Optimal. Leaf size=194 \[ -\frac {(4 A-7 B+10 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac {(5 A-8 B+12 C) \tan (c+d x)}{a^2 d}-\frac {(4 A-7 B+10 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {(4 A-7 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(5 A-8 B+12 C) \tan ^3(c+d x)}{3 a^2 d} \]
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Rubi [A]
time = 0.25, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4169, 4104,
3872, 3853, 3855, 3852} \begin {gather*} \frac {(5 A-8 B+12 C) \tan ^3(c+d x)}{3 a^2 d}+\frac {(5 A-8 B+12 C) \tan (c+d x)}{a^2 d}-\frac {(4 A-7 B+10 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac {(4 A-7 B+10 C) \tan (c+d x) \sec ^3(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac {(4 A-7 B+10 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac {(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4104
Rule 4169
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx &=-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec ^4(c+d x) (-a (A-4 B+4 C)+3 a (A-B+2 C) \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=-\frac {(4 A-7 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {\int \sec ^3(c+d x) \left (-3 a^2 (4 A-7 B+10 C)+3 a^2 (5 A-8 B+12 C) \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {(4 A-7 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {(4 A-7 B+10 C) \int \sec ^3(c+d x) \, dx}{a^2}+\frac {(5 A-8 B+12 C) \int \sec ^4(c+d x) \, dx}{a^2}\\ &=-\frac {(4 A-7 B+10 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {(4 A-7 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac {(4 A-7 B+10 C) \int \sec (c+d x) \, dx}{2 a^2}-\frac {(5 A-8 B+12 C) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {(4 A-7 B+10 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac {(5 A-8 B+12 C) \tan (c+d x)}{a^2 d}-\frac {(4 A-7 B+10 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac {(4 A-7 B+10 C) \sec ^3(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(5 A-8 B+12 C) \tan ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1069\) vs. \(2(194)=388\).
time = 6.48, size = 1069, normalized size = 5.51 \begin {gather*} \frac {4 (4 A-7 B+10 C) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^2}-\frac {4 (4 A-7 B+10 C) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^2}+\frac {\cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec (c) \sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-48 A \sin \left (\frac {d x}{2}\right )+45 B \sin \left (\frac {d x}{2}\right )-6 C \sin \left (\frac {d x}{2}\right )+132 A \sin \left (\frac {3 d x}{2}\right )-201 B \sin \left (\frac {3 d x}{2}\right )+310 C \sin \left (\frac {3 d x}{2}\right )-120 A \sin \left (c-\frac {d x}{2}\right )+195 B \sin \left (c-\frac {d x}{2}\right )-306 C \sin \left (c-\frac {d x}{2}\right )+48 A \sin \left (c+\frac {d x}{2}\right )-51 B \sin \left (c+\frac {d x}{2}\right )+42 C \sin \left (c+\frac {d x}{2}\right )-120 A \sin \left (2 c+\frac {d x}{2}\right )+189 B \sin \left (2 c+\frac {d x}{2}\right )-270 C \sin \left (2 c+\frac {d x}{2}\right )-8 A \sin \left (c+\frac {3 d x}{2}\right )-B \sin \left (c+\frac {3 d x}{2}\right )+50 C \sin \left (c+\frac {3 d x}{2}\right )+72 A \sin \left (2 c+\frac {3 d x}{2}\right )-81 B \sin \left (2 c+\frac {3 d x}{2}\right )+90 C \sin \left (2 c+\frac {3 d x}{2}\right )-68 A \sin \left (3 c+\frac {3 d x}{2}\right )+119 B \sin \left (3 c+\frac {3 d x}{2}\right )-170 C \sin \left (3 c+\frac {3 d x}{2}\right )+84 A \sin \left (c+\frac {5 d x}{2}\right )-129 B \sin \left (c+\frac {5 d x}{2}\right )+198 C \sin \left (c+\frac {5 d x}{2}\right )-9 B \sin \left (2 c+\frac {5 d x}{2}\right )+42 C \sin \left (2 c+\frac {5 d x}{2}\right )+48 A \sin \left (3 c+\frac {5 d x}{2}\right )-57 B \sin \left (3 c+\frac {5 d x}{2}\right )+66 C \sin \left (3 c+\frac {5 d x}{2}\right )-36 A \sin \left (4 c+\frac {5 d x}{2}\right )+63 B \sin \left (4 c+\frac {5 d x}{2}\right )-90 C \sin \left (4 c+\frac {5 d x}{2}\right )+48 A \sin \left (2 c+\frac {7 d x}{2}\right )-75 B \sin \left (2 c+\frac {7 d x}{2}\right )+114 C \sin \left (2 c+\frac {7 d x}{2}\right )+6 A \sin \left (3 c+\frac {7 d x}{2}\right )-15 B \sin \left (3 c+\frac {7 d x}{2}\right )+36 C \sin \left (3 c+\frac {7 d x}{2}\right )+30 A \sin \left (4 c+\frac {7 d x}{2}\right )-39 B \sin \left (4 c+\frac {7 d x}{2}\right )+48 C \sin \left (4 c+\frac {7 d x}{2}\right )-12 A \sin \left (5 c+\frac {7 d x}{2}\right )+21 B \sin \left (5 c+\frac {7 d x}{2}\right )-30 C \sin \left (5 c+\frac {7 d x}{2}\right )+20 A \sin \left (3 c+\frac {9 d x}{2}\right )-32 B \sin \left (3 c+\frac {9 d x}{2}\right )+48 C \sin \left (3 c+\frac {9 d x}{2}\right )+6 A \sin \left (4 c+\frac {9 d x}{2}\right )-12 B \sin \left (4 c+\frac {9 d x}{2}\right )+22 C \sin \left (4 c+\frac {9 d x}{2}\right )+14 A \sin \left (5 c+\frac {9 d x}{2}\right )-20 B \sin \left (5 c+\frac {9 d x}{2}\right )+26 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{48 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.58, size = 260, normalized size = 1.34
method | result | size |
derivativedivides | \(\frac {\frac {A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+5 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-7 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+9 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {6 C -2 B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 A -5 B +10 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (4 A -7 B +10 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {2 C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-6 C +2 B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 A -5 B +10 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (-10 C +7 B -4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {2 C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}}{2 d \,a^{2}}\) | \(260\) |
