Optimal. Leaf size=111 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d e^{3/2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{3/2}}+\frac {2}{a d e \sqrt {e \cot (c+d x)}} \]
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Rubi [A] time = 0.45, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3569, 3653, 3532, 205, 3634, 63} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d e^{3/2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{3/2}}+\frac {2}{a d e \sqrt {e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 3532
Rule 3569
Rule 3634
Rule 3653
Rubi steps
\begin {align*} \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))} \, dx &=\frac {2}{a d e \sqrt {e \cot (c+d x)}}+\frac {2 \int \frac {-\frac {a e^2}{2}-\frac {1}{2} a e^2 \cot (c+d x)-\frac {1}{2} a e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{a e^3}\\ &=\frac {2}{a d e \sqrt {e \cot (c+d x)}}+\frac {\int \frac {-\frac {1}{2} a^2 e^2-\frac {1}{2} a^2 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{a^3 e^3}-\frac {\int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx}{2 e}\\ &=\frac {2}{a d e \sqrt {e \cot (c+d x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{2 d e}-\frac {(a e) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} a^4 e^4-e x^2} \, dx,x,\frac {-\frac {1}{2} a^2 e^2+\frac {1}{2} a^2 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{2 d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{3/2}}+\frac {2}{a d e \sqrt {e \cot (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+\frac {a x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d e^{3/2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d e^{3/2}}+\frac {2}{a d e \sqrt {e \cot (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 2.03, size = 176, normalized size = 1.59 \[ \frac {2 \sin ^4(c+d x) \left (\cot ^4(c+d x)+2 \cot ^2(c+d x)-\sqrt {2} \cot ^{\frac {5}{2}}(c+d x) \csc ^2(2 (c+d x)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )+\sqrt {2} \cot ^{\frac {5}{2}}(c+d x) \csc ^2(2 (c+d x)) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+2 \cot ^{\frac {5}{2}}(c+d x) \csc ^2(2 (c+d x)) \tan ^{-1}\left (\sqrt {\cot (c+d x)}\right )+1\right )}{a d e \sqrt {e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.13, size = 472, normalized size = 4.25 \[ \left [-\frac {\sqrt {2} \sqrt {-e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \log \left (-\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 2 \, \sqrt {-e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) - 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) - 8 \, \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{4 \, {\left (a d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a d e^{2}\right )}}, -\frac {\sqrt {2} \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) - 2 \, \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right ) - 4 \, \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{2 \, {\left (a d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a d e^{2}\right )}}\right ] \]
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 \[ \int \frac {1}{{\left (a \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.81, size = 394, normalized size = 3.55 \[ \frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{a d \,e^{\frac {3}{2}}}+\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{8 d a \,e^{2}}+\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{4 d a \,e^{2}}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{4 d a \,e^{2}}+\frac {\sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{8 d a e \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{4 d a e \left (e^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{4 d a e \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2}{a d e \sqrt {e \cot \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.64, size = 136, normalized size = 1.23 \[ \frac {e {\left (\frac {\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}}}{a e^{2}} + \frac {2 \, \arctan \left (\frac {\sqrt {\frac {e}{\tan \left (d x + c\right )}}}{\sqrt {e}}\right )}{a e^{\frac {5}{2}}} + \frac {4}{a e^{2} \sqrt {\frac {e}{\tan \left (d x + c\right )}}}\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 123, normalized size = 1.11 \[ \frac {2}{a\,d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{a\,d\,e^{3/2}}+\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{4\,a\,d\,e^{3/2}} \]
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 \[ \frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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