Optimal. Leaf size=77 \[ \frac {3 \tanh ^{-1}(\sin (c+d x))}{8 a d}+\frac {1}{8 d (a-a \sin (c+d x))}-\frac {a}{8 d (a+a \sin (c+d x))^2}-\frac {1}{4 d (a+a \sin (c+d x))} \]
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Rubi [A]
time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2746, 46, 212}
\begin {gather*} -\frac {a}{8 d (a \sin (c+d x)+a)^2}+\frac {1}{8 d (a-a \sin (c+d x))}-\frac {1}{4 d (a \sin (c+d x)+a)}+\frac {3 \tanh ^{-1}(\sin (c+d x))}{8 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 212
Rule 2746
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^3 \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \text {Subst}\left (\int \left (\frac {1}{8 a^3 (a-x)^2}+\frac {1}{4 a^2 (a+x)^3}+\frac {1}{4 a^3 (a+x)^2}+\frac {3}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {1}{8 d (a-a \sin (c+d x))}-\frac {a}{8 d (a+a \sin (c+d x))^2}-\frac {1}{4 d (a+a \sin (c+d x))}+\frac {3 \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=\frac {3 \tanh ^{-1}(\sin (c+d x))}{8 a d}+\frac {1}{8 d (a-a \sin (c+d x))}-\frac {a}{8 d (a+a \sin (c+d x))^2}-\frac {1}{4 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 75, normalized size = 0.97 \begin {gather*} -\frac {\sec ^2(c+d x) \left (2-3 \sin (c+d x)-3 \sin ^2(c+d x)+3 \tanh ^{-1}(\sin (c+d x)) (-1+\sin (c+d x)) (1+\sin (c+d x))^2\right )}{8 a d (1+\sin (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 67, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {-\frac {1}{8 \left (\sin \left (d x +c \right )-1\right )}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{16}-\frac {1}{8 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{4 \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{16}}{d a}\) | \(67\) |
default | \(\frac {-\frac {1}{8 \left (\sin \left (d x +c \right )-1\right )}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{16}-\frac {1}{8 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{4 \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{16}}{d a}\) | \(67\) |
risch | \(-\frac {i \left (6 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}-6 i {\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{4 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{2} d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 a d}\) | \(139\) |
norman | \(\frac {\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}+\frac {5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a d}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a d}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 91, normalized size = 1.18 \begin {gather*} -\frac {\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 2\right )}}{a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {3 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {3 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 125, normalized size = 1.62 \begin {gather*} -\frac {6 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (\cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (\cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, \sin \left (d x + c\right ) - 2}{16 \, {\left (a d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{2}\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*} \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.86, size = 96, normalized size = 1.25 \begin {gather*} \frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (3 \, \sin \left (d x + c\right ) - 5\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}} - \frac {9 \, \sin \left (d x + c\right )^{2} + 26 \, \sin \left (d x + c\right ) + 21}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{32 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.66, size = 74, normalized size = 0.96 \begin {gather*} \frac {3\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{8\,a\,d}+\frac {\frac {3\,{\sin \left (c+d\,x\right )}^2}{8}+\frac {3\,\sin \left (c+d\,x\right )}{8}-\frac {1}{4}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^3-a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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