Optimal. Leaf size=247 \[ -\frac {\sqrt {2} a^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {\sqrt {2} a^2 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {a^2 \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}}-\frac {a^2 \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3623, 12,
3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\sqrt {2} a^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {\sqrt {2} a^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d e^{5/2}}+\frac {a^2 \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d e^{5/2}}-\frac {a^2 \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d e^{5/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rule 3623
Rubi steps
\begin {align*} \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{5/2}} \, dx &=\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {\int \frac {2 a^2 e}{(e \cot (c+d x))^{3/2}} \, dx}{e^2}\\ &=\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {\left (2 a^2\right ) \int \frac {1}{(e \cot (c+d x))^{3/2}} \, dx}{e}\\ &=\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}-\frac {\left (2 a^2\right ) \int \sqrt {e \cot (c+d x)} \, dx}{e^3}\\ &=\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \cot (c+d x)\right )}{d e^2}\\ &=\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2}\\ &=\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2}\\ &=\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}}+\frac {a^2 \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2}+\frac {a^2 \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^2}\\ &=\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {a^2 \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}}-\frac {a^2 \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}}+\frac {\left (\sqrt {2} a^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}-\frac {\left (\sqrt {2} a^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}\\ &=-\frac {\sqrt {2} a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {\sqrt {2} a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2}{3 d e (e \cot (c+d x))^{3/2}}+\frac {4 a^2}{d e^2 \sqrt {e \cot (c+d x)}}+\frac {a^2 \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}}-\frac {a^2 \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 1.35, size = 233, normalized size = 0.94 \begin {gather*} \frac {a^2 \left (48 \cos ^2(c+d x) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+\sin (c+d x) \left (8 \cos (c+d x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(c+d x)\right )+3 \sqrt {2} \cot ^{\frac {5}{2}}(c+d x) \left (2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right ) \sin (c+d x)\right )\right ) (1+\tan (c+d x))^2}{12 d e^2 \sqrt {e \cot (c+d x)} (\cos (c+d x)+\sin (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 174, normalized size = 0.70
method | result | size |
derivativedivides | \(-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{3 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2}{e \sqrt {e \cot \left (d x +c \right )}}\right )}{d e}\) | \(174\) |
default | \(-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{3 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2}{e \sqrt {e \cot \left (d x +c \right )}}\right )}{d e}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 147, normalized size = 0.60 \begin {gather*} \frac {{\left (3 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \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} \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^{2} + 4 \, {\left (a^{2} + \frac {6 \, a^{2}}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}\right )} e^{\left (-\frac {5}{2}\right )}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^{2} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {2 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, 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 = 0.70, size = 99, normalized size = 0.40 \begin {gather*} \frac {4\,a^2\,\mathrm {cot}\left (c+d\,x\right )+\frac {2\,a^2}{3}}{d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}+\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,e^{5/2}}-\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,e^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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