Optimal. Leaf size=57 \[ -\frac {\cos (c+d x)}{a^2 d}+\frac {b^2}{a^3 d (b+a \cos (c+d x))}+\frac {2 b \log (b+a \cos (c+d x))}{a^3 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3957, 2912, 12,
45} \begin {gather*} \frac {b^2}{a^3 d (a \cos (c+d x)+b)}+\frac {2 b \log (a \cos (c+d x)+b)}{a^3 d}-\frac {\cos (c+d x)}{a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2912
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin (c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {\text {Subst}\left (\int \left (1+\frac {b^2}{(b-x)^2}-\frac {2 b}{b-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac {\cos (c+d x)}{a^2 d}+\frac {b^2}{a^3 d (b+a \cos (c+d x))}+\frac {2 b \log (b+a \cos (c+d x))}{a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 76, normalized size = 1.33 \begin {gather*} \frac {-a^2 \cos ^2(c+d x)+a b \cos (c+d x) (-1+2 \log (b+a \cos (c+d x)))+b^2 (1+2 \log (b+a \cos (c+d x)))}{a^3 d (b+a \cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 67, normalized size = 1.18
method | result | size |
derivativedivides | \(\frac {-\frac {1}{a^{2} \sec \left (d x +c \right )}-\frac {2 b \ln \left (\sec \left (d x +c \right )\right )}{a^{3}}-\frac {b}{a^{2} \left (a +b \sec \left (d x +c \right )\right )}+\frac {2 b \ln \left (a +b \sec \left (d x +c \right )\right )}{a^{3}}}{d}\) | \(67\) |
default | \(\frac {-\frac {1}{a^{2} \sec \left (d x +c \right )}-\frac {2 b \ln \left (\sec \left (d x +c \right )\right )}{a^{3}}-\frac {b}{a^{2} \left (a +b \sec \left (d x +c \right )\right )}+\frac {2 b \ln \left (a +b \sec \left (d x +c \right )\right )}{a^{3}}}{d}\) | \(67\) |
risch | \(-\frac {2 i b x}{a^{3}}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a^{2} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{2} d}-\frac {4 i b c}{a^{3} d}+\frac {2 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{a^{3} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{3} d}\) | \(138\) |
norman | \(\frac {\frac {2 a^{2}+4 b a +4 b^{2}}{2 a^{2} d b}-\frac {\left (2 a^{2}-4 b a +4 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2} d b}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}-\frac {2 b \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}+\frac {2 b \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}{a^{3} d}\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 55, normalized size = 0.96 \begin {gather*} \frac {\frac {b^{2}}{a^{4} \cos \left (d x + c\right ) + a^{3} b} - \frac {\cos \left (d x + c\right )}{a^{2}} + \frac {2 \, b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{3}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.69, size = 75, normalized size = 1.32 \begin {gather*} -\frac {a^{2} \cos \left (d x + c\right )^{2} + a b \cos \left (d x + c\right ) - b^{2} - 2 \, {\left (a b \cos \left (d x + c\right ) + b^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4} d \cos \left (d x + c\right ) + a^{3} b 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 {\sin {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 61, normalized size = 1.07 \begin {gather*} -\frac {\cos \left (d x + c\right )}{a^{2} d} + \frac {2 \, b \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{3} d} + \frac {b^{2}}{{\left (a \cos \left (d x + c\right ) + b\right )} a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.02, size = 60, normalized size = 1.05 \begin {gather*} \frac {b^2}{d\,\left (\cos \left (c+d\,x\right )\,a^4+b\,a^3\right )}-\frac {\cos \left (c+d\,x\right )}{a^2\,d}+\frac {2\,b\,\ln \left (b+a\,\cos \left (c+d\,x\right )\right )}{a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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