Optimal. Leaf size=117 \[ \frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d \sqrt {e}}-\frac {16 a^3 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 \sqrt {e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e} \]
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Rubi [A]
time = 0.11, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3647, 3711,
3613, 214} \begin {gather*} -\frac {16 a^3 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}+\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 \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 3613
Rule 3647
Rule 3711
Rubi steps
\begin {align*} \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx &=-\frac {2 \sqrt {e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e}-\frac {2 \int \frac {-a^3 e-3 a^3 e \cot (c+d x)-4 a^3 e \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 e}\\ &=-\frac {16 a^3 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 \sqrt {e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e}-\frac {2 \int \frac {3 a^3 e-3 a^3 e \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 e}\\ &=-\frac {16 a^3 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 \sqrt {e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e}+\frac {\left (12 a^6 e\right ) \text {Subst}\left (\int \frac {1}{18 a^6 e^2-e x^2} \, dx,x,\frac {3 a^3 e+3 a^3 e \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 \sqrt {e}}-\frac {16 a^3 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 \sqrt {e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 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 = 5.46, size = 292, normalized size = 2.50 \begin {gather*} -\frac {a^3 (1+\cot (c+d x))^3 \sin (c+d x) \left (4 \cos ^2(c+d x)+8 \cos ^2(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )+6 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sqrt {\cot (c+d x)} \sin ^2(c+d x)-6 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sqrt {\cot (c+d x)} \sin ^2(c+d x)+3 \sqrt {2} \sqrt {\cot (c+d x)} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)-3 \sqrt {2} \sqrt {\cot (c+d x)} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+18 \sin (2 (c+d x))\right )}{6 d \sqrt {e \cot (c+d x)} (\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. \(308\) vs.
\(2(98)=196\).
time = 0.45, size = 309, normalized size = 2.64
method | result | size |
derivativedivides | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(309\) |
default | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(309\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 96, normalized size = 0.82 \begin {gather*} \frac {{\left (3 \, {\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} - \frac {18 \, a^{3}}{\sqrt {\tan \left (d x + c\right )}} - \frac {2 \, a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}\right )} e^{\left (-\frac {1}{2}\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 168, normalized size = 1.44 \begin {gather*} \frac {{\left (3 \, \sqrt {2} a^{3} \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 ) \sin \left (2 \, d x + 2 \, c\right ) - 2 \, {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 9 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) + a^{3}\right )} \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}}\right )} e^{\left (-\frac {1}{2}\right )}}{3 \, d \sin \left (2 \, d x + 2 \, c\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}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {3 \cot {\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {3 \cot ^{2}{\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {\cot ^{3}{\left (c + d x \right )}}{\sqrt {e \cot {\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 = 0.62, size = 100, normalized size = 0.85 \begin {gather*} \frac {2\,\sqrt {2}\,a^3\,\mathrm {atanh}\left (\frac {32\,\sqrt {2}\,a^6\,\sqrt {e}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{32\,a^6\,e+32\,a^6\,e\,\mathrm {cot}\left (c+d\,x\right )}\right )}{d\,\sqrt {e}}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d\,e^2}-\frac {6\,a^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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