Optimal. Leaf size=222 \[ \frac {a^2 \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d e^{3/2}}-\frac {a^2 \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d e^{3/2}}+\frac {\sqrt {2} a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}-\frac {\sqrt {2} a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d e^{3/2}}+\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}} \]
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Rubi [A] time = 0.21, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3542, 12, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {a^2 \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d e^{3/2}}-\frac {a^2 \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d e^{3/2}}+\frac {\sqrt {2} a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}-\frac {\sqrt {2} a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d e^{3/2}}+\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3476
Rule 3542
Rubi steps
\begin {align*} \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{3/2}} \, dx &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {\int \frac {2 a^2 e}{\sqrt {e \cot (c+d x)}} \, dx}{e^2}\\ &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {\left (2 a^2\right ) \int \frac {1}{\sqrt {e \cot (c+d x)}} \, dx}{e}\\ &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (e^2+x^2\right )} \, dx,x,e \cot (c+d x)\right )}{d}\\ &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e}\\ &=\frac {2 a^2}{d e \sqrt {e \cot (c+d x)}}+\frac {a^2 \operatorname {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^{3/2}}+\frac {a^2 \operatorname {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^{3/2}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e}\\ &=\frac {2 a^2}{d e \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^{3/2}}-\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^{3/2}}-\frac {\left (\sqrt {2} a^2\right ) \operatorname {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^{3/2}}+\frac {\left (\sqrt {2} a^2\right ) \operatorname {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^{3/2}}\\ &=\frac {\sqrt {2} a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}-\frac {\sqrt {2} a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}+\frac {2 a^2}{d e \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^{3/2}}-\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^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.81, size = 236, normalized size = 1.06 \[ \frac {a^2 (\cot (c+d x)+1)^2 \left (3 \sin (c+d x) \left (4 \cos (c+d x) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )+\sqrt {2} \sin (c+d x) \cot ^{\frac {3}{2}}(c+d x) \left (\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )\right )\right )-4 \cos ^2(c+d x) \cot (c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )}{6 d (e \cot (c+d x))^{3/2} (\sin (c+d x)+\cos (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {{\left (a \cot \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 195, normalized size = 0.88 \[ -\frac {a^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \,e^{2}}-\frac {a^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{2}}+\frac {a^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \,e^{2}}+\frac {2 a^{2}}{d e \sqrt {e \cot \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 202, normalized size = 0.91 \[ -\frac {e {\left (\frac {\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{e^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{e^{\frac {3}{2}}} + \frac {\sqrt {2} a^{2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{e^{\frac {3}{2}}} - \frac {\sqrt {2} a^{2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{e^{\frac {3}{2}}}}{e} - \frac {4 \, a^{2}}{e^{2} \sqrt {\frac {e}{\tan \left (d x + c\right )}}}\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 86, normalized size = 0.39 \[ \frac {2\,a^2}{d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d\,e^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d\,e^{3/2}} \]
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 \[ a^{2} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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