Optimal. Leaf size=87 \[ \frac {a^6}{6 d (a-a \sin (c+d x))^3}+\frac {a^5}{8 d (a-a \sin (c+d x))^2}+\frac {a^4}{8 d (a-a \sin (c+d x))}+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{8 d} \]
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Rubi [A] time = 0.07, antiderivative size = 87, 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} \[ \frac {a^6}{6 d (a-a \sin (c+d x))^3}+\frac {a^5}{8 d (a-a \sin (c+d x))^2}+\frac {a^4}{8 d (a-a \sin (c+d x))}+\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{8 d} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \sec ^7(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {1}{2 a (a-x)^4}+\frac {1}{4 a^2 (a-x)^3}+\frac {1}{8 a^3 (a-x)^2}+\frac {1}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6}{6 d (a-a \sin (c+d x))^3}+\frac {a^5}{8 d (a-a \sin (c+d x))^2}+\frac {a^4}{8 d (a-a \sin (c+d x))}+\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=\frac {a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^6}{6 d (a-a \sin (c+d x))^3}+\frac {a^5}{8 d (a-a \sin (c+d x))^2}+\frac {a^4}{8 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 67, normalized size = 0.77 \[ -\frac {a^3 (\sin (c+d x)+1)^3 \sec ^6(c+d x) \left (-3 \sin ^2(c+d x)+9 \sin (c+d x)+3 (\sin (c+d x)-1)^3 \tanh ^{-1}(\sin (c+d x))-10\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 185, normalized size = 2.13 \[ \frac {6 \, a^{3} \cos \left (d x + c\right )^{2} + 18 \, a^{3} \sin \left (d x + c\right ) - 26 \, a^{3} + 3 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{48 \, {\left (3 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{2} - 4 \, d\right )} \sin \left (d x + c\right ) - 4 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 90, normalized size = 1.03 \[ \frac {6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac {11 \, a^{3} \sin \left (d x + c\right )^{3} - 45 \, a^{3} \sin \left (d x + c\right )^{2} + 69 \, a^{3} \sin \left (d x + c\right ) - 51 \, a^{3}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 238, normalized size = 2.74 \[ \frac {a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{6 d \cos \left (d x +c \right )^{6}}+\frac {a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{12 d \cos \left (d x +c \right )^{4}}+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{6}}+\frac {3 a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{4}}+\frac {3 a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{16 d \cos \left (d x +c \right )^{2}}+\frac {3 a^{3} \sin \left (d x +c \right )}{16 d}+\frac {a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a^{3}}{2 d \cos \left (d x +c \right )^{6}}+\frac {a^{3} \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{6 d}+\frac {5 a^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{24 d}+\frac {5 a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 96, normalized size = 1.10 \[ \frac {3 \, a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, a^{3} \sin \left (d x + c\right )^{2} - 9 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}\right )}}{\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 1}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 81, normalized size = 0.93 \[ \frac {a^3\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{8\,d}-\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^2}{8}-\frac {3\,a^3\,\sin \left (c+d\,x\right )}{8}+\frac {5\,a^3}{12}}{d\,\left ({\sin \left (c+d\,x\right )}^3-3\,{\sin \left (c+d\,x\right )}^2+3\,\sin \left (c+d\,x\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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