Optimal. Leaf size=161 \[ \frac {3 \sqrt {e \cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+a^3\right )}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2} \]
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Rubi [A] time = 0.59, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3568, 3649, 3654, 3532, 208, 3634, 63, 205} \[ \frac {3 \sqrt {e \cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+a^3\right )}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a \cot (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 208
Rule 3532
Rule 3568
Rule 3634
Rule 3649
Rule 3654
Rubi steps
\begin {align*} \int \frac {\sqrt {e \cot (c+d x)}}{(a+a \cot (c+d x))^3} \, dx &=\frac {\sqrt {e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}-\frac {\int \frac {-\frac {a e}{2}-2 a e \cot (c+d x)+\frac {3}{2} a e \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2} \, dx}{4 a^2}\\ &=\frac {\sqrt {e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac {3 \sqrt {e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}+\frac {\int \frac {\frac {5 a^3 e^2}{2}-\frac {3}{2} a^3 e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{8 a^5 e}\\ &=\frac {\sqrt {e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac {3 \sqrt {e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}+\frac {\int \frac {4 a^4 e^2-4 a^4 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{16 a^7 e}+\frac {e \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{16 a^2}\\ &=\frac {\sqrt {e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac {3 \sqrt {e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}+\frac {e \operatorname {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{16 a^2 d}-\frac {\left (2 a e^3\right ) \operatorname {Subst}\left (\int \frac {1}{32 a^8 e^4-e x^2} \, dx,x,\frac {4 a^4 e^2+4 a^4 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac {3 \sqrt {e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+\frac {a x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{8 a^2 d}\\ &=-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d}-\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d}+\frac {\sqrt {e \cot (c+d x)}}{4 a d (a+a \cot (c+d x))^2}+\frac {3 \sqrt {e \cot (c+d x)}}{8 d \left (a^3+a^3 \cot (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 181, normalized size = 1.12 \[ -\frac {\sqrt {e \cot (c+d x)} \left (\sqrt {\cot (c+d x)} (-3 \sin (2 (c+d x))+5 \cos (2 (c+d x))-5)+2 (\sin (2 (c+d x))+1) \tan ^{-1}\left (\sqrt {\cot (c+d x)}\right )-2 \sqrt {2} (\sin (c+d x)+\cos (c+d x))^2 \left (\log \left (-\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}-1\right )-\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )}{16 a^3 d \sqrt {\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 518, normalized size = 3.22 \[ \left [\frac {4 \, {\left (\sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \sqrt {-e} \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 {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) + \sqrt {-e} {\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) - 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) - \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (5 \, \cos \left (2 \, d x + 2 \, c\right ) - 3 \, \sin \left (2 \, d x + 2 \, c\right ) - 5\right )}}{16 \, {\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}, -\frac {2 \, \sqrt {e} {\left (\sin \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right ) - 2 \, {\left (\sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \sqrt {e} \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 {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (5 \, \cos \left (2 \, d x + 2 \, c\right ) - 3 \, \sin \left (2 \, d x + 2 \, c\right ) - 5\right )}}{16 \, {\left (a^{3} d \sin \left (2 \, d x + 2 \, c\right ) + a^{3} d\right )}}\right ] \]
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 \[ \int \frac {\sqrt {e \cot \left (d x + c\right )}}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.86, size = 423, normalized size = 2.63 \[ \frac {3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{8 d \,a^{3} \left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {5 e^{2} \sqrt {e \cot \left (d x +c \right )}}{8 d \,a^{3} \left (e \cot \left (d x +c \right )+e \right )^{2}}-\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right ) \sqrt {e}}{8 a^{3} d}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{16 d \,a^{3}}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{8 d \,a^{3}}+\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{8 d \,a^{3}}+\frac {e \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{16 d \,a^{3} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {e \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{8 d \,a^{3} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {e \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{8 d \,a^{3} \left (e^{2}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 190, normalized size = 1.18 \[ \frac {e {\left (\frac {5 \, e \sqrt {\frac {e}{\tan \left (d x + c\right )}} + 3 \, \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}{a^{3} e^{2} + \frac {2 \, a^{3} e^{2}}{\tan \left (d x + c\right )} + \frac {a^{3} e^{2}}{\tan \left (d x + c\right )^{2}}} - \frac {\frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}}{a^{3}} - \frac {\arctan \left (\frac {\sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {e}}\right )}{a^{3} \sqrt {e}}\right )}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 151, normalized size = 0.94 \[ \frac {\frac {3\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{8}+\frac {5\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{8}}{d\,a^3\,e^2\,{\mathrm {cot}\left (c+d\,x\right )}^2+2\,d\,a^3\,e^2\,\mathrm {cot}\left (c+d\,x\right )+d\,a^3\,e^2}-\frac {\sqrt {e}\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{8\,a^3\,d}-\frac {\sqrt {2}\,\sqrt {e}\,\mathrm {atanh}\left (\frac {9\,\sqrt {2}\,e^{17/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{32\,\left (\frac {9\,e^9\,\mathrm {cot}\left (c+d\,x\right )}{32}+\frac {9\,e^9}{32}\right )}\right )}{4\,a^3\,d} \]
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 \[ \frac {\int \frac {\sqrt {e \cot {\left (c + d x \right )}}}{\cot ^{3}{\left (c + d x \right )} + 3 \cot ^{2}{\left (c + d x \right )} + 3 \cot {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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