Optimal. Leaf size=162 \[ -\frac {128 a^8 \log (1-\sin (c+d x))}{d}-\frac {64 a^8 \sin (c+d x)}{d}-\frac {16 a^5 (a+a \sin (c+d x))^3}{3 d}-\frac {4 a^3 (a+a \sin (c+d x))^5}{5 d}-\frac {a^2 (a+a \sin (c+d x))^6}{3 d}-\frac {a (a+a \sin (c+d x))^7}{7 d}-\frac {2 \left (a^2+a^2 \sin (c+d x)\right )^4}{d}-\frac {16 \left (a^4+a^4 \sin (c+d x)\right )^2}{d} \]
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Rubi [A]
time = 0.05, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2746, 45}
\begin {gather*} -\frac {64 a^8 \sin (c+d x)}{d}-\frac {128 a^8 \log (1-\sin (c+d x))}{d}-\frac {16 a^5 (a \sin (c+d x)+a)^3}{3 d}-\frac {16 \left (a^4 \sin (c+d x)+a^4\right )^2}{d}-\frac {4 a^3 (a \sin (c+d x)+a)^5}{5 d}-\frac {a^2 (a \sin (c+d x)+a)^6}{3 d}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right )^4}{d}-\frac {a (a \sin (c+d x)+a)^7}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac {a \text {Subst}\left (\int \frac {(a+x)^7}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \text {Subst}\left (\int \left (-64 a^6+\frac {128 a^7}{a-x}-32 a^5 (a+x)-16 a^4 (a+x)^2-8 a^3 (a+x)^3-4 a^2 (a+x)^4-2 a (a+x)^5-(a+x)^6\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {128 a^8 \log (1-\sin (c+d x))}{d}-\frac {64 a^8 \sin (c+d x)}{d}-\frac {16 a^5 (a+a \sin (c+d x))^3}{3 d}-\frac {4 a^3 (a+a \sin (c+d x))^5}{5 d}-\frac {a^2 (a+a \sin (c+d x))^6}{3 d}-\frac {a (a+a \sin (c+d x))^7}{7 d}-\frac {2 \left (a^2+a^2 \sin (c+d x)\right )^4}{d}-\frac {16 \left (a^4+a^4 \sin (c+d x)\right )^2}{d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 95, normalized size = 0.59 \begin {gather*} \frac {a^8 \left (-128 \log (1-\sin (c+d x))-127 \sin (c+d x)-60 \sin ^2(c+d x)-33 \sin ^3(c+d x)-16 \sin ^4(c+d x)-\frac {29}{5} \sin ^5(c+d x)-\frac {4}{3} \sin ^6(c+d x)-\frac {1}{7} \sin ^7(c+d x)\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. \(311\) vs.
\(2(154)=308\).
time = 0.22, size = 312, normalized size = 1.93 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 109, normalized size = 0.67 \begin {gather*} -\frac {15 \, a^{8} \sin \left (d x + c\right )^{7} + 140 \, a^{8} \sin \left (d x + c\right )^{6} + 609 \, a^{8} \sin \left (d x + c\right )^{5} + 1680 \, a^{8} \sin \left (d x + c\right )^{4} + 3465 \, a^{8} \sin \left (d x + c\right )^{3} + 6300 \, a^{8} \sin \left (d x + c\right )^{2} + 13440 \, a^{8} \log \left (\sin \left (d x + c\right ) - 1\right ) + 13335 \, a^{8} \sin \left (d x + c\right )}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 114, normalized size = 0.70 \begin {gather*} \frac {140 \, a^{8} \cos \left (d x + c\right )^{6} - 2100 \, a^{8} \cos \left (d x + c\right )^{4} + 10080 \, a^{8} \cos \left (d x + c\right )^{2} - 13440 \, a^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (5 \, a^{8} \cos \left (d x + c\right )^{6} - 218 \, a^{8} \cos \left (d x + c\right )^{4} + 1576 \, a^{8} \cos \left (d x + c\right )^{2} - 5808 \, a^{8}\right )} \sin \left (d x + c\right )}{105 \, d} \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 = 4.95, size = 288, normalized size = 1.78 \begin {gather*} \frac {2 \, {\left (6720 \, a^{8} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 13440 \, a^{8} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {17424 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 13335 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 134568 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 93870 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 442344 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 265209 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 780640 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 370308 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 780640 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 265209 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 442344 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 93870 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 134568 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 13335 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17424 \, a^{8}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}\right )}}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.65, size = 109, normalized size = 0.67 \begin {gather*} -\frac {128\,a^8\,\ln \left (\sin \left (c+d\,x\right )-1\right )+127\,a^8\,\sin \left (c+d\,x\right )+60\,a^8\,{\sin \left (c+d\,x\right )}^2+33\,a^8\,{\sin \left (c+d\,x\right )}^3+16\,a^8\,{\sin \left (c+d\,x\right )}^4+\frac {29\,a^8\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {4\,a^8\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {a^8\,{\sin \left (c+d\,x\right )}^7}{7}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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