Optimal. Leaf size=217 \[ \frac {1}{8} a^4 (35 A+48 B+52 C) x+\frac {a^4 (B+4 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac {a (A+B) \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+8 B+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A+32 B-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \]
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Rubi [A]
time = 0.44, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {4171, 4102,
4103, 4081, 3855} \begin {gather*} \frac {5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}-\frac {(35 A+32 B-12 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (35 A+48 B+52 C)+\frac {a^4 (B+4 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(7 A+8 B+4 C) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{8 d}+\frac {a (A+B) \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 4081
Rule 4102
Rule 4103
Rule 4171
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 (4 a (A+B)-a (A-4 C) \sec (c+d x)) \, dx}{4 a}\\ &=\frac {a (A+B) \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \left (3 a^2 (7 A+8 B+4 C)-a^2 (7 A+4 B-12 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac {a (A+B) \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+8 B+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^2 \left (2 a^3 (35 A+44 B+36 C)-a^3 (35 A+32 B-12 C) \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {a (A+B) \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+8 B+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A+32 B-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^4 (7 A+8 B+4 C)+24 a^4 (B+4 C) \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac {a (A+B) \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+8 B+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A+32 B-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}-\frac {\int \left (-3 a^5 (35 A+48 B+52 C)-24 a^5 (B+4 C) \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {1}{8} a^4 (35 A+48 B+52 C) x+\frac {5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac {a (A+B) \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+8 B+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A+32 B-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\left (a^4 (B+4 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (35 A+48 B+52 C) x+\frac {a^4 (B+4 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac {a (A+B) \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+8 B+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A+32 B-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1436\) vs. \(2(217)=434\).
time = 6.25, size = 1436, normalized size = 6.62 \begin {gather*} a^4 \left (\frac {(35 A+48 B+52 C) x \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{64 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(-B-4 C) \cos ^2(c+d x) (1+\cos (c+d x))^4 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{8 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(B+4 C) \cos ^2(c+d x) (1+\cos (c+d x))^4 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{8 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(28 A+27 B+16 C) \cos (d x) \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(7 A+4 B+C) \cos (2 d x) \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (2 c)}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(4 A+B) \cos (3 d x) \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (3 c)}{96 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {A \cos (4 d x) \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (4 c)}{256 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(28 A+27 B+16 C) \cos (c) \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (d x)}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(7 A+4 B+C) \cos (2 c) \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (2 d x)}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(4 A+B) \cos (3 c) \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (3 d x)}{96 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {A \cos (4 c) \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (4 d x)}{256 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {C \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin \left (\frac {d x}{2}\right )}{8 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {C \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin \left (\frac {d x}{2}\right )}{8 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.89, size = 289, normalized size = 1.33
method | result | size |
derivativedivides | \(\frac {A \,a^{4} \left (d x +c \right )+a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \tan \left (d x +c \right )+4 A \,a^{4} \sin \left (d x +c \right )+4 a^{4} B \left (d x +c \right )+4 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} B \sin \left (d x +c \right )+6 a^{4} C \left (d x +c \right )+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \sin \left (d x +c \right )+A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(289\) |
default | \(\frac {A \,a^{4} \left (d x +c \right )+a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \tan \left (d x +c \right )+4 A \,a^{4} \sin \left (d x +c \right )+4 a^{4} B \left (d x +c \right )+4 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} B \sin \left (d x +c \right )+6 a^{4} C \left (d x +c \right )+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \sin \left (d x +c \right )+A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(289\) |
