Optimal. Leaf size=74 \[ \frac {\log (1-\cos (c+d x))}{2 (a+b) d}-\frac {\log (1+\cos (c+d x))}{2 (a-b) d}+\frac {b \log (b+a \cos (c+d x))}{\left (a^2-b^2\right ) d} \]
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Rubi [A]
time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3957, 2800,
815} \begin {gather*} \frac {b \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )}+\frac {\log (1-\cos (c+d x))}{2 d (a+b)}-\frac {\log (\cos (c+d x)+1)}{2 d (a-b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 815
Rule 2800
Rule 3957
Rubi steps
\begin {align*} \int \frac {\csc (c+d x)}{a+b \sec (c+d x)} \, dx &=-\int \frac {\cot (c+d x)}{-b-a \cos (c+d x)} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x}{(-b+x) \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{2 (a-b) (a-x)}-\frac {b}{(a-b) (a+b) (b-x)}+\frac {1}{2 (a+b) (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\log (1-\cos (c+d x))}{2 (a+b) d}-\frac {\log (1+\cos (c+d x))}{2 (a-b) d}+\frac {b \log (b+a \cos (c+d x))}{\left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 63, normalized size = 0.85 \begin {gather*} \frac {-\left ((a+b) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+b \log (b+a \cos (c+d x))+(a-b) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a-b) (a+b) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 70, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a -2 b}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{2 a +2 b}+\frac {b \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right ) \left (a -b \right )}}{d}\) | \(70\) |
default | \(\frac {-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a -2 b}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{2 a +2 b}+\frac {b \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right ) \left (a -b \right )}}{d}\) | \(70\) |
norman | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a +b \right )}+\frac {b \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}{d \left (a^{2}-b^{2}\right )}\) | \(72\) |
risch | \(-\frac {i x}{a +b}-\frac {i c}{d \left (a +b \right )}+\frac {i x}{a -b}+\frac {i c}{d \left (a -b \right )}-\frac {2 i b x}{a^{2}-b^{2}}-\frac {2 i b c}{d \left (a^{2}-b^{2}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \left (a +b \right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \left (a -b \right )}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{d \left (a^{2}-b^{2}\right )}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 64, normalized size = 0.86 \begin {gather*} \frac {\frac {2 \, b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} - b^{2}} - \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a - b} + \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a + b}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.76, size = 64, normalized size = 0.86 \begin {gather*} \frac {2 \, b \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a + b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a - b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{2} - b^{2}\right )} d} \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 \frac {\csc {\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 100, normalized size = 1.35 \begin {gather*} \frac {\frac {2 \, b \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{2} - b^{2}} + \frac {\log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a + b}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 68, normalized size = 0.92 \begin {gather*} \frac {\ln \left (\cos \left (c+d\,x\right )-1\right )}{2\,d\,\left (a+b\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )}{2\,d\,\left (a-b\right )}+\frac {b\,\ln \left (b+a\,\cos \left (c+d\,x\right )\right )}{d\,\left (a^2-b^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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