Optimal. Leaf size=137 \[ \frac {1}{2} \left (6 a^2 A b+A b^3+2 a^3 B+3 a b^2 B\right ) x+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \left (9 a A b+8 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 d}+\frac {b^2 (3 A b+5 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {b B (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.22, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3069, 3112,
3102, 2814, 3855} \begin {gather*} \frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \left (8 a^2 B+9 a A b+2 b^2 B\right ) \sin (c+d x)}{3 d}+\frac {1}{2} x \left (2 a^3 B+6 a^2 A b+3 a b^2 B+A b^3\right )+\frac {b^2 (5 a B+3 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {b B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2814
Rule 3069
Rule 3102
Rule 3112
Rule 3855
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx &=\frac {b B (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} \int (a+b \cos (c+d x)) \left (3 a^2 A+\left (6 a A b+3 a^2 B+2 b^2 B\right ) \cos (c+d x)+b (3 A b+5 a B) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {b^2 (3 A b+5 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {b B (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^3 A+3 \left (6 a^2 A b+A b^3+2 a^3 B+3 a b^2 B\right ) \cos (c+d x)+2 b \left (9 a A b+8 a^2 B+2 b^2 B\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {b \left (9 a A b+8 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 d}+\frac {b^2 (3 A b+5 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {b B (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \left (6 a^3 A+3 \left (6 a^2 A b+A b^3+2 a^3 B+3 a b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{2} \left (6 a^2 A b+A b^3+2 a^3 B+3 a b^2 B\right ) x+\frac {b \left (9 a A b+8 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 d}+\frac {b^2 (3 A b+5 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {b B (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\left (a^3 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} \left (6 a^2 A b+A b^3+2 a^3 B+3 a b^2 B\right ) x+\frac {a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \left (9 a A b+8 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 d}+\frac {b^2 (3 A b+5 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {b B (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 159, normalized size = 1.16 \begin {gather*} \frac {6 \left (6 a^2 A b+A b^3+2 a^3 B+3 a b^2 B\right ) (c+d x)-12 a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 b \left (4 a A b+4 a^2 B+b^2 B\right ) \sin (c+d x)+3 b^2 (A b+3 a B) \sin (2 (c+d x))+b^3 B \sin (3 (c+d x))}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 151, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} B \left (d x +c \right )+3 A \,a^{2} b \left (d x +c \right )+3 a^{2} b B \sin \left (d x +c \right )+3 A a \,b^{2} \sin \left (d x +c \right )+3 B a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {b^{3} B \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(151\) |
default | \(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} B \left (d x +c \right )+3 A \,a^{2} b \left (d x +c \right )+3 a^{2} b B \sin \left (d x +c \right )+3 A a \,b^{2} \sin \left (d x +c \right )+3 B a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {b^{3} B \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(151\) |
risch | \(3 x A \,a^{2} b +\frac {x A \,b^{3}}{2}+a^{3} B x +\frac {3 x B a \,b^{2}}{2}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A a \,b^{2}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{2} b B}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} b^{3} B}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A a \,b^{2}}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} b B}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} b^{3} B}{8 d}+\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (3 d x +3 c \right ) b^{3} B}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{3}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B a \,b^{2}}{4 d}\) | \(247\) |
norman | \(\frac {\left (3 A \,a^{2} b +\frac {1}{2} A \,b^{3}+a^{3} B +\frac {3}{2} B a \,b^{2}\right ) x +\left (3 A \,a^{2} b +\frac {1}{2} A \,b^{3}+a^{3} B +\frac {3}{2} B a \,b^{2}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (12 A \,a^{2} b +2 A \,b^{3}+4 a^{3} B +6 B a \,b^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (12 A \,a^{2} b +2 A \,b^{3}+4 a^{3} B +6 B a \,b^{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (18 A \,a^{2} b +3 A \,b^{3}+6 a^{3} B +9 B a \,b^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (6 A a b -A \,b^{2}+6 B \,a^{2}-3 B a b +2 B \,b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {b \left (6 A a b +A \,b^{2}+6 B \,a^{2}+3 B a b +2 B \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {b \left (54 A a b -3 A \,b^{2}+54 B \,a^{2}-9 B a b +10 B \,b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {b \left (54 A a b +3 A \,b^{2}+54 B \,a^{2}+9 B a b +10 B \,b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(426\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 145, normalized size = 1.06 \begin {gather*} \frac {12 \, {\left (d x + c\right )} B a^{3} + 36 \, {\left (d x + c\right )} A a^{2} b + 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{2} + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{3} + 12 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 36 \, B a^{2} b \sin \left (d x + c\right ) + 36 \, A a b^{2} \sin \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 131, normalized size = 0.96 \begin {gather*} \frac {3 \, A a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, A a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (2 \, B a^{3} + 6 \, A a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d x + {\left (2 \, B b^{3} \cos \left (d x + c\right )^{2} + 18 \, B a^{2} b + 18 \, A a b^{2} + 4 \, B b^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \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 (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{3} \sec {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs.
