Optimal. Leaf size=145 \[ \frac {1}{2} \left (3 a^2 b B+2 b^3 B+a^3 C+6 a b^2 C\right ) x+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (2 a^2 B+8 b^2 B+9 a b C\right ) \sin (c+d x)}{3 d}+\frac {a^2 (5 b B+3 a C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a B \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.30, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {4157, 4110,
4159, 4132, 8, 4130, 3855} \begin {gather*} \frac {a \left (2 a^2 B+9 a b C+8 b^2 B\right ) \sin (c+d x)}{3 d}+\frac {a^2 (3 a C+5 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} x \left (a^3 C+3 a^2 b B+6 a b^2 C+2 b^3 B\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3855
Rule 4110
Rule 4130
Rule 4132
Rule 4157
Rule 4159
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (B+C \sec (c+d x)) \, dx\\ &=\frac {a B \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (-a (5 b B+3 a C)-\left (2 a^2 B+3 b^2 B+6 a b C\right ) \sec (c+d x)-3 b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 (5 b B+3 a C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a B \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (2 a^2 B+8 b^2 B+9 a b C\right )+3 \left (3 a^2 b B+2 b^3 B+a^3 C+6 a b^2 C\right ) \sec (c+d x)+6 b^3 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 (5 b B+3 a C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a B \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (2 a^2 B+8 b^2 B+9 a b C\right )+6 b^3 C \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (3 a^2 b B+2 b^3 B+a^3 C+6 a b^2 C\right ) \int 1 \, dx\\ &=\frac {1}{2} \left (3 a^2 b B+2 b^3 B+a^3 C+6 a b^2 C\right ) x+\frac {a \left (2 a^2 B+8 b^2 B+9 a b C\right ) \sin (c+d x)}{3 d}+\frac {a^2 (5 b B+3 a C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a B \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}+\left (b^3 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} \left (3 a^2 b B+2 b^3 B+a^3 C+6 a b^2 C\right ) x+\frac {b^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (2 a^2 B+8 b^2 B+9 a b C\right ) \sin (c+d x)}{3 d}+\frac {a^2 (5 b B+3 a C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {a B \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 159, normalized size = 1.10 \begin {gather*} \frac {6 \left (3 a^2 b B+2 b^3 B+a^3 C+6 a b^2 C\right ) (c+d x)-12 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 b^3 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+9 a \left (a^2 B+4 b^2 B+4 a b C\right ) \sin (c+d x)+3 a^2 (3 b B+a C) \sin (2 (c+d x))+a^3 B \sin (3 (c+d x))}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 151, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {b^{3} B \left (d x +c \right )+C \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a \,b^{2} B \sin \left (d x +c \right )+3 C \,b^{2} a \left (d x +c \right )+3 a^{2} b B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b C \sin \left (d x +c \right )+\frac {a^{3} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(151\) |
default | \(\frac {b^{3} B \left (d x +c \right )+C \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a \,b^{2} B \sin \left (d x +c \right )+3 C \,b^{2} a \left (d x +c \right )+3 a^{2} b B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b C \sin \left (d x +c \right )+\frac {a^{3} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(151\) |
risch | \(\frac {3 B \,a^{2} b x}{2}+x \,b^{3} B +\frac {C \,a^{3} x}{2}+3 C a \,b^{2} x -\frac {3 i a^{3} B \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a \,b^{2} B}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{2} b C}{2 d}+\frac {3 i a^{3} B \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a \,b^{2} B}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{2} b C}{2 d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {a^{3} B \sin \left (3 d x +3 c \right )}{12 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b B}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}\) | \(247\) |
norman | \(\frac {\left (\frac {3}{2} a^{2} b B +b^{3} B +\frac {1}{2} a^{3} C +3 C \,b^{2} a \right ) x +\left (\frac {3}{2} a^{2} b B +b^{3} B +\frac {1}{2} a^{3} C +3 C \,b^{2} a \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6 a^{2} b B -4 b^{3} B -2 a^{3} C -12 C \,b^{2} a \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-6 a^{2} b B -4 b^{3} B -2 a^{3} C -12 C \,b^{2} a \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (9 a^{2} b B +6 b^{3} B +3 a^{3} C +18 C \,b^{2} a \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a \left (2 a^{2} B -3 a b B -18 b^{2} B -a^{2} C -18 a b C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (2 a^{2} B -3 a b B +6 b^{2} B -a^{2} C +6 a b C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (2 a^{2} B +3 a b B -18 b^{2} B +a^{2} C -18 a b C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (2 a^{2} B +3 a b B +6 b^{2} B +a^{2} C +6 a b C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (2 a^{2} B -45 a b B +54 b^{2} B -15 a^{2} C +54 a b C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a \left (2 a^{2} B +45 a b B +54 b^{2} B +15 a^{2} C +54 a b C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a \left (14 a^{2} B -27 a b B +18 b^{2} B -9 a^{2} C +18 a b C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {a \left (14 a^{2} B +27 a b B +18 b^{2} B +9 a^{2} C +18 a b C \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} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {C \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {C \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(622\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 152, normalized size = 1.05 \begin {gather*} -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b - 36 \, {\left (d x + c\right )} C a b^{2} - 12 \, {\left (d x + c\right )} B b^{3} - 6 \, C b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, C a^{2} b \sin \left (d x + c\right ) - 36 \, B 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 = 2.88, size = 131, normalized size = 0.90 \begin {gather*} \frac {3 \, C b^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, C b^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (C a^{3} + 3 \, B a^{2} b + 6 \, C a b^{2} + 2 \, B b^{3}\right )} d x + {\left (2 \, B a^{3} \cos \left (d x + c\right )^{2} + 4 \, B a^{3} + 18 \, C a^{2} b + 18 \, B a b^{2} + 3 \, {\left (C a^{3} + 3 \, B a^{2} b\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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.
