Optimal. Leaf size=183 \[ \frac {3 (4 A-4 B+5 C) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac {(3 A-4 B+4 C) \tan (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {(4 A-4 B+5 C) \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(3 A-4 B+4 C) \tan ^3(c+d x)}{3 a d} \]
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Rubi [A]
time = 0.16, antiderivative size = 183, 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 = {4169, 3872,
3852, 3853, 3855} \begin {gather*} -\frac {(3 A-4 B+4 C) \tan ^3(c+d x)}{3 a d}-\frac {(3 A-4 B+4 C) \tan (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac {(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{d (a \sec (c+d x)+a)}+\frac {(4 A-4 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac {3 (4 A-4 B+5 C) \tan (c+d x) \sec (c+d x)}{8 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rule 3853
Rule 3855
Rule 3872
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)} \, dx &=-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}+\frac {\int \sec ^4(c+d x) (-a (3 A-4 B+4 C)+a (4 A-4 B+5 C) \sec (c+d x)) \, dx}{a^2}\\ &=-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(3 A-4 B+4 C) \int \sec ^4(c+d x) \, dx}{a}+\frac {(4 A-4 B+5 C) \int \sec ^5(c+d x) \, dx}{a}\\ &=\frac {(4 A-4 B+5 C) \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}+\frac {(3 (4 A-4 B+5 C)) \int \sec ^3(c+d x) \, dx}{4 a}+\frac {(3 A-4 B+4 C) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d}\\ &=-\frac {(3 A-4 B+4 C) \tan (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {(4 A-4 B+5 C) \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(3 A-4 B+4 C) \tan ^3(c+d x)}{3 a d}+\frac {(3 (4 A-4 B+5 C)) \int \sec (c+d x) \, dx}{8 a}\\ &=\frac {3 (4 A-4 B+5 C) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac {(3 A-4 B+4 C) \tan (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {(4 A-4 B+5 C) \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac {(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac {(3 A-4 B+4 C) \tan ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1099\) vs. \(2(183)=366\).
time = 6.44, size = 1099, normalized size = 6.01 \begin {gather*} -\frac {3 (4 A-4 B+5 C) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos (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 ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))}+\frac {3 (4 A-4 B+5 C) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos (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 ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))}+\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 (-60 A \sin \left (\frac {d x}{2}\right )+108 B \sin \left (\frac {d x}{2}\right )-75 C \sin \left (\frac {d x}{2}\right )-60 A \sin \left (\frac {3 d x}{2}\right )+124 B \sin \left (\frac {3 d x}{2}\right )-91 C \sin \left (\frac {3 d x}{2}\right )+204 A \sin \left (c-\frac {d x}{2}\right )-252 B \sin \left (c-\frac {d x}{2}\right )+219 C \sin \left (c-\frac {d x}{2}\right )-60 A \sin \left (c+\frac {d x}{2}\right )+12 B \sin \left (c+\frac {d x}{2}\right )+21 C \sin \left (c+\frac {d x}{2}\right )+84 A \sin \left (2 c+\frac {d x}{2}\right )-132 B \sin \left (2 c+\frac {d x}{2}\right )+165 C \sin \left (2 c+\frac {d x}{2}\right )+36 A \sin \left (c+\frac {3 d x}{2}\right )+28 B \sin \left (c+\frac {3 d x}{2}\right )+5 C \sin \left (c+\frac {3 d x}{2}\right )+36 A \sin \left (2 c+\frac {3 d x}{2}\right )-36 B \sin \left (2 c+\frac {3 d x}{2}\right )+69 C \sin \left (2 c+\frac {3 d x}{2}\right )+132 A \sin \left (3 c+\frac {3 d x}{2}\right )-132 B \sin \left (3 c+\frac {3 d x}{2}\right )+165 C \sin \left (3 c+\frac {3 d x}{2}\right )-156 A \sin \left (c+\frac {5 d x}{2}\right )+220 B \sin \left (c+\frac {5 d x}{2}\right )-211 C \sin \left (c+\frac {5 d x}{2}\right )-60 A \sin \left (2 c+\frac {5 d x}{2}\right )+124 B \sin \left (2 c+\frac {5 d x}{2}\right )-115 C \sin \left (2 c+\frac {5 d x}{2}\right )-60 A \sin \left (3 c+\frac {5 d x}{2}\right )+60 B \sin \left (3 c+\frac {5 d x}{2}\right )-51 C \sin \left (3 c+\frac {5 d x}{2}\right )+36 A \sin \left (4 c+\frac {5 d x}{2}\right )-36 B \sin \left (4 c+\frac {5 d x}{2}\right )+45 C \sin \left (4 c+\frac {5 d x}{2}\right )-12 A \sin \left (2 c+\frac {7 d x}{2}\right )+28 B \sin \left (2 c+\frac {7 d x}{2}\right )-19 C \sin \left (2 c+\frac {7 d x}{2}\right )+12 A \sin \left (3 c+\frac {7 d x}{2}\right )+4 B \sin \left (3 c+\frac {7 d x}{2}\right )+5 C \sin \left (3 c+\frac {7 d x}{2}\right )+12 A \sin \left (4 c+\frac {7 d x}{2}\right )-12 B \sin \left (4 c+\frac {7 d x}{2}\right )+21 C \sin \left (4 c+\frac {7 d x}{2}\right )+36 A \sin \left (5 c+\frac {7 d x}{2}\right )-36 B \sin \left (5 c+\frac {7 d x}{2}\right )+45 C \sin \left (5 c+\frac {7 d x}{2}\right )-48 A \sin \left (3 c+\frac {9 d x}{2}\right )+64 B \sin \left (3 c+\frac {9 d x}{2}\right )-64 C \sin \left (3 c+\frac {9 d x}{2}\right )-24 A \sin \left (4 c+\frac {9 d x}{2}\right )+40 B \sin \left (4 c+\frac {9 d x}{2}\right )-40 C \sin \left (4 c+\frac {9 d x}{2}\right )-24 A \sin \left (5 c+\frac {9 d x}{2}\right )+24 B \sin \left (5 c+\frac {9 d x}{2}\right )-24 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{192 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.56, size = 260, normalized size = 1.42
method | result | size |
norman | \(\frac {-\frac {\left (A -B +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {\left (8 A -16 B +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (32 A -40 B +45 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}+\frac {2 \left (33 A -43 B +43 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {\left (66 A -98 B +95 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}-\frac {\left (120 A -152 B +155 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {3 \left (4 A -4 B +5 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a d}+\frac {3 \left (4 A -4 B +5 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a d}\) | \(247\) |
