Optimal. Leaf size=99 \[ -\frac {\sqrt {2} a \tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac {2 a}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3529, 3532, 205} \[ \frac {2 a}{d e^2 \sqrt {e \cot (c+d x)}}-\frac {\sqrt {2} a \tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 3529
Rule 3532
Rubi steps
\begin {align*} \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{5/2}} \, dx &=\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}+\frac {\int \frac {a e-a e \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx}{e^2}\\ &=\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}+\frac {2 a}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {\int \frac {-a e^2-a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{e^4}\\ &=\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}+\frac {2 a}{d e^2 \sqrt {e \cot (c+d x)}}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2 a^2 e^4-e x^2} \, dx,x,\frac {-a e^2+a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=-\frac {\sqrt {2} a \tan ^{-1}\left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{5/2}}+\frac {2 a}{3 d e (e \cot (c+d x))^{3/2}}+\frac {2 a}{d e^2 \sqrt {e \cot (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.42, size = 203, normalized size = 2.05 \[ \frac {a \left (-8 \tan ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(c+d x)\right )+6 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-6 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )+8 \tan ^{\frac {3}{2}}(c+d x)+24 \sqrt {\tan (c+d x)}+3 \sqrt {2} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-3 \sqrt {2} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{12 d \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 358, normalized size = 3.62 \[ \left [\frac {3 \, \sqrt {2} {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {-\frac {1}{e}} \log \left (\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) - 4 \, {\left (a \cos \left (2 \, d x + 2 \, c\right ) - 3 \, a \sin \left (2 \, d x + 2 \, c\right ) - a\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )}}, -\frac {\frac {3 \, \sqrt {2} {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + a e\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right )}{\sqrt {e}} + 2 \, {\left (a \cos \left (2 \, d x + 2 \, c\right ) - 3 \, a \sin \left (2 \, d x + 2 \, c\right ) - a\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, {\left (d e^{3} \cos \left (2 \, d x + 2 \, c\right ) + d e^{3}\right )}}\right ] \]
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 \[ \int \frac {a \cot \left (d x + c\right ) + a}{\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 374, normalized size = 3.78 \[ \frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d \,e^{3}}+\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \,e^{3}}-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \,e^{3}}+\frac {a \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \,e^{2} \left (e^{2}\right )^{\frac {1}{4}}}+\frac {2 a}{d \,e^{2} \sqrt {e \cot \left (d x +c \right )}}+\frac {2 a}{3 d e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 123, normalized size = 1.24 \[ \frac {e {\left (\frac {3 \, a {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}}\right )}}{e^{3}} + \frac {2 \, {\left (a e + \frac {3 \, a e}{\tan \left (d x + c\right )}\right )}}{e^{3} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 103, normalized size = 1.04 \[ \frac {2\,a}{d\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {2\,a}{3\,d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1-\mathrm {i}\right )}{d\,e^{5/2}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (-1-\mathrm {i}\right )}{d\,e^{5/2}} \]
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 \[ a \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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