Optimal. Leaf size=246 \[ \frac{4 (34 A-19 B+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(13 A-8 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.57554, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{4 (34 A-19 B+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{(13 A-8 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3} \]
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))^3} \, dx &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{\int \frac{(a (8 A-3 B+3 C)-5 a (A-B) \cos (c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(13 A-8 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac{\int \frac{\left (3 a^2 (21 A-11 B+6 C)-4 a^2 (13 A-8 B+3 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(13 A-8 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(23 A-13 B+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\int \left (12 a^3 (34 A-19 B+9 C)-15 a^3 (23 A-13 B+6 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{15 a^6}\\ &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(13 A-8 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(23 A-13 B+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{(23 A-13 B+6 C) \int \sec ^3(c+d x) \, dx}{a^3}+\frac{(4 (34 A-19 B+9 C)) \int \sec ^4(c+d x) \, dx}{5 a^3}\\ &=-\frac{(23 A-13 B+6 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(13 A-8 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(23 A-13 B+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{(23 A-13 B+6 C) \int \sec (c+d x) \, dx}{2 a^3}-\frac{(4 (34 A-19 B+9 C)) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 a^3 d}\\ &=-\frac{(23 A-13 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{4 (34 A-19 B+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(13 A-8 B+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac{(23 A-13 B+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{4 (34 A-19 B+9 C) \tan ^3(c+d x)}{15 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.989298, size = 270, normalized size = 1.1 \[ \frac{960 (23 A-13 B+6 C) \cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+2 \sin \left (\frac{1}{2} (c+d x)\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) ((7814 A-4274 B+2124 C) \cos (c+d x)+8 (691 A-381 B+186 C) \cos (2 (c+d x))+3098 A \cos (3 (c+d x))+1287 A \cos (4 (c+d x))+272 A \cos (5 (c+d x))+4321 A-1718 B \cos (3 (c+d x))-717 B \cos (4 (c+d x))-152 B \cos (5 (c+d x))-2331 B+828 C \cos (3 (c+d x))+342 C \cos (4 (c+d x))+72 C \cos (5 (c+d x))+1146 C)}{240 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.085, size = 566, normalized size = 2.3 \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 = 1.09323, size = 851, normalized size = 3.46 \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 = 2.11088, size = 910, normalized size = 3.7 \begin{align*} -\frac{15 \,{\left ({\left (23 \, A - 13 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 3 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (23 \, A - 13 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left ({\left (23 \, A - 13 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 3 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (23 \, A - 13 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (16 \,{\left (34 \, A - 19 \, B + 9 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \,{\left (429 \, A - 239 \, B + 114 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (869 \, A - 479 \, B + 234 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \,{\left (19 \, A - 9 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{2} - 15 \,{\left (A - B\right )} \cos \left (d x + c\right ) + 10 \, A\right )} \sin \left (d x + c\right )}{60 \,{\left (a^{3} d \cos \left (d x + c\right )^{6} + 3 \, a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + a^{3} 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 = 1.24013, size = 481, normalized size = 1.96 \begin{align*} -\frac{\frac{30 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{30 \,{\left (23 \, A - 13 \, B + 6 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{20 \,{\left (51 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 21 \, 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} - 76 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 36 \, 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} + 33 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, 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^{3}} - \frac{3 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 50 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 30 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 465 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 255 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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