Optimal. Leaf size=247 \[ \frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {17 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac {a^2}{24 d (a-a \sin (c+d x))^3}+\frac {125 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac {109 a}{512 d (a-a \sin (c+d x))^2}-\frac {515 a}{512 d (a \sin (c+d x)+a)^2}-\frac {203}{256 d (a-a \sin (c+d x))}+\frac {5}{2 d (a \sin (c+d x)+a)}-\frac {\sin (c+d x)}{a d}-\frac {437 \log (1-\sin (c+d x))}{512 a d}+\frac {949 \log (\sin (c+d x)+1)}{512 a d} \]
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Rubi [A] time = 0.26, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ \frac {a^4}{160 d (a \sin (c+d x)+a)^5}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {17 a^3}{256 d (a \sin (c+d x)+a)^4}-\frac {a^2}{24 d (a-a \sin (c+d x))^3}+\frac {125 a^2}{384 d (a \sin (c+d x)+a)^3}+\frac {109 a}{512 d (a-a \sin (c+d x))^2}-\frac {515 a}{512 d (a \sin (c+d x)+a)^2}-\frac {203}{256 d (a-a \sin (c+d x))}+\frac {5}{2 d (a \sin (c+d x)+a)}-\frac {\sin (c+d x)}{a d}-\frac {437 \log (1-\sin (c+d x))}{512 a d}+\frac {949 \log (\sin (c+d x)+1)}{512 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x) \tan ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^9 \operatorname {Subst}\left (\int \frac {x^{11}}{a^{11} (a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^{11}}{(a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+\frac {a^5}{64 (a-x)^5}-\frac {a^4}{8 (a-x)^4}+\frac {109 a^3}{256 (a-x)^3}-\frac {203 a^2}{256 (a-x)^2}+\frac {437 a}{512 (a-x)}-\frac {a^6}{32 (a+x)^6}+\frac {17 a^5}{64 (a+x)^5}-\frac {125 a^4}{128 (a+x)^4}+\frac {515 a^3}{256 (a+x)^3}-\frac {5 a^2}{2 (a+x)^2}+\frac {949 a}{512 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac {437 \log (1-\sin (c+d x))}{512 a d}+\frac {949 \log (1+\sin (c+d x))}{512 a d}-\frac {\sin (c+d x)}{a d}+\frac {a^3}{256 d (a-a \sin (c+d x))^4}-\frac {a^2}{24 d (a-a \sin (c+d x))^3}+\frac {109 a}{512 d (a-a \sin (c+d x))^2}-\frac {203}{256 d (a-a \sin (c+d x))}+\frac {a^4}{160 d (a+a \sin (c+d x))^5}-\frac {17 a^3}{256 d (a+a \sin (c+d x))^4}+\frac {125 a^2}{384 d (a+a \sin (c+d x))^3}-\frac {515 a}{512 d (a+a \sin (c+d x))^2}+\frac {5}{2 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.19, size = 159, normalized size = 0.64 \[ -\frac {7680 \sin (c+d x)+\frac {6090}{1-\sin (c+d x)}-\frac {19200}{\sin (c+d x)+1}-\frac {1635}{(1-\sin (c+d x))^2}+\frac {7725}{(\sin (c+d x)+1)^2}+\frac {320}{(1-\sin (c+d x))^3}-\frac {2500}{(\sin (c+d x)+1)^3}-\frac {30}{(1-\sin (c+d x))^4}+\frac {510}{(\sin (c+d x)+1)^4}-\frac {48}{(\sin (c+d x)+1)^5}+6555 \log (1-\sin (c+d x))-14235 \log (\sin (c+d x)+1)}{7680 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 207, normalized size = 0.84 \[ \frac {7680 \, \cos \left (d x + c\right )^{10} + 17610 \, \cos \left (d x + c\right )^{8} - 27630 \, \cos \left (d x + c\right )^{6} + 15828 \, \cos \left (d x + c\right )^{4} - 5584 \, \cos \left (d x + c\right )^{2} + 14235 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6555 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (3840 \, \cos \left (d x + c\right )^{8} + 3045 \, \cos \left (d x + c\right )^{6} - 1170 \, \cos \left (d x + c\right )^{4} + 344 \, \cos \left (d x + c\right )^{2} - 48\right )} \sin \left (d x + c\right ) + 864}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 167, normalized size = 0.68 \[ \frac {\frac {56940 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {26220 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {30720 \, \sin \left (d x + c\right )}{a} + \frac {5 \, {\left (10925 \, \sin \left (d x + c\right )^{4} - 38828 \, \sin \left (d x + c\right )^{3} + 52242 \, \sin \left (d x + c\right )^{2} - 31444 \, \sin \left (d x + c\right ) + 7129\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {130013 \, \sin \left (d x + c\right )^{5} + 573265 \, \sin \left (d x + c\right )^{4} + 1023830 \, \sin \left (d x + c\right )^{3} + 922030 \, \sin \left (d x + c\right )^{2} + 417605 \, \sin \left (d x + c\right ) + 75961}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 212, normalized size = 0.86 \[ -\frac {\sin \left (d x +c \right )}{a d}+\frac {1}{256 a d \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{24 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {109}{512 a d \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {203}{256 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {437 \ln \left (\sin \left (d x +c \right )-1\right )}{512 a d}+\frac {1}{160 a d \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {17}{256 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {125}{384 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {515}{512 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{2 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {949 \ln \left (1+\sin \left (d x +c \right )\right )}{512 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 225, normalized size = 0.91 \[ \frac {\frac {2 \, {\left (12645 \, \sin \left (d x + c\right )^{8} + 3045 \, \sin \left (d x + c\right )^{7} - 36765 \, \sin \left (d x + c\right )^{6} - 7965 \, \sin \left (d x + c\right )^{5} + 42339 \, \sin \left (d x + c\right )^{4} + 7139 \, \sin \left (d x + c\right )^{3} - 22171 \, \sin \left (d x + c\right )^{2} - 2171 \, \sin \left (d x + c\right ) + 4384\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} + \frac {14235 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {6555 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} - \frac {7680 \, \sin \left (d x + c\right )}{a}}{7680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.85, size = 595, normalized size = 2.41 \[ \text {result too large to display} \]
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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