default | \(\frac {\frac {A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+5 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-7 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+9 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {6 C -2 B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 A -5 B +10 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (4 A -7 B +10 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {2 C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-6 C +2 B}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 A -5 B +10 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (-10 C +7 B -4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {2 C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}}{2 d \,a^{2}}\) | \(260\) |
norman | \(\frac {\frac {\left (A -B +C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}+\frac {\left (5 A -8 B +11 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {\left (9 A -13 B +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {2 \left (47 A -77 B +115 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {\left (61 A -94 B +143 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {\left (77 A -125 B +185 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}-\frac {\left (217 A -349 B +521 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5} a}+\frac {\left (4 A -7 B +10 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{2} d}-\frac {\left (4 A -7 B +10 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{2} d}\) | \(279\) |
risch | \(\frac {i \left (48 C +20 A -32 B +36 A \,{\mathrm e}^{7 i \left (d x +c \right )}-63 B \,{\mathrm e}^{7 i \left (d x +c \right )}+90 C \,{\mathrm e}^{7 i \left (d x +c \right )}-195 B \,{\mathrm e}^{4 i \left (d x +c \right )}-129 B \,{\mathrm e}^{2 i \left (d x +c \right )}+68 A \,{\mathrm e}^{6 i \left (d x +c \right )}+120 A \,{\mathrm e}^{5 i \left (d x +c \right )}-189 B \,{\mathrm e}^{5 i \left (d x +c \right )}-75 B \,{\mathrm e}^{i \left (d x +c \right )}-119 B \,{\mathrm e}^{6 i \left (d x +c \right )}-201 B \,{\mathrm e}^{3 i \left (d x +c \right )}+170 C \,{\mathrm e}^{6 i \left (d x +c \right )}+270 C \,{\mathrm e}^{5 i \left (d x +c \right )}+310 C \,{\mathrm e}^{3 i \left (d x +c \right )}+12 A \,{\mathrm e}^{8 i \left (d x +c \right )}-21 B \,{\mathrm e}^{8 i \left (d x +c \right )}+30 C \,{\mathrm e}^{8 i \left (d x +c \right )}+132 A \,{\mathrm e}^{3 i \left (d x +c \right )}+48 \,{\mathrm e}^{i \left (d x +c \right )} A +114 C \,{\mathrm e}^{i \left (d x +c \right )}+120 A \,{\mathrm e}^{4 i \left (d x +c \right )}+84 A \,{\mathrm e}^{2 i \left (d x +c \right )}+198 C \,{\mathrm e}^{2 i \left (d x +c \right )}+306 C \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{2} d}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 a^{2} d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{2} d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{2} d}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 a^{2} d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{2} d}\) | \(467\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 567 vs.
\(2 (184) = 368\).
time = 0.29, size = 567, normalized size = 2.92 \begin {gather*} \frac {C {\left (\frac {4 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {30 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {30 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )} - B {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )} + A {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.15, size = 271, normalized size = 1.40 \begin {gather*} -\frac {3 \, {\left ({\left (4 \, A - 7 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, A - 7 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, A - 7 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (4 \, A - 7 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, A - 7 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, A - 7 \, B + 10 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (5 \, A - 8 \, B + 12 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (28 \, A - 43 \, B + 66 \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (A - B + 2 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, B - 2 \, C\right )} \cos \left (d x + c\right ) + 2 \, C\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}} \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*} \frac {\int \frac {A \sec ^{4}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{6}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 303, normalized size = 1.56 \begin {gather*} -\frac {\frac {3 \, {\left (4 \, A - 7 \, B + 10 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {3 \, {\left (4 \, A - 7 \, B + 10 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {2 \, {\left (6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, C \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} a^{2}} - \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.43, size = 218, normalized size = 1.12 \begin {gather*} \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-3\,B+5\,C}{2\,a^2}+\frac {2\,\left (A-B+C\right )}{a^2}\right )}{d}-\frac {\left (2\,A-5\,B+10\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (8\,B-4\,A-\frac {40\,C}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A-3\,B+6\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^2\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,A-7\,B+10\,C\right )}{a^2\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A-B+C\right )}{6\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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