risch | \(\frac {35 a^{4} A x}{8}+6 a^{4} x B +\frac {13 a^{4} x C}{2}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} B}{2 d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} a^{4} B}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} C}{8 d}-\frac {7 i A \,a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {7 i A \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {2 i a^{4} C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} C}{8 d}+\frac {7 i A \,a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} B}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} B}{2 d}+\frac {7 i A \,a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {A \,a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {A \,a^{4} \sin \left (3 d x +3 c \right )}{3 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} B}{12 d}\) | \(415\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 290, normalized size = 1.34 \begin {gather*} -\frac {128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 96 \, {\left (d x + c\right )} A a^{4} + 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 384 \, {\left (d x + c\right )} B a^{4} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 576 \, {\left (d x + c\right )} C a^{4} - 48 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 192 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 384 \, A a^{4} \sin \left (d x + c\right ) - 576 \, B a^{4} \sin \left (d x + c\right ) - 384 \, C a^{4} \sin \left (d x + c\right ) - 96 \, C a^{4} \tan \left (d x + c\right )}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.04, size = 179, normalized size = 0.82 \begin {gather*} \frac {3 \, {\left (35 \, A + 48 \, B + 52 \, C\right )} a^{4} d x \cos \left (d x + c\right ) + 12 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{4} + 8 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (27 \, A + 16 \, B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, {\left (5 \, A + 5 \, B + 3 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, C a^{4}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \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 [A]
time = 0.53, size = 332, normalized size = 1.53 \begin {gather*} -\frac {\frac {48 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - 3 \, {\left (35 \, A a^{4} + 48 \, B a^{4} + 52 \, C a^{4}\right )} {\left (d x + c\right )} - 24 \, {\left (B a^{4} + 4 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 24 \, {\left (B a^{4} + 4 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 385 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 424 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 276 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 511 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 520 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 300 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 279 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 216 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 108 \, C a^{4} \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 )}^{4}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.60, size = 1244, normalized size = 5.73 \begin {gather*} -\frac {\left (\frac {35\,A\,a^4}{4}+10\,B\,a^4+5\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {70\,A\,a^4}{3}+\frac {76\,B\,a^4}{3}+8\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {21\,A\,a^4}{2}+8\,B\,a^4-10\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {58\,A\,a^4}{3}-\frac {76\,B\,a^4}{3}-24\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-\frac {93\,A\,a^4}{4}-18\,B\,a^4-11\,C\,a^4\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 )}^{10}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,{\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 {a^4\,\mathrm {atan}\left (\frac {\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1225\,A^2\,a^8}{2}+1680\,A\,B\,a^8+1820\,A\,C\,a^8+1184\,B^2\,a^8+2752\,B\,C\,a^8+1864\,C^2\,a^8\right )-\frac {a^4\,\left (35\,A+48\,B+52\,C\right )\,\left (140\,A\,a^4+224\,B\,a^4+336\,C\,a^4\right )\,1{}\mathrm {i}}{8}\right )\,\left (35\,A+48\,B+52\,C\right )}{8}+\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1225\,A^2\,a^8}{2}+1680\,A\,B\,a^8+1820\,A\,C\,a^8+1184\,B^2\,a^8+2752\,B\,C\,a^8+1864\,C^2\,a^8\right )+\frac {a^4\,\left (35\,A+48\,B+52\,C\right )\,\left (140\,A\,a^4+224\,B\,a^4+336\,C\,a^4\right )\,1{}\mathrm {i}}{8}\right )\,\left (35\,A+48\,B+52\,C\right )}{8}}{1920\,B^3\,a^{12}+4160\,C^3\,a^{12}+3080\,A\,B^2\,a^{12}+1225\,A^2\,B\,a^{12}+10080\,A\,C^2\,a^{12}+4900\,A^2\,C\,a^{12}+13200\,B\,C^2\,a^{12}+10720\,B^2\,C\,a^{12}+14840\,A\,B\,C\,a^{12}-\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1225\,A^2\,a^8}{2}+1680\,A\,B\,a^8+1820\,A\,C\,a^8+1184\,B^2\,a^8+2752\,B\,C\,a^8+1864\,C^2\,a^8\right )-\frac {a^4\,\left (35\,A+48\,B+52\,C\right )\,\left (140\,A\,a^4+224\,B\,a^4+336\,C\,a^4\right )\,1{}\mathrm {i}}{8}\right )\,\left (35\,A+48\,B+52\,C\right )\,1{}\mathrm {i}}{8}+\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1225\,A^2\,a^8}{2}+1680\,A\,B\,a^8+1820\,A\,C\,a^8+1184\,B^2\,a^8+2752\,B\,C\,a^8+1864\,C^2\,a^8\right )+\frac {a^4\,\left (35\,A+48\,B+52\,C\right )\,\left (140\,A\,a^4+224\,B\,a^4+336\,C\,a^4\right )\,1{}\mathrm {i}}{8}\right )\,\left (35\,A+48\,B+52\,C\right )\,1{}\mathrm {i}}{8}}\right )\,\left (35\,A+48\,B+52\,C\right )}{4\,d}-\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1225\,A^2\,a^8}{2}+1680\,A\,B\,a^8+1820\,A\,C\,a^8+1184\,B^2\,a^8+2752\,B\,C\,a^8+1864\,C^2\,a^8\right )+a^4\,\left (B+4\,C\right )\,\left (140\,A\,a^4+224\,B\,a^4+336\,C\,a^4\right )\right )\,\left (B+4\,C\right )\,1{}\mathrm {i}+a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1225\,A^2\,a^8}{2}+1680\,A\,B\,a^8+1820\,A\,C\,a^8+1184\,B^2\,a^8+2752\,B\,C\,a^8+1864\,C^2\,a^8\right )-a^4\,\left (B+4\,C\right )\,\left (140\,A\,a^4+224\,B\,a^4+336\,C\,a^4\right )\right )\,\left (B+4\,C\right )\,1{}\mathrm {i}}{1920\,B^3\,a^{12}+4160\,C^3\,a^{12}+3080\,A\,B^2\,a^{12}+1225\,A^2\,B\,a^{12}+10080\,A\,C^2\,a^{12}+4900\,A^2\,C\,a^{12}+13200\,B\,C^2\,a^{12}+10720\,B^2\,C\,a^{12}+a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1225\,A^2\,a^8}{2}+1680\,A\,B\,a^8+1820\,A\,C\,a^8+1184\,B^2\,a^8+2752\,B\,C\,a^8+1864\,C^2\,a^8\right )+a^4\,\left (B+4\,C\right )\,\left (140\,A\,a^4+224\,B\,a^4+336\,C\,a^4\right )\right )\,\left (B+4\,C\right )-a^4\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1225\,A^2\,a^8}{2}+1680\,A\,B\,a^8+1820\,A\,C\,a^8+1184\,B^2\,a^8+2752\,B\,C\,a^8+1864\,C^2\,a^8\right )-a^4\,\left (B+4\,C\right )\,\left (140\,A\,a^4+224\,B\,a^4+336\,C\,a^4\right )\right )\,\left (B+4\,C\right )+14840\,A\,B\,C\,a^{12}}\right )\,\left (B+4\,C\right )\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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