Optimal. Leaf size=289 \[ \frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))} \]
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Rubi [A]
time = 0.71, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2803, 3134,
3080, 3855, 2739, 632, 210} \begin {gather*} \frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2803
Rule 3080
Rule 3134
Rule 3855
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^3(c+d x) \left (2 \left (3 a^2-10 b^2\right )-2 a b \sin (c+d x)-3 \left (a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{6 a^2 b}\\ &=\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (12 \left (a^4-6 a^2 b^2+5 b^4\right )-5 a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (3 a^4-23 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^3 b \left (a^2-b^2\right )}\\ &=-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 b \left (17 a^4-77 a^2 b^2+60 b^4\right )+20 a b^2 \left (a^2-b^2\right ) \sin (c+d x)+12 b \left (a^4-6 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^4 b \left (a^2-b^2\right )}\\ &=\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (6 b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right )+12 a b \left (a^4-6 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^5 b \left (a^2-b^2\right )}\\ &=\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (b \left (9 a^4-29 a^2 b^2+20 b^4\right )\right ) \int \csc (c+d x) \, dx}{2 a^6 \left (a^2-b^2\right )}+\frac {\left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )}\\ &=-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 d}\\ &=-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}-\frac {\left (2 \left (2 a^4-19 a^2 b^2+20 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 d}\\ &=\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 6.17, size = 459, normalized size = 1.59 \begin {gather*} \frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} d}+\frac {\left (2 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{3 a^5 d}+\frac {3 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}+\frac {\left (-9 a^2 b+20 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\left (9 a^2 b-20 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}-\frac {3 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-2 a^2 \sin \left (\frac {1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))^2}+\frac {3 a^2 b \cos (c+d x)-8 b^3 \cos (c+d x)}{2 a^5 d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 360, normalized size = 1.25 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 972 vs.
\(2 (274) = 548\).
time = 0.65, size = 2027, normalized size = 7.01 \begin {gather*} \text {Too large to display} \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 {\cot ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.21, size = 451, normalized size = 1.56 \begin {gather*} \frac {\frac {12 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} + \frac {24 \, {\left (2 \, a^{4} - 19 \, a^{2} b^{2} + 20 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{6}} + \frac {24 \, {\left (5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 18 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 26 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{4} b - 9 \, a^{2} b^{3}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} a^{6}} + \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {198 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 440 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.31, size = 1261, normalized size = 4.36 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^3\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (a^2+4\,b^2\right )}{8\,a^5}+\frac {1}{4\,a^3}-\frac {9\,b^2}{2\,a^5}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (5\,a^4+16\,a^2\,b^2-80\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {29\,a^4}{3}+72\,a^2\,b^2-304\,b^4\right )-\frac {a^4}{3}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {13\,a^4}{3}-\frac {40\,a^2\,b^2}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (156\,a\,b^3-\frac {170\,a^3\,b}{3}\right )-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (-55\,a^4\,b+104\,a^2\,b^3+144\,b^5\right )}{a}+\frac {5\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}}{d\,\left (8\,a^7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,a^7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (16\,a^7+32\,a^5\,b^2\right )+32\,a^6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+32\,a^6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (9\,a^2\,b-20\,b^3\right )}{2\,a^6\,d}-\frac {3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^4\,d}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )\,\left (\frac {2\,a^{10}-28\,a^8\,b^2+40\,a^6\,b^4}{a^{10}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,a^8\,b-76\,a^6\,b^3+80\,a^4\,b^5\right )}{a^9}+\frac {\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^{12}-8\,a^{10}\,b^2\right )}{a^9}\right )\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )}{a^8-a^6\,b^2}\right )\,1{}\mathrm {i}}{a^8-a^6\,b^2}+\frac {\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )\,\left (\frac {2\,a^{10}-28\,a^8\,b^2+40\,a^6\,b^4}{a^{10}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,a^8\,b-76\,a^6\,b^3+80\,a^4\,b^5\right )}{a^9}-\frac {\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^{12}-8\,a^{10}\,b^2\right )}{a^9}\right )\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )}{a^8-a^6\,b^2}\right )\,1{}\mathrm {i}}{a^8-a^6\,b^2}}{\frac {18\,a^6\,b-211\,a^4\,b^3+560\,a^2\,b^5-400\,b^7}{a^{10}}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^6-58\,a^4\,b^2+230\,a^2\,b^4-200\,b^6\right )}{a^9}-\frac {\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )\,\left (\frac {2\,a^{10}-28\,a^8\,b^2+40\,a^6\,b^4}{a^{10}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,a^8\,b-76\,a^6\,b^3+80\,a^4\,b^5\right )}{a^9}+\frac {\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^{12}-8\,a^{10}\,b^2\right )}{a^9}\right )\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )}{a^8-a^6\,b^2}\right )}{a^8-a^6\,b^2}+\frac {\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )\,\left (\frac {2\,a^{10}-28\,a^8\,b^2+40\,a^6\,b^4}{a^{10}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,a^8\,b-76\,a^6\,b^3+80\,a^4\,b^5\right )}{a^9}-\frac {\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^{12}-8\,a^{10}\,b^2\right )}{a^9}\right )\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )}{a^8-a^6\,b^2}\right )}{a^8-a^6\,b^2}}\right )\,\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (a^4-\frac {19\,a^2\,b^2}{2}+10\,b^4\right )\,2{}\mathrm {i}}{d\,\left (a^8-a^6\,b^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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