Optimal. Leaf size=317 \[ -\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {2 \left (a^2-b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d} \]
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Rubi [A]
time = 0.23, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3624, 3609,
3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {e^{3/2} \left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {e^{3/2} \left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}-\frac {e^{3/2} \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 {e^{3/2} \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 {2 e \left (a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3624
Rubi steps
\begin {align*} \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx &=-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}+\int (e \cot (c+d x))^{3/2} \left (a^2-b^2+2 a b \cot (c+d x)\right ) \, dx\\ &=-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}+\int \sqrt {e \cot (c+d x)} \left (-2 a b e+\left (a^2-b^2\right ) e \cot (c+d x)\right ) \, dx\\ &=-\frac {2 \left (a^2-b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}+\int \frac {-\left (a^2-b^2\right ) e^2-2 a b e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx\\ &=-\frac {2 \left (a^2-b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac {2 \text {Subst}\left (\int \frac {\left (a^2-b^2\right ) e^3+2 a b e^2 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {2 \left (a^2-b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}+\frac {\left (\left (a^2-2 a b-b^2\right ) e^2\right ) \text {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^2\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {2 \left (a^2-b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac {\left (\left (a^2-2 a b-b^2\right ) e^{3/2}\right ) \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 )}{2 \sqrt {2} d}-\frac {\left (\left (a^2-2 a b-b^2\right ) e^{3/2}\right ) \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 )}{2 \sqrt {2} d}+\frac {\left (\left (a^2+2 a b-b^2\right ) e^2\right ) \text {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^2\right ) \text {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 {2 \left (a^2-b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \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 ) e^{3/2} \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 ) e^{3/2}\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 )}{\sqrt {2} d}-\frac {\left (\left (a^2+2 a b-b^2\right ) e^{3/2}\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 )}{\sqrt {2} d}\\ &=-\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \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 ) e^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {2 \left (a^2-b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \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 ) e^{3/2} \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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 2.10, size = 224, normalized size = 0.71 \begin {gather*} -\frac {(e \cot (c+d x))^{3/2} \left (\frac {2}{5} b^2 \cot ^{\frac {5}{2}}(c+d x)-\frac {4}{3} a b \cot ^{\frac {3}{2}}(c+d x) \left (-1+\, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )\right )+\frac {1}{4} \left (a^2-b^2\right ) \left (2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{d \cot ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 360, normalized size = 1.14
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a e b \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+a^{2} e^{2} \sqrt {e \cot \left (d x +c \right )}-b^{2} e^{2} \sqrt {e \cot \left (d x +c \right )}-e^{3} \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\) | \(360\) |
default | \(-\frac {2 \left (\frac {b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a e b \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+a^{2} e^{2} \sqrt {e \cot \left (d x +c \right )}-b^{2} e^{2} \sqrt {e \cot \left (d x +c \right )}-e^{3} \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 211, normalized size = 0.67 \begin {gather*} \frac {{\left (30 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 30 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 15 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 15 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {80 \, a b}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {120 \, {\left (a^{2} - b^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}} - \frac {24 \, b^{2}}{\tan \left (d x + c\right )^{\frac {5}{2}}}\right )} e^{\frac {3}{2}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \cot {\left (c + d x \right )}\right )^{2}\, dx \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 = 2.47, size = 1274, normalized size = 4.02 \begin {gather*} -\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (\frac {2\,a^2\,e}{d}-\frac {2\,b^2\,e}{d}\right )-\frac {2\,b^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e}-\frac {4\,a\,b\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}+\mathrm {atan}\left (\frac {a^4\,e^6\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3\,e^3}{d^2}-\frac {b^4\,e^3\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,e^3\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b\,e^3}{d^2}+\frac {a^2\,b^2\,e^3\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {a^6\,e^8\,16{}\mathrm {i}}{d}-\frac {b^6\,e^8\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5\,e^8}{d}+\frac {32\,a^5\,b\,e^8}{d}+\frac {a^2\,b^4\,e^8\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^8}{d}-\frac {a^4\,b^2\,e^8\,112{}\mathrm {i}}{d}}+\frac {b^4\,e^6\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3\,e^3}{d^2}-\frac {b^4\,e^3\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,e^3\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b\,e^3}{d^2}+\frac {a^2\,b^2\,e^3\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {a^6\,e^8\,16{}\mathrm {i}}{d}-\frac {b^6\,e^8\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5\,e^8}{d}+\frac {32\,a^5\,b\,e^8}{d}+\frac {a^2\,b^4\,e^8\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^8}{d}-\frac {a^4\,b^2\,e^8\,112{}\mathrm {i}}{d}}-\frac {a^2\,b^2\,e^6\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3\,e^3}{d^2}-\frac {b^4\,e^3\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,e^3\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b\,e^3}{d^2}+\frac {a^2\,b^2\,e^3\,3{}\mathrm {i}}{2\,d^2}}\,192{}\mathrm {i}}{\frac {a^6\,e^8\,16{}\mathrm {i}}{d}-\frac {b^6\,e^8\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5\,e^8}{d}+\frac {32\,a^5\,b\,e^8}{d}+\frac {a^2\,b^4\,e^8\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^8}{d}-\frac {a^4\,b^2\,e^8\,112{}\mathrm {i}}{d}}\right )\,\sqrt {-\frac {a^4\,e^3\,1{}\mathrm {i}+4\,a^3\,b\,e^3-a^2\,b^2\,e^3\,6{}\mathrm {i}-4\,a\,b^3\,e^3+b^4\,e^3\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {a^4\,e^6\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,e^3\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,e^3\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3\,e^3}{d^2}-\frac {a^3\,b\,e^3}{d^2}-\frac {a^2\,b^2\,e^3\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {32\,a\,b^5\,e^8}{d}+\frac {b^6\,e^8\,16{}\mathrm {i}}{d}-\frac {a^6\,e^8\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b\,e^8}{d}-\frac {a^2\,b^4\,e^8\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^8}{d}+\frac {a^4\,b^2\,e^8\,112{}\mathrm {i}}{d}}+\frac {b^4\,e^6\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,e^3\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,e^3\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3\,e^3}{d^2}-\frac {a^3\,b\,e^3}{d^2}-\frac {a^2\,b^2\,e^3\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {32\,a\,b^5\,e^8}{d}+\frac {b^6\,e^8\,16{}\mathrm {i}}{d}-\frac {a^6\,e^8\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b\,e^8}{d}-\frac {a^2\,b^4\,e^8\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^8}{d}+\frac {a^4\,b^2\,e^8\,112{}\mathrm {i}}{d}}-\frac {a^2\,b^2\,e^6\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,e^3\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,e^3\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3\,e^3}{d^2}-\frac {a^3\,b\,e^3}{d^2}-\frac {a^2\,b^2\,e^3\,3{}\mathrm {i}}{2\,d^2}}\,192{}\mathrm {i}}{\frac {32\,a\,b^5\,e^8}{d}+\frac {b^6\,e^8\,16{}\mathrm {i}}{d}-\frac {a^6\,e^8\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b\,e^8}{d}-\frac {a^2\,b^4\,e^8\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3\,e^8}{d}+\frac {a^4\,b^2\,e^8\,112{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a^4\,e^3\,1{}\mathrm {i}-4\,a^3\,b\,e^3-a^2\,b^2\,e^3\,6{}\mathrm {i}+4\,a\,b^3\,e^3+b^4\,e^3\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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