Optimal. Leaf size=116 \[ \frac {\left (2 a^2 A+6 a b B+3 A b^2\right ) \tan (c+d x)}{3 d}+\frac {\left (a^2 B+2 a A b+2 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2 A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac {a (a B+2 A b) \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.27, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2988, 3021, 2748, 3767, 8, 3770} \[ \frac {\left (2 a^2 A+6 a b B+3 A b^2\right ) \tan (c+d x)}{3 d}+\frac {\left (a^2 B+2 a A b+2 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2 A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac {a (a B+2 A b) \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2988
Rule 3021
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx &=\frac {a^2 A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{3} \int \left (-3 a (2 A b+a B)-\left (2 a^2 A+3 A b^2+6 a b B\right ) \cos (c+d x)-3 b^2 B \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {a (2 A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{6} \int \left (-2 \left (2 a^2 A+3 A b^2+6 a b B\right )-3 \left (2 a A b+a^2 B+2 b^2 B\right ) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a (2 A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {1}{3} \left (-2 a^2 A-3 A b^2-6 a b B\right ) \int \sec ^2(c+d x) \, dx-\frac {1}{2} \left (-2 a A b-a^2 B-2 b^2 B\right ) \int \sec (c+d x) \, dx\\ &=\frac {\left (2 a A b+a^2 B+2 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a (2 A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 A \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {\left (2 a^2 A+3 A b^2+6 a b B\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {\left (2 a A b+a^2 B+2 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {\left (2 a^2 A+3 A b^2+6 a b B\right ) \tan (c+d x)}{3 d}+\frac {a (2 A b+a B) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^2 A \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 92, normalized size = 0.79 \[ \frac {3 \left (a^2 B+2 a A b+2 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (2 \left (a^2 A \tan ^2(c+d x)+3 a^2 A+6 a b B+3 A b^2\right )+3 a (a B+2 A b) \sec (c+d x)\right )}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 150, normalized size = 1.29 \[ \frac {3 \, {\left (B a^{2} + 2 \, A a b + 2 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (B a^{2} + 2 \, A a b + 2 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a^{2} + 2 \, {\left (2 \, A a^{2} + 6 \, B a b + 3 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.82, size = 294, normalized size = 2.53 \[ \frac {3 \, {\left (B a^{2} + 2 \, A a b + 2 \, B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (B a^{2} + 2 \, A a b + 2 \, B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 174, normalized size = 1.50 \[ \frac {2 a^{2} A \tan \left (d x +c \right )}{3 d}+\frac {a^{2} A \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{3 d}+\frac {B \,a^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {A a b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {A a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 B a b \tan \left (d x +c \right )}{d}+\frac {A \,b^{2} \tan \left (d x +c \right )}{d}+\frac {b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 172, normalized size = 1.48 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} - 3 \, B a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, A a b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, B b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B a b \tan \left (d x + c\right ) + 12 \, A b^{2} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 227, normalized size = 1.96 \[ \frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B\,a^2}{2}+A\,a\,b+B\,b^2\right )}{2\,B\,a^2+4\,A\,a\,b+4\,B\,b^2}\right )\,\left (B\,a^2+2\,A\,a\,b+2\,B\,b^2\right )}{d}-\frac {\left (2\,A\,a^2+2\,A\,b^2-B\,a^2-2\,A\,a\,b+4\,B\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {4\,A\,a^2}{3}-8\,B\,a\,b-4\,A\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^2+2\,A\,b^2+B\,a^2+2\,A\,a\,b+4\,B\,a\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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