Optimal. Leaf size=110 \[ -\frac {80 a^8 \log (1-\sin (c+d x))}{d}-\frac {31 a^8 \sin (c+d x)}{d}-\frac {4 a^8 \sin ^2(c+d x)}{d}-\frac {a^8 \sin ^3(c+d x)}{3 d}+\frac {16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac {80 a^9}{d (a-a \sin (c+d x))} \]
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Rubi [A]
time = 0.06, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45}
\begin {gather*} \frac {16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac {80 a^9}{d (a-a \sin (c+d x))}-\frac {a^8 \sin ^3(c+d x)}{3 d}-\frac {4 a^8 \sin ^2(c+d x)}{d}-\frac {31 a^8 \sin (c+d x)}{d}-\frac {80 a^8 \log (1-\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac {a^5 \text {Subst}\left (\int \frac {(a+x)^5}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \text {Subst}\left (\int \left (-31 a^2+\frac {32 a^5}{(a-x)^3}-\frac {80 a^4}{(a-x)^2}+\frac {80 a^3}{a-x}-8 a x-x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {80 a^8 \log (1-\sin (c+d x))}{d}-\frac {31 a^8 \sin (c+d x)}{d}-\frac {4 a^8 \sin ^2(c+d x)}{d}-\frac {a^8 \sin ^3(c+d x)}{3 d}+\frac {16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac {80 a^9}{d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 73, normalized size = 0.66 \begin {gather*} \frac {a^8 \left (-80 \log (1-\sin (c+d x))-31 \sin (c+d x)-4 \sin ^2(c+d x)-\frac {1}{3} \sin ^3(c+d x)+\frac {16 (-4+5 \sin (c+d x))}{(-1+\sin (c+d x))^2}\right )}{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. \(526\) vs.
\(2(108)=216\).
time = 0.20, size = 527, normalized size = 4.79 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 95, normalized size = 0.86 \begin {gather*} -\frac {a^{8} \sin \left (d x + c\right )^{3} + 12 \, a^{8} \sin \left (d x + c\right )^{2} + 240 \, a^{8} \log \left (\sin \left (d x + c\right ) - 1\right ) + 93 \, a^{8} \sin \left (d x + c\right ) - \frac {48 \, {\left (5 \, a^{8} \sin \left (d x + c\right ) - 4 \, a^{8}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 139, normalized size = 1.26 \begin {gather*} \frac {10 \, a^{8} \cos \left (d x + c\right )^{4} + 160 \, a^{8} \cos \left (d x + c\right )^{2} + 16 \, a^{8} - 240 \, {\left (a^{8} \cos \left (d x + c\right )^{2} + 2 \, a^{8} \sin \left (d x + c\right ) - 2 \, a^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (a^{8} \cos \left (d x + c\right )^{4} - 72 \, a^{8} \cos \left (d x + c\right )^{2} - 64 \, a^{8}\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs.
\(2 (110) = 220\).
time = 6.20, size = 243, normalized size = 2.21 \begin {gather*} \frac {2 \, {\left (120 \, a^{8} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 240 \, a^{8} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {220 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 93 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 684 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 190 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 684 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 93 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 220 \, a^{8}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}} + \frac {4 \, {\left (125 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 536 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 846 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 536 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 125 \, a^{8}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{4}}\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.60, size = 96, normalized size = 0.87 \begin {gather*} -\frac {80\,a^8\,\ln \left (\sin \left (c+d\,x\right )-1\right )+31\,a^8\,\sin \left (c+d\,x\right )-\frac {80\,a^8\,\sin \left (c+d\,x\right )-64\,a^8}{{\sin \left (c+d\,x\right )}^2-2\,\sin \left (c+d\,x\right )+1}+4\,a^8\,{\sin \left (c+d\,x\right )}^2+\frac {a^8\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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