Optimal. Leaf size=82 \[ \frac {a^2 (2 A+3 B) \tan (c+d x)}{2 d}+\frac {a^2 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 A x+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3917, 3914, 3767, 8, 3770} \[ \frac {a^2 (2 A+3 B) \tan (c+d x)}{2 d}+\frac {a^2 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 A x+\frac {B \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3914
Rule 3917
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+a \sec (c+d x)) (2 a A+a (2 A+3 B) \sec (c+d x)) \, dx\\ &=a^2 A x+\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^2 (2 A+3 B)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (a^2 (4 A+3 B)\right ) \int \sec (c+d x) \, dx\\ &=a^2 A x+\frac {a^2 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d}-\frac {\left (a^2 (2 A+3 B)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d}\\ &=a^2 A x+\frac {a^2 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2 (2 A+3 B) \tan (c+d x)}{2 d}+\frac {B \left (a^2+a^2 \sec (c+d x)\right ) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 1.32, size = 307, normalized size = 3.74 \[ \frac {a^2 \cos ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left (\frac {4 (A+2 B) \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 (A+2 B) \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {2 (4 A+3 B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {2 (4 A+3 B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+4 A x+\frac {B}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {B}{d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}\right )}{16 (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 119, normalized size = 1.45 \[ \frac {4 \, A a^{2} d x \cos \left (d x + c\right )^{2} + {\left (4 \, A + 3 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (4 \, A + 3 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right ) + B a^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 154, normalized size = 1.88 \[ \frac {2 \, {\left (d x + c\right )} A a^{2} + {\left (4 \, A a^{2} + 3 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (4 \, A a^{2} + 3 \, B a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 113, normalized size = 1.38 \[ a^{2} A x +\frac {A \,a^{2} c}{d}+\frac {3 B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 a^{2} B \tan \left (d x +c \right )}{d}+\frac {a^{2} A \tan \left (d x +c \right )}{d}+\frac {a^{2} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 128, normalized size = 1.56 \[ \frac {4 \, {\left (d x + c\right )} A a^{2} - B a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, A a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, B a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, A a^{2} \tan \left (d x + c\right ) + 8 \, B a^{2} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.01, size = 162, normalized size = 1.98 \[ \frac {2\,A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,B\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {2\,B\,a^2\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {B\,a^2\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2} \]
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 \[ a^{2} \left (\int A\, dx + \int 2 A \sec {\left (c + d x \right )}\, dx + \int A \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sec {\left (c + d x \right )}\, dx + \int 2 B \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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