Optimal. Leaf size=84 \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a \sin (c+d x)+a)}+\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d} \]
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Rubi [A] time = 0.06, antiderivative size = 84, 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 = {2667, 44, 206} \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a \sin (c+d x)+a)}+\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \left (\frac {1}{4 a^2 (a-x)^3}+\frac {1}{4 a^3 (a-x)^2}+\frac {1}{8 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 {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a+a \sin (c+d x))}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 68, normalized size = 0.81 \[ \frac {a \sec ^4(c+d x)}{4 d}+\frac {a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 136, normalized size = 1.62 \[ -\frac {6 \, a \cos \left (d x + c\right )^{2} - 3 \, {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 6 \, a \sin \left (d x + c\right ) - 2 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 92, normalized size = 1.10 \[ \frac {6 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 6 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, a \sin \left (d x + c\right ) + 5 \, a\right )}}{\sin \left (d x + c\right ) + 1} + \frac {9 \, a \sin \left (d x + c\right )^{2} - 26 \, a \sin \left (d x + c\right ) + 21 \, a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 74, normalized size = 0.88 \[ \frac {a}{4 d \cos \left (d x +c \right )^{4}}+\frac {a \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 86, normalized size = 1.02 \[ \frac {3 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, a \sin \left (d x + c\right )^{2} - 3 \, a \sin \left (d x + c\right ) - 2 \, a\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.50, size = 71, normalized size = 0.85 \[ \frac {3\,a\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{8\,d}-\frac {-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{8}+\frac {3\,a\,\sin \left (c+d\,x\right )}{8}+\frac {a}{4}}{d\,\left (-{\sin \left (c+d\,x\right )}^3+{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )-1\right )} \]
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 \[ a \left (\int \sin {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx + \int \sec ^{5}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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