derivativedivides | \(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {C}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-\frac {5 C}{2}+B}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\left (\frac {15 C}{8}-\frac {3 B}{2}+\frac {3 A}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {-\frac {25 C}{8}+\frac {5 B}{2}-\frac {3 A}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {\frac {15 C}{4}-2 B +A}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {C}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-\frac {5 C}{2}+B}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\left (-\frac {15 C}{8}+\frac {3 B}{2}-\frac {3 A}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {-\frac {25 C}{8}+\frac {5 B}{2}-\frac {3 A}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\frac {-\frac {15 C}{4}+2 B -A}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{d a}\) | \(260\) |
default | \(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {C}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-\frac {5 C}{2}+B}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\left (\frac {15 C}{8}-\frac {3 B}{2}+\frac {3 A}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {-\frac {25 C}{8}+\frac {5 B}{2}-\frac {3 A}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {\frac {15 C}{4}-2 B +A}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {C}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-\frac {5 C}{2}+B}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\left (-\frac {15 C}{8}+\frac {3 B}{2}-\frac {3 A}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {-\frac {25 C}{8}+\frac {5 B}{2}-\frac {3 A}{2}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\frac {-\frac {15 C}{4}+2 B -A}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{d a}\) | \(260\) |
risch | \(-\frac {i \left (64 C +48 A -64 B +36 A \,{\mathrm e}^{7 i \left (d x +c \right )}-36 B \,{\mathrm e}^{7 i \left (d x +c \right )}+45 C \,{\mathrm e}^{7 i \left (d x +c \right )}-252 B \,{\mathrm e}^{4 i \left (d x +c \right )}-220 B \,{\mathrm e}^{2 i \left (d x +c \right )}+132 A \,{\mathrm e}^{6 i \left (d x +c \right )}+84 A \,{\mathrm e}^{5 i \left (d x +c \right )}-132 B \,{\mathrm e}^{5 i \left (d x +c \right )}-28 B \,{\mathrm e}^{i \left (d x +c \right )}-132 B \,{\mathrm e}^{6 i \left (d x +c \right )}-124 B \,{\mathrm e}^{3 i \left (d x +c \right )}+165 C \,{\mathrm e}^{6 i \left (d x +c \right )}+165 C \,{\mathrm e}^{5 i \left (d x +c \right )}+91 C \,{\mathrm e}^{3 i \left (d x +c \right )}+36 A \,{\mathrm e}^{8 i \left (d x +c \right )}-36 B \,{\mathrm e}^{8 i \left (d x +c \right )}+45 C \,{\mathrm e}^{8 i \left (d x +c \right )}+60 A \,{\mathrm e}^{3 i \left (d x +c \right )}+12 \,{\mathrm e}^{i \left (d x +c \right )} A +19 C \,{\mathrm e}^{i \left (d x +c \right )}+204 A \,{\mathrm e}^{4 i \left (d x +c \right )}+156 A \,{\mathrm e}^{2 i \left (d x +c \right )}+211 C \,{\mathrm e}^{2 i \left (d x +c \right )}+219 C \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{12 d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 a d}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 a d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 a d}\) | \(467\) |
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. 611 vs.
\(2 (175) = 350\).
time = 0.29, size = 611, normalized size = 3.34 \begin {gather*} -\frac {C {\left (\frac {2 \, {\left (\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a - \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {45 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {45 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {24 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 4 \, B {\left (\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16 \, \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 - \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {6 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 12 \, A {\left (\frac {2 \, {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {2 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.63, size = 216, normalized size = 1.18 \begin {gather*} \frac {9 \, {\left ({\left (4 \, A - 4 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (4 \, A - 4 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, {\left ({\left (4 \, A - 4 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (4 \, A - 4 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (3 \, A - 4 \, B + 4 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (12 \, A - 28 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{3} - {\left (12 \, A - 4 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (4 \, B - C\right )} \cos \left (d x + c\right ) - 6 \, C\right )} \sin \left (d x + c\right )}{48 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\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 {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{5}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 285, normalized size = 1.56 \begin {gather*} \frac {\frac {9 \, {\left (4 \, A - 4 \, B + 5 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {9 \, {\left (4 \, A - 4 \, B + 5 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {24 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} + \frac {2 \, {\left (36 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 60 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 75 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 84 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 124 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 115 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 100 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 109 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, 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 )}^{4} a}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.29, size = 203, normalized size = 1.11 \begin {gather*} \frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,A-4\,B+5\,C\right )}{4\,a\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B+C\right )}{a\,d}-\frac {\left (5\,B-3\,A-\frac {25\,C}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (7\,A-\frac {31\,B}{3}+\frac {115\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {25\,B}{3}-5\,A-\frac {109\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A-3\,B+\frac {7\,C}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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