Optimal. Leaf size=109 \[ \frac {a^5}{12 d (a-a \sin (c+d x))^3}+\frac {a^4}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{16 d (a-a \sin (c+d x))}-\frac {a^3}{16 d (a \sin (c+d x)+a)}+\frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d} \]
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Rubi [A] time = 0.09, antiderivative size = 109, 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} \[ \frac {a^5}{12 d (a-a \sin (c+d x))^3}+\frac {a^4}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{16 d (a-a \sin (c+d x))}-\frac {a^3}{16 d (a \sin (c+d x)+a)}+\frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d} \]
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))^2 \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {1}{4 a^2 (a-x)^4}+\frac {1}{4 a^3 (a-x)^3}+\frac {3}{16 a^4 (a-x)^2}+\frac {1}{16 a^4 (a+x)^2}+\frac {1}{4 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5}{12 d (a-a \sin (c+d x))^3}+\frac {a^4}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{16 d (a-a \sin (c+d x))}-\frac {a^3}{16 d (a+a \sin (c+d x))}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=\frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a^5}{12 d (a-a \sin (c+d x))^3}+\frac {a^4}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{16 d (a-a \sin (c+d x))}-\frac {a^3}{16 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 85, normalized size = 0.78 \[ -\frac {a^2 (\sin (c+d x)+1)^2 \sec ^6(c+d x) \left (-3 \sin ^3(c+d x)+6 \sin ^2(c+d x)-\sin (c+d x)+3 (\sin (c+d x)+1) (\sin (c+d x)-1)^3 \tanh ^{-1}(\sin (c+d x))-4\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 203, normalized size = 1.86 \[ -\frac {12 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2} - 3 \, {\left (a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2}\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 119, normalized size = 1.09 \[ \frac {6 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 6 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {3 \, {\left (2 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}\right )}}{\sin \left (d x + c\right ) + 1} + \frac {11 \, a^{2} \sin \left (d x + c\right )^{3} - 42 \, a^{2} \sin \left (d x + c\right )^{2} + 57 \, a^{2} \sin \left (d x + c\right ) - 30 \, a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 190, normalized size = 1.74 \[ \frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{6 d \cos \left (d x +c \right )^{6}}+\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{4}}+\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{16 d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \sin \left (d x +c \right )}{16 d}+\frac {a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {a^{2}}{3 d \cos \left (d x +c \right )^{6}}+\frac {a^{2} \tan \left (d x +c \right ) \left (\sec ^{5}\left (d x +c \right )\right )}{6 d}+\frac {5 a^{2} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{24 d}+\frac {5 a^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 108, normalized size = 0.99 \[ \frac {3 \, a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, a^{2} \sin \left (d x + c\right )^{3} - 6 \, a^{2} \sin \left (d x + c\right )^{2} + a^{2} \sin \left (d x + c\right ) + 4 \, a^{2}\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{3} + 2 \, \sin \left (d x + c\right ) - 1}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.34, size = 94, normalized size = 0.86 \[ \frac {a^2\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{4\,d}-\frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^3}{4}-\frac {a^2\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {a^2\,\sin \left (c+d\,x\right )}{12}+\frac {a^2}{3}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^3+2\,\sin \left (c+d\,x\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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