\(2 (129) = 258\).
time = 0.48, size = 314, normalized size = 2.29 \begin {gather*} \frac {6 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (2 \, B a^{3} + 6 \, A a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.91, size = 1924, normalized size = 14.04 \begin {gather*} \frac {\left (2\,B\,b^3-A\,b^3+6\,A\,a\,b^2-3\,B\,a\,b^2+6\,B\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (12\,B\,a^2\,b+12\,A\,a\,b^2+\frac {4\,B\,b^3}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A\,b^3+2\,B\,b^3+6\,A\,a\,b^2+3\,B\,a\,b^2+6\,B\,a^2\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atan}\left (\frac {\left (\left (1{}\mathrm {i}\,B\,a^3+3{}\mathrm {i}\,A\,a^2\,b+\frac {3{}\mathrm {i}\,B\,a\,b^2}{2}+\frac {1{}\mathrm {i}\,A\,b^3}{2}\right )\,\left (32\,A\,a^3+16\,A\,b^3+32\,B\,a^3+96\,A\,a^2\,b+48\,B\,a\,b^2\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,A^2\,a^6+288\,A^2\,a^4\,b^2+96\,A^2\,a^2\,b^4+8\,A^2\,b^6+192\,A\,B\,a^5\,b+320\,A\,B\,a^3\,b^3+48\,A\,B\,a\,b^5+32\,B^2\,a^6+96\,B^2\,a^4\,b^2+72\,B^2\,a^2\,b^4\right )\right )\,\left (1{}\mathrm {i}\,B\,a^3+3{}\mathrm {i}\,A\,a^2\,b+\frac {3{}\mathrm {i}\,B\,a\,b^2}{2}+\frac {1{}\mathrm {i}\,A\,b^3}{2}\right )\,1{}\mathrm {i}-\left (\left (1{}\mathrm {i}\,B\,a^3+3{}\mathrm {i}\,A\,a^2\,b+\frac {3{}\mathrm {i}\,B\,a\,b^2}{2}+\frac {1{}\mathrm {i}\,A\,b^3}{2}\right )\,\left (32\,A\,a^3+16\,A\,b^3+32\,B\,a^3+96\,A\,a^2\,b+48\,B\,a\,b^2\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,A^2\,a^6+288\,A^2\,a^4\,b^2+96\,A^2\,a^2\,b^4+8\,A^2\,b^6+192\,A\,B\,a^5\,b+320\,A\,B\,a^3\,b^3+48\,A\,B\,a\,b^5+32\,B^2\,a^6+96\,B^2\,a^4\,b^2+72\,B^2\,a^2\,b^4\right )\right )\,\left (1{}\mathrm {i}\,B\,a^3+3{}\mathrm {i}\,A\,a^2\,b+\frac {3{}\mathrm {i}\,B\,a\,b^2}{2}+\frac {1{}\mathrm {i}\,A\,b^3}{2}\right )\,1{}\mathrm {i}}{\left (\left (1{}\mathrm {i}\,B\,a^3+3{}\mathrm {i}\,A\,a^2\,b+\frac {3{}\mathrm {i}\,B\,a\,b^2}{2}+\frac {1{}\mathrm {i}\,A\,b^3}{2}\right )\,\left (32\,A\,a^3+16\,A\,b^3+32\,B\,a^3+96\,A\,a^2\,b+48\,B\,a\,b^2\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,A^2\,a^6+288\,A^2\,a^4\,b^2+96\,A^2\,a^2\,b^4+8\,A^2\,b^6+192\,A\,B\,a^5\,b+320\,A\,B\,a^3\,b^3+48\,A\,B\,a\,b^5+32\,B^2\,a^6+96\,B^2\,a^4\,b^2+72\,B^2\,a^2\,b^4\right )\right )\,\left (1{}\mathrm {i}\,B\,a^3+3{}\mathrm {i}\,A\,a^2\,b+\frac {3{}\mathrm {i}\,B\,a\,b^2}{2}+\frac {1{}\mathrm {i}\,A\,b^3}{2}\right )+\left (\left (1{}\mathrm {i}\,B\,a^3+3{}\mathrm {i}\,A\,a^2\,b+\frac {3{}\mathrm {i}\,B\,a\,b^2}{2}+\frac {1{}\mathrm {i}\,A\,b^3}{2}\right )\,\left (32\,A\,a^3+16\,A\,b^3+32\,B\,a^3+96\,A\,a^2\,b+48\,B\,a\,b^2\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,A^2\,a^6+288\,A^2\,a^4\,b^2+96\,A^2\,a^2\,b^4+8\,A^2\,b^6+192\,A\,B\,a^5\,b+320\,A\,B\,a^3\,b^3+48\,A\,B\,a\,b^5+32\,B^2\,a^6+96\,B^2\,a^4\,b^2+72\,B^2\,a^2\,b^4\right )\right )\,\left (1{}\mathrm {i}\,B\,a^3+3{}\mathrm {i}\,A\,a^2\,b+\frac {3{}\mathrm {i}\,B\,a\,b^2}{2}+\frac {1{}\mathrm {i}\,A\,b^3}{2}\right )+64\,A\,B^2\,a^9-64\,A^2\,B\,a^9-192\,A^3\,a^8\,b+16\,A^3\,a^3\,b^6+192\,A^3\,a^5\,b^4-32\,A^3\,a^6\,b^3+576\,A^3\,a^7\,b^2+384\,A^2\,B\,a^8\,b+144\,A\,B^2\,a^5\,b^4+192\,A\,B^2\,a^7\,b^2+96\,A^2\,B\,a^4\,b^5+640\,A^2\,B\,a^6\,b^3-96\,A^2\,B\,a^7\,b^2}\right )\,\left (2\,B\,a^3+6\,A\,a^2\,b+3\,B\,a\,b^2+A\,b^3\right )}{d}-\frac {A\,a^3\,\mathrm {atan}\left (\frac {A\,a^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,A^2\,a^6+288\,A^2\,a^4\,b^2+96\,A^2\,a^2\,b^4+8\,A^2\,b^6+192\,A\,B\,a^5\,b+320\,A\,B\,a^3\,b^3+48\,A\,B\,a\,b^5+32\,B^2\,a^6+96\,B^2\,a^4\,b^2+72\,B^2\,a^2\,b^4\right )+A\,a^3\,\left (32\,A\,a^3+16\,A\,b^3+32\,B\,a^3+96\,A\,a^2\,b+48\,B\,a\,b^2\right )\right )\,1{}\mathrm {i}+A\,a^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,A^2\,a^6+288\,A^2\,a^4\,b^2+96\,A^2\,a^2\,b^4+8\,A^2\,b^6+192\,A\,B\,a^5\,b+320\,A\,B\,a^3\,b^3+48\,A\,B\,a\,b^5+32\,B^2\,a^6+96\,B^2\,a^4\,b^2+72\,B^2\,a^2\,b^4\right )-A\,a^3\,\left (32\,A\,a^3+16\,A\,b^3+32\,B\,a^3+96\,A\,a^2\,b+48\,B\,a\,b^2\right )\right )\,1{}\mathrm {i}}{64\,A\,B^2\,a^9-64\,A^2\,B\,a^9-192\,A^3\,a^8\,b+A\,a^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,A^2\,a^6+288\,A^2\,a^4\,b^2+96\,A^2\,a^2\,b^4+8\,A^2\,b^6+192\,A\,B\,a^5\,b+320\,A\,B\,a^3\,b^3+48\,A\,B\,a\,b^5+32\,B^2\,a^6+96\,B^2\,a^4\,b^2+72\,B^2\,a^2\,b^4\right )+A\,a^3\,\left (32\,A\,a^3+16\,A\,b^3+32\,B\,a^3+96\,A\,a^2\,b+48\,B\,a\,b^2\right )\right )-A\,a^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,A^2\,a^6+288\,A^2\,a^4\,b^2+96\,A^2\,a^2\,b^4+8\,A^2\,b^6+192\,A\,B\,a^5\,b+320\,A\,B\,a^3\,b^3+48\,A\,B\,a\,b^5+32\,B^2\,a^6+96\,B^2\,a^4\,b^2+72\,B^2\,a^2\,b^4\right )-A\,a^3\,\left (32\,A\,a^3+16\,A\,b^3+32\,B\,a^3+96\,A\,a^2\,b+48\,B\,a\,b^2\right )\right )+16\,A^3\,a^3\,b^6+192\,A^3\,a^5\,b^4-32\,A^3\,a^6\,b^3+576\,A^3\,a^7\,b^2+384\,A^2\,B\,a^8\,b+144\,A\,B^2\,a^5\,b^4+192\,A\,B^2\,a^7\,b^2+96\,A^2\,B\,a^4\,b^5+640\,A^2\,B\,a^6\,b^3-96\,A^2\,B\,a^7\,b^2}\right )\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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