Optimal. Leaf size=313 \[ -\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}} \]
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Rubi [A]
time = 0.29, antiderivative size = 313, 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 = {3646, 3711,
3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{3/2}}+\frac {(a-b) \left (a^2+4 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 e^{3/2}}-\frac {(a-b) \left (a^2+4 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 e^{3/2}}-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3646
Rule 3711
Rubi steps
\begin {align*} \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx &=\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \int \frac {-2 a^2 b e^2+\frac {1}{2} a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)-\frac {1}{2} b \left (a^2+b^2\right ) e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{e^3}\\ &=-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \int \frac {-\frac {1}{2} b \left (3 a^2-b^2\right ) e^2+\frac {1}{2} a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{e^3}\\ &=-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}-\frac {4 \text {Subst}\left (\int \frac {\frac {1}{2} b \left (3 a^2-b^2\right ) e^3-\frac {1}{2} a \left (a^2-3 b^2\right ) e^2 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e^3}\\ &=-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d e}\\ &=-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 e^{3/2}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 e^{3/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 e}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 e}\\ &=-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 e^{3/2}}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 e^{3/2}}\\ &=-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{3/2}}-\frac {2 b \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 a^2 (a+b \cot (c+d x))}{d e \sqrt {e \cot (c+d x)}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 3.55, size = 193, normalized size = 0.62 \begin {gather*} \frac {2 \left (3 a b^2-b^3 \cot (c+d x)+a \left (a^2-3 b^2\right ) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )-\frac {b \left (-3 a^2+b^2\right ) \sqrt {\cot (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 )}{4 \sqrt {2}}\right )}{d e \sqrt {e \cot (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 332, normalized size = 1.06
method | result | size |
derivativedivides | \(-\frac {2 \left (b^{3} \sqrt {e \cot \left (d x +c \right )}-\frac {a^{3} e}{\sqrt {e \cot \left (d x +c \right )}}-e \left (\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(332\) |
default | \(-\frac {2 \left (b^{3} \sqrt {e \cot \left (d x +c \right )}-\frac {a^{3} e}{\sqrt {e \cot \left (d x +c \right )}}-e \left (\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(332\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 220, normalized size = 0.70 \begin {gather*} \frac {{\left (8 \, a^{3} \sqrt {\tan \left (d x + c\right )} + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \frac {8 \, b^{3}}{\sqrt {\tan \left (d x + c\right )}}\right )} e^{\left (-\frac {3}{2}\right )}}{4 \, 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 \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{3}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{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 = 1.20, size = 1951, normalized size = 6.23 \begin {gather*} \frac {2\,a^3}{d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}-\frac {2\,b^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d\,e^2}-\mathrm {atan}\left (\frac {\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )+\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,1{}\mathrm {i}+\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )-\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,1{}\mathrm {i}}{48\,a\,b^8\,d^2\,e^4+\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )-\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}-16\,a^9\,d^2\,e^4-\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )+\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}+128\,a^3\,b^6\,d^2\,e^4+96\,a^5\,b^4\,d^2\,e^4}\right )\,\sqrt {-\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )+\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,1{}\mathrm {i}+\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )-\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,1{}\mathrm {i}}{48\,a\,b^8\,d^2\,e^4+\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )-\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}-16\,a^9\,d^2\,e^4-\left (\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (16\,a^6\,d^3\,e^5-240\,a^4\,b^2\,d^3\,e^5+240\,a^2\,b^4\,d^3\,e^5-16\,b^6\,d^3\,e^5\right )+\left (32\,b^3\,d^4\,e^7-96\,a^2\,b\,d^4\,e^7\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}+128\,a^3\,b^6\,d^2\,e^4+96\,a^5\,b^4\,d^2\,e^4}\right )\,\sqrt {-\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e^3}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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