &=\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}+\frac {\left (28 a^6 e\right ) \text {Subst}\left (\int \frac {1}{98 a^6 e^{10}-e x^2} \, dx,x,\frac {7 a^3 e^5+7 a^3 e^5 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=\frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{9/2}}+\frac {32 a^3}{35 d e^2 (e \cot (c+d x))^{5/2}}+\frac {4 a^3}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {4 a^3}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{7 d e (e \cot (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 2.12, size = 174, normalized size = 1.05 \begin {gather*} \frac {2 a^3 \cos (c+d x) (1+\cot (c+d x))^3 \left (35 \cos ^2(c+d x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(c+d x)\right )+35 \cos ^2(c+d x) \cot (c+d x) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+5 \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\cot ^2(c+d x)\right ) \sin ^2(c+d x)+\frac {21}{2} \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\cot ^2(c+d x)\right ) \sin (2 (c+d x))\right )}{35 d (e \cot (c+d x))^{9/2} (\cos (c+d x)+\sin (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(337\) vs.
\(2(140)=280\).
time = 0.43, size = 338, normalized size = 2.05
method | result | size |
derivativedivides | \(-\frac {2 a^{3} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}-\frac {e}{7 \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {3}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {2}{e^{2} \sqrt {e \cot \left (d x +c \right )}}-\frac {2}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d \,e^{2}}\) | \(338\) |
default | \(-\frac {2 a^{3} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}-\frac {e}{7 \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {3}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {2}{e^{2} \sqrt {e \cot \left (d x +c \right )}}-\frac {2}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d \,e^{2}}\) | \(338\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 125, normalized size = 0.76 \begin {gather*} \frac {{\left (2 \, {\left (15 \, a^{3} + \frac {63 \, a^{3}}{\tan \left (d x + c\right )} + \frac {70 \, a^{3}}{\tan \left (d x + c\right )^{2}} - \frac {210 \, a^{3}}{\tan \left (d x + c\right )^{3}}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} + 105 \, {\left (\sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3}\right )} e^{\left (-\frac {9}{2}\right )}}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 252 vs.
\(2 (125) = 250\).
time = 3.91, size = 252, normalized size = 1.53 \begin {gather*} \frac {105 \, {\left (\sqrt {2} a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \sqrt {2} a^{3} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt {2} a^{3}\right )} \log \left (-{\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) - \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} + 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) - 2 \, {\left (55 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} + 30 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} + 21 \, {\left (13 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 7 \, a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}}}{105 \, {\left (d \cos \left (2 \, d x + 2 \, c\right )^{2} e^{\frac {9}{2}} + 2 \, d \cos \left (2 \, d x + 2 \, c\right ) e^{\frac {9}{2}} + d e^{\frac {9}{2}}\right )}} \end {gather*}
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 \begin {gather*} a^{3} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx + \int \frac {3 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx + \int \frac {3 \cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx + \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.92, size = 129, normalized size = 0.78 \begin {gather*} \frac {-4\,e\,a^3\,{\mathrm {cot}\left (c+d\,x\right )}^3+\frac {4\,e\,a^3\,{\mathrm {cot}\left (c+d\,x\right )}^2}{3}+\frac {6\,e\,a^3\,\mathrm {cot}\left (c+d\,x\right )}{5}+\frac {2\,e\,a^3}{7}}{d\,e^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}}+\frac {2\,\sqrt {2}\,a^3\,\mathrm {atanh}\left (\frac {32\,\sqrt {2}\,a^6\,d\,e^{9/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{32\,a^6\,d\,e^5+32\,a^6\,d\,e^5\,\mathrm {cot}\left (c+d\,x\right )}\right )}{d\,e^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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