Optimal. Leaf size=222 \[ -\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}-\frac {a^2 \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d \sqrt {e}}+\frac {a^2 \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d \sqrt {e}}+\frac {\sqrt {2} a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}-\frac {\sqrt {2} a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d \sqrt {e}} \]
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Rubi [A] time = 0.20, antiderivative size = 222, 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 = {3543, 12, 16, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}-\frac {a^2 \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d \sqrt {e}}+\frac {a^2 \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{\sqrt {2} d \sqrt {e}}+\frac {\sqrt {2} a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}-\frac {\sqrt {2} a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{d \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3476
Rule 3543
Rubi steps
\begin {align*} \int \frac {(a+a \cot (c+d x))^2}{\sqrt {e \cot (c+d x)}} \, dx &=-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}+\int \frac {2 a^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx\\ &=-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}+\left (2 a^2\right ) \int \frac {\cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx\\ &=-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}+\frac {\left (2 a^2\right ) \int \sqrt {e \cot (c+d x)} \, dx}{e}\\ &=-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \cot (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}-\frac {\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{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}-\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}\\ &=-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{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}-\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}-\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 \sqrt {e}}-\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 \sqrt {e}}\\ &=-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}-\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 \sqrt {e}}+\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 \sqrt {e}}-\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 \sqrt {e}}+\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 \sqrt {e}}\\ &=\frac {\sqrt {2} a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}-\frac {\sqrt {2} a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}-\frac {2 a^2 \sqrt {e \cot (c+d x)}}{d e}-\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 \sqrt {e}}+\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 \sqrt {e}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 53, normalized size = 0.24 \[ -\frac {2 a^2 \sqrt {e \cot (c+d x)} \left (2 \cot (c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )+3\right )}{3 d e} \]
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 \[ \int \frac {{\left (a \cot \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \cot \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 186, normalized size = 0.84 \[ -\frac {2 a^{2} \sqrt {e \cot \left (d x +c \right )}}{d e}-\frac {a^{2} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a^{2} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{d \left (e^{2}\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 193, normalized size = 0.87 \[ -\frac {{\left (\frac {a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}\right )}}{e} + \frac {4 \, a^{2} \sqrt {\frac {e}{\tan \left (d x + c\right )}}}{e^{2}}\right )} e}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 86, normalized size = 0.39 \[ \frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,\sqrt {e}}-\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d\,\sqrt {e}}-\frac {2\,a^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d\,e} \]
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 \[ a^{2} \left (\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {2 \cot {\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {\cot ^{2}{\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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