Optimal. Leaf size=183 \[ -\frac {(3 A-4 B+4 C) \tan ^3(c+d x)}{3 a d}-\frac {(3 A-4 B+4 C) \tan (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac {(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{d (a \sec (c+d x)+a)}+\frac {(4 A-4 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac {3 (4 A-4 B+5 C) \tan (c+d x) \sec (c+d x)}{8 a d} \]
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Rubi [A] time = 0.22, 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 = {4084, 3787, 3767, 3768, 3770} \[ -\frac {(3 A-4 B+4 C) \tan ^3(c+d x)}{3 a d}-\frac {(3 A-4 B+4 C) \tan (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac {(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{d (a \sec (c+d x)+a)}+\frac {(4 A-4 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac {3 (4 A-4 B+5 C) \tan (c+d x) \sec (c+d x)}{8 a d} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 3768
Rule 3770
Rule 3787
Rule 4084
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) \operatorname {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] time = 6.42, size = 1099, normalized size = 6.01 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 216, normalized size = 1.18 \[ \frac {9 \, {\left ({\left (4 \, A - 4 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (4 \, A - 4 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, {\left ({\left (4 \, A - 4 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (4 \, A - 4 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (3 \, A - 4 \, B + 4 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (12 \, A - 28 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{3} - {\left (12 \, A - 4 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (4 \, B - C\right )} \cos \left (d x + c\right ) - 6 \, C\right )} \sin \left (d x + c\right )}{48 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 285, normalized size = 1.56 \[ \frac {\frac {9 \, {\left (4 \, A - 4 \, B + 5 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {9 \, {\left (4 \, A - 4 \, B + 5 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {24 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} + \frac {2 \, {\left (36 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 60 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 75 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 84 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 124 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 115 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 100 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 109 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} a}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.89, size = 576, normalized size = 3.15 \[ \frac {25 C}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {B}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {B}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) C}{8 a d}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) B}{2 a d}+\frac {3 A}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) B}{2 a d}+\frac {3 A}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {C}{4 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {5 C}{6 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {B}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {15 C}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}-\frac {C}{4 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {3 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d}+\frac {15 C}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {25 C}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {5 C}{6 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {B}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) C}{8 a d}-\frac {5 B}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {A}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {5 B}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {A}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 611, normalized size = 3.34 \[ -\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} \]
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 \[ \frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,A-4\,B+5\,C\right )}{4\,a\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B+C\right )}{a\,d}-\frac {\left (5\,B-3\,A-\frac {25\,C}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (7\,A-\frac {31\,B}{3}+\frac {115\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {25\,B}{3}-5\,A-\frac {109\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A-3\,B+\frac {7\,C}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \]
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 \[ \frac {\int \frac {A \sec ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{5}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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