Optimal. Leaf size=77 \[ -\frac {a}{8 d (a \sin (c+d x)+a)^2}+\frac {1}{8 d (a-a \sin (c+d x))}-\frac {1}{4 d (a \sin (c+d x)+a)}+\frac {3 \tanh ^{-1}(\sin (c+d x))}{8 a d} \]
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Rubi [A] time = 0.08, 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 = {2667, 44, 206} \[ -\frac {a}{8 d (a \sin (c+d x)+a)^2}+\frac {1}{8 d (a-a \sin (c+d x))}-\frac {1}{4 d (a \sin (c+d x)+a)}+\frac {3 \tanh ^{-1}(\sin (c+d x))}{8 a d} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {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 \operatorname {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 = 0.11, size = 75, normalized size = 0.97 \[ -\frac {\sec ^2(c+d x) \left (-3 \sin ^2(c+d x)-3 \sin (c+d x)+3 (\sin (c+d x)-1) (\sin (c+d x)+1)^2 \tanh ^{-1}(\sin (c+d x))+2\right )}{8 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 125, normalized size = 1.62 \[ -\frac {6 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (\cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (\cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, \sin \left (d x + c\right ) - 2}{16 \, {\left (a d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.99, size = 96, normalized size = 1.25 \[ \frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (3 \, \sin \left (d x + c\right ) - 5\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}} - \frac {9 \, \sin \left (d x + c\right )^{2} + 26 \, \sin \left (d x + c\right ) + 21}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 90, normalized size = 1.17 \[ -\frac {1}{8 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{16 a d}-\frac {1}{8 a d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{4 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{16 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 91, normalized size = 1.18 \[ -\frac {\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 2\right )}}{a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {3 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {3 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 74, normalized size = 0.96 \[ \frac {3\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{8\,a\,d}+\frac {\frac {3\,{\sin \left (c+d\,x\right )}^2}{8}+\frac {3\,\sin \left (c+d\,x\right )}{8}-\frac {1}{4}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^3-a\,{\sin \left (c+d\,x\right )}^2+a\,\sin \left (c+d\,x\right )+a\right )} \]
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 \[ \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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