\(2 (137) = 274\).
time = 0.50, size = 314, normalized size = 2.17 \begin {gather*} \frac {6 \, C b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6 \, C b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3 \, {\left (C a^{3} + 3 \, B a^{2} b + 6 \, C a b^{2} + 2 \, B b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a b^{2} \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 = 5.42, size = 1924, normalized size = 13.27 \begin {gather*} \frac {\left (2\,B\,a^3-C\,a^3+6\,B\,a\,b^2-3\,B\,a^2\,b+6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {4\,B\,a^3}{3}+12\,C\,a^2\,b+12\,B\,a\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B\,a^3+C\,a^3+6\,B\,a\,b^2+3\,B\,a^2\,b+6\,C\,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 (\frac {1{}\mathrm {i}\,C\,a^3}{2}+\frac {3{}\mathrm {i}\,B\,a^2\,b}{2}+3{}\mathrm {i}\,C\,a\,b^2+1{}\mathrm {i}\,B\,b^3\right )\,\left (32\,B\,b^3+16\,C\,a^3+32\,C\,b^3+48\,B\,a^2\,b+96\,C\,a\,b^2\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,B^2\,a^4\,b^2+96\,B^2\,a^2\,b^4+32\,B^2\,b^6+48\,B\,C\,a^5\,b+320\,B\,C\,a^3\,b^3+192\,B\,C\,a\,b^5+8\,C^2\,a^6+96\,C^2\,a^4\,b^2+288\,C^2\,a^2\,b^4+32\,C^2\,b^6\right )\right )\,\left (\frac {1{}\mathrm {i}\,C\,a^3}{2}+\frac {3{}\mathrm {i}\,B\,a^2\,b}{2}+3{}\mathrm {i}\,C\,a\,b^2+1{}\mathrm {i}\,B\,b^3\right )\,1{}\mathrm {i}-\left (\left (\frac {1{}\mathrm {i}\,C\,a^3}{2}+\frac {3{}\mathrm {i}\,B\,a^2\,b}{2}+3{}\mathrm {i}\,C\,a\,b^2+1{}\mathrm {i}\,B\,b^3\right )\,\left (32\,B\,b^3+16\,C\,a^3+32\,C\,b^3+48\,B\,a^2\,b+96\,C\,a\,b^2\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,B^2\,a^4\,b^2+96\,B^2\,a^2\,b^4+32\,B^2\,b^6+48\,B\,C\,a^5\,b+320\,B\,C\,a^3\,b^3+192\,B\,C\,a\,b^5+8\,C^2\,a^6+96\,C^2\,a^4\,b^2+288\,C^2\,a^2\,b^4+32\,C^2\,b^6\right )\right )\,\left (\frac {1{}\mathrm {i}\,C\,a^3}{2}+\frac {3{}\mathrm {i}\,B\,a^2\,b}{2}+3{}\mathrm {i}\,C\,a\,b^2+1{}\mathrm {i}\,B\,b^3\right )\,1{}\mathrm {i}}{\left (\left (\frac {1{}\mathrm {i}\,C\,a^3}{2}+\frac {3{}\mathrm {i}\,B\,a^2\,b}{2}+3{}\mathrm {i}\,C\,a\,b^2+1{}\mathrm {i}\,B\,b^3\right )\,\left (32\,B\,b^3+16\,C\,a^3+32\,C\,b^3+48\,B\,a^2\,b+96\,C\,a\,b^2\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,B^2\,a^4\,b^2+96\,B^2\,a^2\,b^4+32\,B^2\,b^6+48\,B\,C\,a^5\,b+320\,B\,C\,a^3\,b^3+192\,B\,C\,a\,b^5+8\,C^2\,a^6+96\,C^2\,a^4\,b^2+288\,C^2\,a^2\,b^4+32\,C^2\,b^6\right )\right )\,\left (\frac {1{}\mathrm {i}\,C\,a^3}{2}+\frac {3{}\mathrm {i}\,B\,a^2\,b}{2}+3{}\mathrm {i}\,C\,a\,b^2+1{}\mathrm {i}\,B\,b^3\right )+\left (\left (\frac {1{}\mathrm {i}\,C\,a^3}{2}+\frac {3{}\mathrm {i}\,B\,a^2\,b}{2}+3{}\mathrm {i}\,C\,a\,b^2+1{}\mathrm {i}\,B\,b^3\right )\,\left (32\,B\,b^3+16\,C\,a^3+32\,C\,b^3+48\,B\,a^2\,b+96\,C\,a\,b^2\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,B^2\,a^4\,b^2+96\,B^2\,a^2\,b^4+32\,B^2\,b^6+48\,B\,C\,a^5\,b+320\,B\,C\,a^3\,b^3+192\,B\,C\,a\,b^5+8\,C^2\,a^6+96\,C^2\,a^4\,b^2+288\,C^2\,a^2\,b^4+32\,C^2\,b^6\right )\right )\,\left (\frac {1{}\mathrm {i}\,C\,a^3}{2}+\frac {3{}\mathrm {i}\,B\,a^2\,b}{2}+3{}\mathrm {i}\,C\,a\,b^2+1{}\mathrm {i}\,B\,b^3\right )-64\,B\,C^2\,b^9+64\,B^2\,C\,b^9-192\,C^3\,a\,b^8+576\,C^3\,a^2\,b^7-32\,C^3\,a^3\,b^6+192\,C^3\,a^4\,b^5+16\,C^3\,a^6\,b^3+384\,B\,C^2\,a\,b^8-96\,B\,C^2\,a^2\,b^7+640\,B\,C^2\,a^3\,b^6+96\,B\,C^2\,a^5\,b^4+192\,B^2\,C\,a^2\,b^7+144\,B^2\,C\,a^4\,b^5}\right )\,\left (C\,a^3+3\,B\,a^2\,b+6\,C\,a\,b^2+2\,B\,b^3\right )}{d}-\frac {C\,b^3\,\mathrm {atan}\left (\frac {C\,b^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,B^2\,a^4\,b^2+96\,B^2\,a^2\,b^4+32\,B^2\,b^6+48\,B\,C\,a^5\,b+320\,B\,C\,a^3\,b^3+192\,B\,C\,a\,b^5+8\,C^2\,a^6+96\,C^2\,a^4\,b^2+288\,C^2\,a^2\,b^4+32\,C^2\,b^6\right )+C\,b^3\,\left (32\,B\,b^3+16\,C\,a^3+32\,C\,b^3+48\,B\,a^2\,b+96\,C\,a\,b^2\right )\right )\,1{}\mathrm {i}+C\,b^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,B^2\,a^4\,b^2+96\,B^2\,a^2\,b^4+32\,B^2\,b^6+48\,B\,C\,a^5\,b+320\,B\,C\,a^3\,b^3+192\,B\,C\,a\,b^5+8\,C^2\,a^6+96\,C^2\,a^4\,b^2+288\,C^2\,a^2\,b^4+32\,C^2\,b^6\right )-C\,b^3\,\left (32\,B\,b^3+16\,C\,a^3+32\,C\,b^3+48\,B\,a^2\,b+96\,C\,a\,b^2\right )\right )\,1{}\mathrm {i}}{64\,B^2\,C\,b^9-64\,B\,C^2\,b^9-192\,C^3\,a\,b^8+576\,C^3\,a^2\,b^7-32\,C^3\,a^3\,b^6+192\,C^3\,a^4\,b^5+16\,C^3\,a^6\,b^3+C\,b^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,B^2\,a^4\,b^2+96\,B^2\,a^2\,b^4+32\,B^2\,b^6+48\,B\,C\,a^5\,b+320\,B\,C\,a^3\,b^3+192\,B\,C\,a\,b^5+8\,C^2\,a^6+96\,C^2\,a^4\,b^2+288\,C^2\,a^2\,b^4+32\,C^2\,b^6\right )+C\,b^3\,\left (32\,B\,b^3+16\,C\,a^3+32\,C\,b^3+48\,B\,a^2\,b+96\,C\,a\,b^2\right )\right )-C\,b^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (72\,B^2\,a^4\,b^2+96\,B^2\,a^2\,b^4+32\,B^2\,b^6+48\,B\,C\,a^5\,b+320\,B\,C\,a^3\,b^3+192\,B\,C\,a\,b^5+8\,C^2\,a^6+96\,C^2\,a^4\,b^2+288\,C^2\,a^2\,b^4+32\,C^2\,b^6\right )-C\,b^3\,\left (32\,B\,b^3+16\,C\,a^3+32\,C\,b^3+48\,B\,a^2\,b+96\,C\,a\,b^2\right )\right )+384\,B\,C^2\,a\,b^8-96\,B\,C^2\,a^2\,b^7+640\,B\,C^2\,a^3\,b^6+96\,B\,C^2\,a^5\,b^4+192\,B^2\,C\,a^2\,b^7+144\,B^2\,C\,a^4\,b^5}\right )\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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