Optimal. Leaf size=58 \[ \frac {a A \sin (c+d x)}{d}+\frac {a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (A+2 C)+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.13, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4075, 4047, 8, 4045, 3770} \[ \frac {a A \sin (c+d x)}{d}+\frac {a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (A+2 C)+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3770
Rule 4045
Rule 4047
Rule 4075
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {a A \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} \int \cos (c+d x) \left (-2 a A-a (A+2 C) \sec (c+d x)-2 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a A \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} \int \cos (c+d x) \left (-2 a A-2 a C \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} (a (A+2 C)) \int 1 \, dx\\ &=\frac {1}{2} a (A+2 C) x+\frac {a A \sin (c+d x)}{d}+\frac {a A \cos (c+d x) \sin (c+d x)}{2 d}+(a C) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a (A+2 C) x+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \sin (c+d x)}{d}+\frac {a A \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 52, normalized size = 0.90 \[ \frac {a \left (4 A \sin (c+d x)+A \sin (2 (c+d x))+2 A c+2 A d x+4 C \tanh ^{-1}(\sin (c+d x))+4 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 63, normalized size = 1.09 \[ \frac {{\left (A + 2 \, C\right )} a d x + C a \log \left (\sin \left (d x + c\right ) + 1\right ) - C a \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (A a \cos \left (d x + c\right ) + 2 \, A a\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 99, normalized size = 1.71 \[ \frac {2 \, C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, C a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (A a + 2 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 77, normalized size = 1.33 \[ \frac {a A \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {a A x}{2}+\frac {A a c}{2 d}+a C x +\frac {C a c}{d}+\frac {a A \sin \left (d x +c \right )}{d}+\frac {a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 70, normalized size = 1.21 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 4 \, {\left (d x + c\right )} C a + 2 \, C a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A a \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.75, size = 115, normalized size = 1.98 \[ \frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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 \[ a \left (\int A \cos ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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