Optimal. Leaf size=194 \[ \frac{(12 A-8 B+5 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(12 A-8 B+5 C) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.387534, antiderivative size = 194, normalized size of antiderivative = 1., 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 = {3041, 2978, 2748, 3767, 3768, 3770} \[ \frac{(12 A-8 B+5 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(12 A-8 B+5 C) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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))^2} \, dx &=-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{(3 a (2 A-B+C)-a (4 A-4 B+C) \cos (c+d x)) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{(10 A-7 B+4 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \left (3 a^2 (12 A-8 B+5 C)-3 a^2 (10 A-7 B+4 C) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx}{3 a^4}\\ &=-\frac{(10 A-7 B+4 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(10 A-7 B+4 C) \int \sec ^3(c+d x) \, dx}{a^2}+\frac{(12 A-8 B+5 C) \int \sec ^4(c+d x) \, dx}{a^2}\\ &=-\frac{(10 A-7 B+4 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(10 A-7 B+4 C) \int \sec (c+d x) \, dx}{2 a^2}-\frac{(12 A-8 B+5 C) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a^2 d}\\ &=-\frac{(10 A-7 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(12 A-8 B+5 C) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac{(A-B+C) \sec ^2(c+d x) \tan (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{(12 A-8 B+5 C) \tan ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [B] time = 6.21015, size = 763, normalized size = 3.93 \[ \frac{4 \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (11 A \sin \left (\frac{1}{2} (c+d x)\right )-6 B \sin \left (\frac{1}{2} (c+d x)\right )+3 C \sin \left (\frac{1}{2} (c+d x)\right )\right )}{3 d (a \cos (c+d x)+a)^2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{4 \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (11 A \sin \left (\frac{1}{2} (c+d x)\right )-6 B \sin \left (\frac{1}{2} (c+d x)\right )+3 C \sin \left (\frac{1}{2} (c+d x)\right )\right )}{3 d (a \cos (c+d x)+a)^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 (10 A-7 B+4 C) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d (a \cos (c+d x)+a)^2}-\frac{2 (10 A-7 B+4 C) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d (a \cos (c+d x)+a)^2}+\frac{2 \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \left (A \sin \left (\frac{1}{2} (c+d x)\right )-B \sin \left (\frac{1}{2} (c+d x)\right )+C \sin \left (\frac{1}{2} (c+d x)\right )\right )}{3 d (a \cos (c+d x)+a)^2}+\frac{4 \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (13 A \sin \left (\frac{1}{2} (c+d x)\right )-10 B \sin \left (\frac{1}{2} (c+d x)\right )+7 C \sin \left (\frac{1}{2} (c+d x)\right )\right )}{3 d (a \cos (c+d x)+a)^2}+\frac{(3 B-5 A) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (a \cos (c+d x)+a)^2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{(5 A-3 B) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (a \cos (c+d x)+a)^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{2 A \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (a \cos (c+d x)+a)^2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{2 A \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (a \cos (c+d x)+a)^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.078, size = 506, normalized size = 2.6 \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 = 1.04391, size = 765, normalized size = 3.94 \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 = 2.02162, size = 689, normalized size = 3.55 \begin{align*} -\frac{3 \,{\left ({\left (10 \, A - 7 \, B + 4 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (10 \, A - 7 \, B + 4 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (10 \, A - 7 \, B + 4 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left ({\left (10 \, A - 7 \, B + 4 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (10 \, A - 7 \, B + 4 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (10 \, A - 7 \, B + 4 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (4 \,{\left (12 \, A - 8 \, B + 5 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (66 \, A - 43 \, B + 28 \, C\right )} \cos \left (d x + c\right )^{3} + 6 \,{\left (2 \, A - B + C\right )} \cos \left (d x + c\right )^{2} -{\left (2 \, A - 3 \, B\right )} \cos \left (d x + c\right ) + 2 \, A\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{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 = 1.21765, size = 409, normalized size = 2.11 \begin{align*} -\frac{\frac{3 \,{\left (10 \, A - 7 \, B + 4 \, 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 (10 \, A - 7 \, B + 4 \, 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 (30 \, 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} + 6 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 40 \, 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} - 12 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 18 \, 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 ) + 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^{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} + 27 \, 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 ) + 15 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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