Optimal. Leaf size=288 \[ -\frac {\sqrt {e} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {\sqrt {e} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {\sqrt {e} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {\sqrt {e} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e} \]
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Rubi [A] time = 0.27, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3543, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\sqrt {e} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {\sqrt {e} \left (a^2+2 a b-b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {\sqrt {e} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {\sqrt {e} \left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3528
Rule 3534
Rule 3543
Rubi steps
\begin {align*} \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2 \, dx &=-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}+\int \sqrt {e \cot (c+d x)} \left (a^2-b^2+2 a b \cot (c+d x)\right ) \, dx\\ &=-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}+\int \frac {-2 a b e+\left (a^2-b^2\right ) e \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx\\ &=-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}+\frac {2 \operatorname {Subst}\left (\int \frac {2 a b e^2-\left (a^2-b^2\right ) e x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac {\left (\left (a^2-2 a b-b^2\right ) e\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 (\left (a^2+2 a b-b^2\right ) e\right ) \operatorname {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac {\left (\left (a^2+2 a b-b^2\right ) \sqrt {e}\right ) \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 )}{2 \sqrt {2} d}-\frac {\left (\left (a^2+2 a b-b^2\right ) \sqrt {e}\right ) \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 )}{2 \sqrt {2} d}-\frac {\left (\left (a^2-2 a b-b^2\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d}-\frac {\left (\left (a^2-2 a b-b^2\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d}\\ &=-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac {\left (a^2+2 a b-b^2\right ) \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (\left (a^2-2 a b-b^2\right ) \sqrt {e}\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 )}{\sqrt {2} d}+\frac {\left (\left (a^2-2 a b-b^2\right ) \sqrt {e}\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 )}{\sqrt {2} d}\\ &=\frac {\left (a^2-2 a b-b^2\right ) \sqrt {e} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \sqrt {e} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {4 a b \sqrt {e \cot (c+d x)}}{d}-\frac {2 b^2 (e \cot (c+d x))^{3/2}}{3 d e}-\frac {\left (a^2+2 a b-b^2\right ) \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}\\ \end {align*}
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Mathematica [C] time = 0.57, size = 220, normalized size = 0.76 \[ -\frac {\sqrt {e \cot (c+d x)} \left (4 \left (a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )+b \left (24 a \sqrt {\cot (c+d x)}+3 \sqrt {2} a \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )-3 \sqrt {2} a \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+6 \sqrt {2} a \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-6 \sqrt {2} a \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )+4 b \cot ^{\frac {3}{2}}(c+d x)\right )\right )}{6 d \sqrt {\cot (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int {\left (b \cot \left (d x + c\right ) + a\right )}^{2} \sqrt {e \cot \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 534, normalized size = 1.85 \[ -\frac {2 b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d e}-\frac {4 a b \sqrt {e \cot \left (d x +c \right )}}{d}+\frac {a b \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}+\frac {a b \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}-\frac {a b \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}-\frac {e \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 ) a^{2}}{4 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {e \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 ) b^{2}}{4 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {e \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{2 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {e \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{2 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {e \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{2 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {e \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{2 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 = 0.45, size = 257, normalized size = 0.89 \[ -\frac {{\left (\frac {6 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {6 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {3 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {3 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {8 \, {\left (6 \, a b e \sqrt {\frac {e}{\tan \left (d x + c\right )}} + b^{2} \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}\right )}}{e^{2}}\right )} e}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 1157, normalized size = 4.02 \[ \text {result too large to display} \]
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 \[ \int \sqrt {e \cot {\left (c + d x \right )}} \left (a + b \cot {\left (c + d x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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