Optimal. Leaf size=156 \[ a^3 (3 A+B) x+\frac {a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {5 a^3 (B+C) \tan (c+d x)}{2 d}-\frac {(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac {(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d} \]
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Rubi [A]
time = 0.21, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4171, 4002,
3999, 3852, 8, 3855} \begin {gather*} \frac {a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {(6 A-3 B-5 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+a^3 x (3 A+B)+\frac {5 a^3 (B+C) \tan (c+d x)}{2 d}-\frac {(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 a d}+\frac {A \sin (c+d x) (a \sec (c+d x)+a)^3}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3999
Rule 4002
Rule 4171
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {\int (a+a \sec (c+d x))^3 (a (3 A+B)-a (3 A-C) \sec (c+d x)) \, dx}{a}\\ &=\frac {A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}+\frac {\int (a+a \sec (c+d x))^2 \left (3 a^2 (3 A+B)-a^2 (6 A-3 B-5 C) \sec (c+d x)\right ) \, dx}{3 a}\\ &=\frac {A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac {(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac {\int (a+a \sec (c+d x)) \left (6 a^3 (3 A+B)+15 a^3 (B+C) \sec (c+d x)\right ) \, dx}{6 a}\\ &=a^3 (3 A+B) x+\frac {A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac {(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac {1}{2} \left (5 a^3 (B+C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (a^3 (6 A+7 B+5 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 (3 A+B) x+\frac {a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}-\frac {(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac {(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}-\frac {\left (5 a^3 (B+C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d}\\ &=a^3 (3 A+B) x+\frac {a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}+\frac {5 a^3 (B+C) \tan (c+d x)}{2 d}-\frac {(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac {(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1503\) vs. \(2(156)=312\).
time = 6.50, size = 1503, normalized size = 9.63 \begin {gather*} \frac {(3 A+B) x \cos ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(-6 A-7 B-5 C) \cos ^5(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \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 {(6 A+7 B+5 C) \cos ^5(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \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 {A \cos (d x) \cos ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{4 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {A \cos (c) \cos ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (d x)}{4 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {C \cos ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin \left (\frac {d x}{2}\right )}{24 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 )^3}+\frac {\cos ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (3 B \cos \left (\frac {c}{2}\right )+10 C \cos \left (\frac {c}{2}\right )-3 B \sin \left (\frac {c}{2}\right )-8 C \sin \left (\frac {c}{2}\right )\right )}{48 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 )^2}+\frac {\cos ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (3 A \sin \left (\frac {d x}{2}\right )+9 B \sin \left (\frac {d x}{2}\right )+11 C \sin \left (\frac {d x}{2}\right )\right )}{12 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 ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin \left (\frac {d x}{2}\right )}{24 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 )^3}+\frac {\cos ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-3 B \cos \left (\frac {c}{2}\right )-10 C \cos \left (\frac {c}{2}\right )-3 B \sin \left (\frac {c}{2}\right )-8 C \sin \left (\frac {c}{2}\right )\right )}{48 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 )^2}+\frac {\cos ^5(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (3 A \sin \left (\frac {d x}{2}\right )+9 B \sin \left (\frac {d x}{2}\right )+11 C \sin \left (\frac {d x}{2}\right )\right )}{12 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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 1.02, size = 229, normalized size = 1.47
method | result | size |
derivativedivides | \(\frac {A \,a^{3} \tan \left (d x +c \right )+a^{3} B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} B \tan \left (d x +c \right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{3} \left (d x +c \right )+3 a^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} C \tan \left (d x +c \right )+A \,a^{3} \sin \left (d x +c \right )+a^{3} B \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(229\) |
default | \(\frac {A \,a^{3} \tan \left (d x +c \right )+a^{3} B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} B \tan \left (d x +c \right )+3 a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 A \,a^{3} \left (d x +c \right )+3 a^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+3 a^{3} C \tan \left (d x +c \right )+A \,a^{3} \sin \left (d x +c \right )+a^{3} B \left (d x +c \right )+a^{3} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(229\) |
risch | \(3 a^{3} A x +a^{3} B x -\frac {i A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i A \,a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{3} \left (3 B \,{\mathrm e}^{5 i \left (d x +c \right )}+9 C \,{\mathrm e}^{5 i \left (d x +c \right )}-6 A \,{\mathrm e}^{4 i \left (d x +c \right )}-18 B \,{\mathrm e}^{4 i \left (d x +c \right )}-18 C \,{\mathrm e}^{4 i \left (d x +c \right )}-12 A \,{\mathrm e}^{2 i \left (d x +c \right )}-36 B \,{\mathrm e}^{2 i \left (d x +c \right )}-48 C \,{\mathrm e}^{2 i \left (d x +c \right )}-3 B \,{\mathrm e}^{i \left (d x +c \right )}-9 C \,{\mathrm e}^{i \left (d x +c \right )}-6 A -18 B -22 C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {7 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}\) | \(337\) |
norman | \(\frac {\left (3 A \,a^{3}+a^{3} B \right ) x +\left (-9 A \,a^{3}-3 a^{3} B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-9 A \,a^{3}-3 a^{3} B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 A \,a^{3}+a^{3} B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A \,a^{3}+2 a^{3} B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 A \,a^{3}+2 a^{3} B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a^{3} \left (4 A +7 B +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {5 a^{3} \left (B +C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{3} \left (3 A -9 B -10 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a^{3} \left (6 A -B -3 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{3} \left (9 B +9 A +10 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {a^{3} \left (6 A +7 B +5 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{3} \left (6 A +7 B +5 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(372\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 274, normalized size = 1.76 \begin {gather*} \frac {36 \, {\left (d x + c\right )} A a^{3} + 12 \, {\left (d x + c\right )} B a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 3 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 9 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, A a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, B a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} \tan \left (d x + c\right ) + 36 \, B a^{3} \tan \left (d x + c\right ) + 36 \, C a^{3} \tan \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 = 2.71, size = 168, normalized size = 1.08 \begin {gather*} \frac {12 \, {\left (3 \, A + B\right )} a^{3} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (6 \, A + 7 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (6 \, A + 7 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 2 \, {\left (3 \, A + 9 \, B + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 2 \, C a^{3}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \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*} a^{3} \left (\int A \cos {\left (c + d x \right )}\, dx + \int 3 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 B \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 C \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 C \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int C \cos {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 288, normalized size = 1.85 \begin {gather*} \frac {\frac {12 \, A a^{3} \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} + 6 \, {\left (3 \, A a^{3} + B a^{3}\right )} {\left (d x + c\right )} + 3 \, {\left (6 \, A a^{3} + 7 \, B a^{3} + 5 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (6 \, A a^{3} + 7 \, B a^{3} + 5 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C a^{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 = 4.44, size = 523, normalized size = 3.35 \begin {gather*} \frac {\frac {A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {A\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {3\,B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {3\,C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {11\,C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12}+\frac {A\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {5\,C\,a^3\,\sin \left (c+d\,x\right )}{4}+\frac {9\,A\,a^3\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\frac {9\,A\,a^3\,\cos \left (c+d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\frac {3\,B\,a^3\,\cos \left (c+d\,x\right )\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\frac {21\,B\,a^3\,\cos \left (c+d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}+\frac {15\,C\,a^3\,\cos \left (c+d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}+\frac {3\,A\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {3\,A\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {B\,a^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{2}+\frac {7\,B\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{4}+\frac {5\,C\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (3\,c+3\,d\,x\right )}{4}}{d\,\left (\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {\cos \left (3\,c+3\,d\,x\right )}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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