Optimal. Leaf size=28 \[ \frac {d \log (\cos (a+b x))}{b^2}+\frac {(c+d x) \tan (a+b x)}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4269, 3556}
\begin {gather*} \frac {d \log (\cos (a+b x))}{b^2}+\frac {(c+d x) \tan (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 4269
Rubi steps
\begin {align*} \int (c+d x) \sec ^2(a+b x) \, dx &=\frac {(c+d x) \tan (a+b x)}{b}-\frac {d \int \tan (a+b x) \, dx}{b}\\ &=\frac {d \log (\cos (a+b x))}{b^2}+\frac {(c+d x) \tan (a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.29 \begin {gather*} \frac {d \log (\cos (a+b x))}{b^2}+\frac {c \tan (a+b x)}{b}+\frac {d x \tan (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 52, normalized size = 1.86
method | result | size |
derivativedivides | \(\frac {-\frac {d a \tan \left (b x +a \right )}{b}+c \tan \left (b x +a \right )+\frac {d \left (\left (b x +a \right ) \tan \left (b x +a \right )+\ln \left (\cos \left (b x +a \right )\right )\right )}{b}}{b}\) | \(52\) |
default | \(\frac {-\frac {d a \tan \left (b x +a \right )}{b}+c \tan \left (b x +a \right )+\frac {d \left (\left (b x +a \right ) \tan \left (b x +a \right )+\ln \left (\cos \left (b x +a \right )\right )\right )}{b}}{b}\) | \(52\) |
risch | \(-\frac {2 i d x}{b}-\frac {2 i d a}{b^{2}}+\frac {2 i \left (d x +c \right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}+\frac {d \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b^{2}}\) | \(59\) |
norman | \(\frac {-\frac {2 c \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {2 d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}}{\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1}+\frac {d \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b^{2}}+\frac {d \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b^{2}}-\frac {d \ln \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}\) | \(104\) |
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. 159 vs.
\(2 (28) = 56\).
time = 0.52, size = 159, normalized size = 5.68 \begin {gather*} \frac {2 \, c \tan \left (b x + a\right ) - \frac {2 \, a d \tan \left (b x + a\right )}{b} + \frac {{\left ({\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 4 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} b}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 45, normalized size = 1.61 \begin {gather*} \frac {d \cos \left (b x + a\right ) \log \left (-\cos \left (b x + a\right )\right ) + {\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2} \cos \left (b x + a\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 (c + d x\right ) \sec ^{2}{\left (a + b 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. 1459 vs.
\(2 (28) = 56\).
time = 0.66, size = 1459, normalized size = 52.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.83, size = 55, normalized size = 1.96 \begin {gather*} \frac {d\,\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+1\right )}{b^2}+\frac {\left (c+d\,x\right )\,2{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )}-\frac {d\,x\,2{}\mathrm {i}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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