Optimal. Leaf size=60 \[ \frac {a^2 A \sin (c+d x)}{d}+\frac {b (2 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a B+2 A b)+\frac {b^2 B \tan (c+d x)}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4024, 3770, 3767, 8} \[ \frac {a^2 A \sin (c+d x)}{d}+\frac {b (2 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a B+2 A b)+\frac {b^2 B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 4024
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac {a^2 A \sin (c+d x)}{d}-\int \left (-a (2 A b+a B)+\left (-A b^2-2 a b B\right ) \sec (c+d x)-b^2 B \sec ^2(c+d x)\right ) \, dx\\ &=a (2 A b+a B) x+\frac {a^2 A \sin (c+d x)}{d}+\left (b^2 B\right ) \int \sec ^2(c+d x) \, dx+(b (A b+2 a B)) \int \sec (c+d x) \, dx\\ &=a (2 A b+a B) x+\frac {b (A b+2 a B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 A \sin (c+d x)}{d}-\frac {\left (b^2 B\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a (2 A b+a B) x+\frac {b (A b+2 a B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 A \sin (c+d x)}{d}+\frac {b^2 B \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 109, normalized size = 1.82 \[ \frac {a^2 A \sin (c+d x)+a (c+d x) (a B+2 A b)-b (2 a B+A b) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+b (2 a B+A b) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+b^2 B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 117, normalized size = 1.95 \[ \frac {2 \, {\left (B a^{2} + 2 \, A a b\right )} d x \cos \left (d x + c\right ) + {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A a^{2} \cos \left (d x + c\right ) + B b^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 154, normalized size = 2.57 \[ \frac {{\left (B a^{2} + 2 \, A a b\right )} {\left (d x + c\right )} + {\left (2 \, B a b + A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, B a b + A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.87, size = 104, normalized size = 1.73 \[ 2 A x a b +a^{2} B x +\frac {a^{2} A \sin \left (d x +c \right )}{d}+\frac {A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 A a b c}{d}+\frac {b^{2} B \tan \left (d x +c \right )}{d}+\frac {2 B a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {B \,a^{2} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 103, normalized size = 1.72 \[ \frac {2 \, {\left (d x + c\right )} B a^{2} + 4 \, {\left (d x + c\right )} A a b + 2 \, B a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + A b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a^{2} \sin \left (d x + c\right ) + 2 \, B b^{2} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.55, size = 163, normalized size = 2.72 \[ \frac {B\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {2\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d\,\cos \left (c+d\,x\right )}+\frac {4\,A\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,B\,a\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{2} \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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