Optimal. Leaf size=87 \[ \frac {a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^6}{6 d (a-a \sin (c+d x))^3}+\frac {a^5}{8 d (a-a \sin (c+d x))^2}+\frac {a^4}{8 d (a-a \sin (c+d x))} \]
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Rubi [A]
time = 0.05, 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 = {2746, 46, 212}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 212
Rule 2746
Rubi steps
\begin {align*} \int \sec ^7(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {a^7 \text {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \text {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 \text {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 = 0.07, size = 67, normalized size = 0.77 \begin {gather*} -\frac {a^3 \sec ^6(c+d x) (1+\sin (c+d x))^3 \left (-10+3 \tanh ^{-1}(\sin (c+d x)) (-1+\sin (c+d x))^3+9 \sin (c+d x)-3 \sin ^2(c+d x)\right )}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs.
\(2(79)=158\).
time = 0.18, size = 202, normalized size = 2.32
method | result | size |
risch | \(-\frac {i a^{3} \left (-18 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}+18 i {\mathrm e}^{2 i \left (d x +c \right )}-46 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}\) | \(125\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )+3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {a^{3}}{2 \cos \left (d x +c \right )^{6}}+a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(202\) |
default | \(\frac {a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )+3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {a^{3}}{2 \cos \left (d x +c \right )^{6}}+a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 96, normalized size = 1.10 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs.
\(2 (82) = 164\).
time = 0.37, size = 185, normalized size = 2.13 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.33, size = 90, normalized size = 1.03 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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