Optimal. Leaf size=56 \[ \frac {a (B+C) \tan (c+d x)}{d}+\frac {a (B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a B \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.19, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {3029, 2968, 3021, 2748, 3767, 8, 3770} \[ \frac {a (B+C) \tan (c+d x)}{d}+\frac {a (B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a B \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2968
Rule 3021
Rule 3029
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=\int (a+a \cos (c+d x)) (B+C \cos (c+d x)) \sec ^3(c+d x) \, dx\\ &=\int \left (a B+(a B+a C) \cos (c+d x)+a C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {a B \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (2 a (B+C)+a (B+2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac {a B \sec (c+d x) \tan (c+d x)}{2 d}+(a (B+C)) \int \sec ^2(c+d x) \, dx+\frac {1}{2} (a (B+2 C)) \int \sec (c+d x) \, dx\\ &=\frac {a (B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a B \sec (c+d x) \tan (c+d x)}{2 d}-\frac {(a (B+C)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {a (B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a (B+C) \tan (c+d x)}{d}+\frac {a B \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 1.34 \[ \frac {a B \tan (c+d x)}{d}+\frac {a B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a B \tan (c+d x) \sec (c+d x)}{2 d}+\frac {a C \tan (c+d x)}{d}+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 89, normalized size = 1.59 \[ \frac {{\left (B + 2 \, C\right )} a \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (B + 2 \, C\right )} a \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, {\left (B + C\right )} a \cos \left (d x + c\right ) + B a\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 124, normalized size = 2.21 \[ \frac {{\left (B a + 2 \, C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (B a + 2 \, C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 86, normalized size = 1.54 \[ \frac {a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a B \tan \left (d x +c \right )}{d}+\frac {a C \tan \left (d x +c \right )}{d}+\frac {a B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 95, normalized size = 1.70 \[ -\frac {B a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2 \, C a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, B a \tan \left (d x + c\right ) - 4 \, C a \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 94, normalized size = 1.68 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,B\,a+2\,C\,a\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (B\,a+2\,C\,a\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (B+2\,C\right )}{d} \]
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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