Optimal. Leaf size=35 \[ a B x+\frac {(b B+a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b C \tan (c+d x)}{d} \]
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Rubi [A]
time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {4157, 3999,
3852, 8, 3855} \begin {gather*} \frac {(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac {b C \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3999
Rule 4157
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=a B x+(b C) \int \sec ^2(c+d x) \, dx+(b B+a C) \int \sec (c+d x) \, dx\\ &=a B x+\frac {(b B+a C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {(b C) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a B x+\frac {(b B+a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b C \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 1.23 \begin {gather*} a B x+\frac {b B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b C \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 57, normalized size = 1.63
method | result | size |
derivativedivides | \(\frac {b B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C b \tan \left (d x +c \right )+B a \left (d x +c \right )+a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(57\) |
default | \(\frac {b B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C b \tan \left (d x +c \right )+B a \left (d x +c \right )+a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(57\) |
risch | \(a B x +\frac {2 i C b}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b B}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b B}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(105\) |
norman | \(\frac {a B x +a B x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 C b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-a B x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a B x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (b B +a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {\left (b B +a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (35) = 70\).
time = 0.28, size = 73, normalized size = 2.09 \begin {gather*} \frac {2 \, {\left (d x + c\right )} B a + C a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + B b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C b \tan \left (d x + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (35) = 70\).
time = 2.92, size = 85, normalized size = 2.43 \begin {gather*} \frac {2 \, B a d x \cos \left (d x + c\right ) + {\left (C a + B b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (C a + B b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C b \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end {gather*}
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 \begin {gather*} \int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs.
\(2 (35) = 70\).
time = 0.49, size = 84, normalized size = 2.40 \begin {gather*} \frac {{\left (d x + c\right )} B a + {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.87, size = 114, normalized size = 3.26 \begin {gather*} \frac {2\,B\,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 {C\,b\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}-\frac {B\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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