Optimal. Leaf size=186 \[ \frac {2 \sqrt {2} a^3 e^{5/2} \text {ArcTan}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {4 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}+\frac {4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac {4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e} \]
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Rubi [A]
time = 0.22, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3647, 3711,
3609, 3613, 211} \begin {gather*} \frac {2 \sqrt {2} a^3 e^{5/2} \text {ArcTan}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {4 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{7/2}}{9 d e}-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {4 a^3 (e \cot (c+d x))^{5/2}}{5 d}+\frac {4 a^3 e (e \cot (c+d x))^{3/2}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 3609
Rule 3613
Rule 3647
Rule 3711
Rubi steps
\begin {align*} \int (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \, dx &=-\frac {2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac {2 \int (e \cot (c+d x))^{5/2} \left (-a^3 e-9 a^3 e \cot (c+d x)-10 a^3 e \cot ^2(c+d x)\right ) \, dx}{9 e}\\ &=-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac {2 \int (e \cot (c+d x))^{5/2} \left (9 a^3 e-9 a^3 e \cot (c+d x)\right ) \, dx}{9 e}\\ &=-\frac {4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac {2 \int (e \cot (c+d x))^{3/2} \left (9 a^3 e^2+9 a^3 e^2 \cot (c+d x)\right ) \, dx}{9 e}\\ &=\frac {4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac {4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (-9 a^3 e^3+9 a^3 e^3 \cot (c+d x)\right ) \, dx}{9 e}\\ &=\frac {4 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}+\frac {4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac {4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}-\frac {2 \int \frac {-9 a^3 e^4-9 a^3 e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{9 e}\\ &=\frac {4 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}+\frac {4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac {4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}+\frac {\left (36 a^6 e^7\right ) \text {Subst}\left (\int \frac {1}{-162 a^6 e^8-e x^2} \, dx,x,\frac {-9 a^3 e^4+9 a^3 e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=\frac {2 \sqrt {2} a^3 e^{5/2} \tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {4 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}+\frac {4 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac {4 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {40 a^3 (e \cot (c+d x))^{7/2}}{63 d e}-\frac {2 (e \cot (c+d x))^{7/2} \left (a^3+a^3 \cot (c+d x)\right )}{9 d e}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 6.11, size = 729, normalized size = 3.92 \begin {gather*} -\frac {2 \cos ^2(c+d x) (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \sin (c+d x)}{9 d (\cos (c+d x)+\sin (c+d x))^3}-\frac {6 \cos (c+d x) (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \sin ^2(c+d x)}{7 d (\cos (c+d x)+\sin (c+d x))^3}-\frac {4 (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \sin ^3(c+d x)}{5 d (\cos (c+d x)+\sin (c+d x))^3}+\frac {\sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \sin ^3(c+d x)}{d \cot ^{\frac {5}{2}}(c+d x) (\cos (c+d x)+\sin (c+d x))^3}-\frac {\sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \sin ^3(c+d x)}{d \cot ^{\frac {5}{2}}(c+d x) (\cos (c+d x)+\sin (c+d x))^3}+\frac {(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^3(c+d x)}{\sqrt {2} d \cot ^{\frac {5}{2}}(c+d x) (\cos (c+d x)+\sin (c+d x))^3}-\frac {(e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^3(c+d x)}{\sqrt {2} d \cot ^{\frac {5}{2}}(c+d x) (\cos (c+d x)+\sin (c+d x))^3}+\frac {4 (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \sin ^3(c+d x) \tan (c+d x)}{3 d (\cos (c+d x)+\sin (c+d x))^3}-\frac {4 (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right ) \sin ^3(c+d x) \tan (c+d x)}{3 d (\cos (c+d x)+\sin (c+d x))^3}+\frac {4 (e \cot (c+d x))^{5/2} (a+a \cot (c+d x))^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d (\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. \(353\) vs.
\(2(157)=314\).
time = 0.48, size = 354, normalized size = 1.90
method | result | size |
derivativedivides | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {9}{2}}}{9}+\frac {3 e \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {2 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {2 e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 e^{4} \sqrt {e \cot \left (d x +c \right )}+2 e^{5} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(354\) |
default | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {9}{2}}}{9}+\frac {3 e \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {2 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {2 e^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 e^{4} \sqrt {e \cot \left (d x +c \right )}+2 e^{5} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(354\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 127, normalized size = 0.68 \begin {gather*} -\frac {2 \, {\left (315 \, {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right )\right )} a^{3} - \frac {630 \, a^{3}}{\sqrt {\tan \left (d x + c\right )}} - \frac {210 \, a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {126 \, a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}} + \frac {135 \, a^{3}}{\tan \left (d x + c\right )^{\frac {7}{2}}} + \frac {35 \, a^{3}}{\tan \left (d x + c\right )^{\frac {9}{2}}}\right )} e^{\frac {5}{2}}}{315 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.53, size = 258, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left (315 \, {\left (\sqrt {2} a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} e^{\frac {5}{2}} - 2 \, \sqrt {2} a^{3} \cos \left (2 \, d x + 2 \, c\right ) e^{\frac {5}{2}} + \sqrt {2} a^{3} e^{\frac {5}{2}}\right )} \arctan \left (-\frac {{\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 \, {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right ) + {\left (721 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} e^{\frac {5}{2}} - 1330 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) e^{\frac {5}{2}} + 469 \, a^{3} e^{\frac {5}{2}} - 15 \, {\left (23 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) e^{\frac {5}{2}} - 5 \, a^{3} e^{\frac {5}{2}}\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 )}}\right )}}{315 \, {\left (d \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x + 2 \, c\right ) + d\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 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot {\left (c + d x \right )}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.44, size = 177, normalized size = 0.95 \begin {gather*} \frac {4\,a^3\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {4\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d}-\frac {6\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}}{7\,d\,e}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{9/2}}{9\,d\,e^2}+\frac {4\,a^3\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {\sqrt {2}\,a^3\,e^{5/2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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