Optimal. Leaf size=131 \[ \frac{(4 B-3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 B-3 C) \tan (c+d x)}{a d}-\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)} \]
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Rubi [A] time = 0.264745, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3029, 2978, 2748, 3767, 3768, 3770} \[ \frac{(4 B-3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 B-3 C) \tan (c+d x)}{a d}-\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2978
Rule 2748
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{a+a \cos (c+d x)} \, dx &=\int \frac{(B+C \cos (c+d x)) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx\\ &=-\frac{(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int (a (4 B-3 C)-3 a (B-C) \cos (c+d x)) \sec ^4(c+d x) \, dx}{a^2}\\ &=-\frac{(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac{(4 B-3 C) \int \sec ^4(c+d x) \, dx}{a}-\frac{(3 (B-C)) \int \sec ^3(c+d x) \, dx}{a}\\ &=-\frac{3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac{(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 (B-C)) \int \sec (c+d x) \, dx}{2 a}-\frac{(4 B-3 C) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d}\\ &=-\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(4 B-3 C) \tan (c+d x)}{a d}-\frac{3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac{(B-C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac{(4 B-3 C) \tan ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [B] time = 4.39558, size = 490, normalized size = 3.74 \[ \frac{\cos \left (\frac{1}{2} (c+d x)\right ) \left (\sec \left (\frac{c}{2}\right ) \sec (c) \sec ^3(c+d x) \left (-24 B \sin \left (c-\frac{d x}{2}\right )-6 B \sin \left (c+\frac{d x}{2}\right )-24 B \sin \left (2 c+\frac{d x}{2}\right )+21 B \sin \left (c+\frac{3 d x}{2}\right )+9 B \sin \left (2 c+\frac{3 d x}{2}\right )-9 B \sin \left (3 c+\frac{3 d x}{2}\right )+7 B \sin \left (c+\frac{5 d x}{2}\right )+B \sin \left (2 c+\frac{5 d x}{2}\right )-3 B \sin \left (3 c+\frac{5 d x}{2}\right )-9 B \sin \left (4 c+\frac{5 d x}{2}\right )+16 B \sin \left (2 c+\frac{7 d x}{2}\right )+10 B \sin \left (3 c+\frac{7 d x}{2}\right )+6 B \sin \left (4 c+\frac{7 d x}{2}\right )+6 (B+C) \sin \left (\frac{d x}{2}\right )+3 (13 B-9 C) \sin \left (\frac{3 d x}{2}\right )+12 C \sin \left (c-\frac{d x}{2}\right )+6 C \sin \left (c+\frac{d x}{2}\right )+24 C \sin \left (2 c+\frac{d x}{2}\right )-9 C \sin \left (c+\frac{3 d x}{2}\right )-9 C \sin \left (2 c+\frac{3 d x}{2}\right )+9 C \sin \left (3 c+\frac{3 d x}{2}\right )-3 C \sin \left (c+\frac{5 d x}{2}\right )+3 C \sin \left (2 c+\frac{5 d x}{2}\right )+3 C \sin \left (3 c+\frac{5 d x}{2}\right )+9 C \sin \left (4 c+\frac{5 d x}{2}\right )-12 C \sin \left (2 c+\frac{7 d x}{2}\right )-6 C \sin \left (3 c+\frac{7 d x}{2}\right )-6 C \sin \left (4 c+\frac{7 d x}{2}\right )\right )+144 (B-C) \cos \left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{48 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 340, normalized size = 2.6 \begin{align*}{\frac{B}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{C}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{B}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}+{\frac{C}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{B}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{3\,B}{2\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{3\,C}{2\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{5\,B}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{3\,C}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{B}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+{\frac{B}{da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{C}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{3\,B}{2\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{3\,C}{2\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{5\,B}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3\,C}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20924, size = 497, normalized size = 3.79 \begin{align*} \frac{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 )} - 3 \, C{\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 )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76172, size = 419, normalized size = 3.2 \begin{align*} -\frac{9 \,{\left ({\left (B - C\right )} \cos \left (d x + c\right )^{4} +{\left (B - C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \,{\left ({\left (B - C\right )} \cos \left (d x + c\right )^{4} +{\left (B - C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (4 \,{\left (4 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (7 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (B - 3 \, C\right )} \cos \left (d x + c\right ) + 2 \, B\right )} \sin \left (d x + c\right )}{12 \,{\left (a d \cos \left (d x + c\right )^{4} + a 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 = 1.34002, size = 246, normalized size = 1.88 \begin{align*} -\frac{\frac{9 \,{\left (B - C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac{9 \,{\left (B - C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac{6 \,{\left (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 (15 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 9 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 16 \, 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} + 9 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, 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}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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