Optimal. Leaf size=287 \[ \frac{4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac{(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac{(178 A-87 B+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(16 A-9 B+2 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.798291, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {3041, 2978, 2748, 3767, 3768, 3770} \[ \frac{4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac{(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac{(178 A-87 B+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(16 A-9 B+2 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 3041
Rule 2978
Rule 2748
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{\int \frac{(a (10 A-3 B+3 C)-a (6 A-6 B-C) \cos (c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (7 a^2 (14 A-6 B+3 C)-5 a^2 (16 A-9 B+2 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (3 a^3 (276 A-129 B+52 C)-4 a^3 (178 A-87 B+31 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(44 A-21 B+8 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{\int \left (12 a^4 (454 A-216 B+83 C)-105 a^4 (44 A-21 B+8 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{105 a^8}\\ &=-\frac{(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(44 A-21 B+8 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac{(44 A-21 B+8 C) \int \sec ^3(c+d x) \, dx}{a^4}+\frac{(4 (454 A-216 B+83 C)) \int \sec ^4(c+d x) \, dx}{35 a^4}\\ &=-\frac{(44 A-21 B+8 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac{(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(44 A-21 B+8 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac{(44 A-21 B+8 C) \int \sec (c+d x) \, dx}{2 a^4}-\frac{(4 (454 A-216 B+83 C)) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 a^4 d}\\ &=-\frac{(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac{(44 A-21 B+8 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac{(178 A-87 B+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(16 A-9 B+2 C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(44 A-21 B+8 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}\\ \end{align*}
Mathematica [A] time = 1.65462, size = 304, normalized size = 1.06 \[ \frac{26880 (44 A-21 B+8 C) \cos ^8\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) (14 (28252 A-13353 B+5224 C) \cos (c+d x)+56 (5218 A-2472 B+961 C) \cos (2 (c+d x))+173316 A \cos (3 (c+d x))+79264 A \cos (4 (c+d x))+24436 A \cos (5 (c+d x))+3632 A \cos (6 (c+d x))+217696 A-82239 B \cos (3 (c+d x))-37656 B \cos (4 (c+d x))-11619 B \cos (5 (c+d x))-1728 B \cos (6 (c+d x))-102504 B+31832 C \cos (3 (c+d x))+14528 C \cos (4 (c+d x))+4472 C \cos (5 (c+d x))+664 C \cos (6 (c+d x))+39952 C)}{3360 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.092, size = 626, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.40792, size = 932, normalized size = 3.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18403, size = 1135, normalized size = 3.95 \begin{align*} -\frac{105 \,{\left ({\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{7} + 4 \,{\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \,{\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \,{\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left ({\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{7} + 4 \,{\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \,{\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \,{\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (44 \, A - 21 \, B + 8 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (16 \,{\left (454 \, A - 216 \, B + 83 \, C\right )} \cos \left (d x + c\right )^{6} +{\left (24436 \, A - 11619 \, B + 4472 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \,{\left (7184 \, A - 3411 \, B + 1318 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (3196 \, A - 1509 \, B + 592 \, C\right )} \cos \left (d x + c\right )^{3} + 70 \,{\left (14 \, A - 6 \, B + 3 \, C\right )} \cos \left (d x + c\right )^{2} - 35 \,{\left (4 \, A - 3 \, B\right )} \cos \left (d x + c\right ) + 70 \, A\right )} \sin \left (d x + c\right )}{420 \,{\left (a^{4} d \cos \left (d x + c\right )^{7} + 4 \, a^{4} d \cos \left (d x + c\right )^{6} + 6 \, a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + a^{4} d \cos \left (d x + c\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2168, size = 549, normalized size = 1.91 \begin{align*} -\frac{\frac{420 \,{\left (44 \, A - 21 \, B + 8 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{420 \,{\left (44 \, A - 21 \, B + 8 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac{280 \,{\left (78 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 27 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 124 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 48 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 54 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 21 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, 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^{4}} - \frac{15 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 231 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 189 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 147 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 21945 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 11655 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5145 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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