Optimal. Leaf size=60 \[ \frac {a^2 B \sin (c+d x)}{d}+\frac {b (2 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a C+2 b B)+\frac {b^2 C \tan (c+d x)}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4072, 4024, 3770, 3767, 8} \[ \frac {a^2 B \sin (c+d x)}{d}+\frac {b (2 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a C+2 b B)+\frac {b^2 C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 4024
Rule 4072
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos (c+d x) (a+b \sec (c+d x))^2 (B+C \sec (c+d x)) \, dx\\ &=\frac {a^2 B \sin (c+d x)}{d}-\int \left (-a (2 b B+a C)+\left (-b^2 B-2 a b C\right ) \sec (c+d x)-b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=a (2 b B+a C) x+\frac {a^2 B \sin (c+d x)}{d}+\left (b^2 C\right ) \int \sec ^2(c+d x) \, dx+(b (b B+2 a C)) \int \sec (c+d x) \, dx\\ &=a (2 b B+a C) x+\frac {b (b B+2 a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 B \sin (c+d x)}{d}-\frac {\left (b^2 C\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a (2 b B+a C) x+\frac {b (b B+2 a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a^2 B \sin (c+d x)}{d}+\frac {b^2 C \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 109, normalized size = 1.82 \[ \frac {a^2 B \sin (c+d x)+a (c+d x) (a C+2 b B)-b (2 a C+b B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+b (2 a C+b B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+b^2 C \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 117, normalized size = 1.95 \[ \frac {2 \, {\left (C a^{2} + 2 \, B a b\right )} d x \cos \left (d x + c\right ) + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (B a^{2} \cos \left (d x + c\right ) + C b^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 154, normalized size = 2.57 \[ \frac {{\left (C a^{2} + 2 \, B a b\right )} {\left (d x + c\right )} + {\left (2 \, C a b + B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, C a b + B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.88, size = 104, normalized size = 1.73 \[ 2 B x a b +a^{2} C x +\frac {B \,a^{2} \sin \left (d x +c \right )}{d}+\frac {b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 B a b c}{d}+\frac {b^{2} C \tan \left (d x +c \right )}{d}+\frac {2 C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,a^{2} c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 103, normalized size = 1.72 \[ \frac {2 \, {\left (d x + c\right )} C a^{2} + 4 \, {\left (d x + c\right )} B a b + 2 \, C a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + B b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a^{2} \sin \left (d x + c\right ) + 2 \, C b^{2} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.28, size = 163, normalized size = 2.72 \[ \frac {C\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {2\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,B\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d\,\cos \left (c+d\,x\right )}+\frac {4\,B\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,C\,a\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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 \[ \int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{2} \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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