Optimal. Leaf size=101 \[ -\frac {2 a^2 \cos (c+d x)}{d}-\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {11 a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^2 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {7 a^2 x}{2} \]
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Rubi [A] time = 0.21, antiderivative size = 120, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2708, 2765, 2977, 2734} \[ -\frac {16 a^2 \cos (c+d x)}{3 d}+\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {8 a^2 \sin ^2(c+d x) \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {7 a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {7 a^2 x}{2} \]
Antiderivative was successfully verified.
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Rule 2708
Rule 2734
Rule 2765
Rule 2977
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^2 \tan ^4(c+d x) \, dx &=a^4 \int \frac {\sin ^4(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{3} a^2 \int \frac {\sin ^2(c+d x) (-3 a-5 a \sin (c+d x))}{a-a \sin (c+d x)} \, dx\\ &=-\frac {8 a^2 \cos (c+d x) \sin ^2(c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {1}{3} \int \sin (c+d x) \left (-16 a^2-21 a^2 \sin (c+d x)\right ) \, dx\\ &=\frac {7 a^2 x}{2}-\frac {16 a^2 \cos (c+d x)}{3 d}-\frac {7 a^2 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {8 a^2 \cos (c+d x) \sin ^2(c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{3 d (a-a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 1.26, size = 159, normalized size = 1.57 \[ -\frac {a^2 \left (-21 (12 c+12 d x+7) \cos \left (\frac {1}{2} (c+d x)\right )+(84 c+84 d x+239) \cos \left (\frac {3}{2} (c+d x)\right )+3 \left (-5 \cos \left (\frac {5}{2} (c+d x)\right )+\cos \left (\frac {7}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right ) ((28 c+28 d x-27) \cos (c+d x)-6 \cos (2 (c+d x))-\cos (3 (c+d x))+56 c+56 d x+50)\right )\right )}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 196, normalized size = 1.94 \[ \frac {3 \, a^{2} \cos \left (d x + c\right )^{4} - 6 \, a^{2} \cos \left (d x + c\right )^{3} - 42 \, a^{2} d x + {\left (21 \, a^{2} d x + 31 \, a^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (21 \, a^{2} d x - 38 \, a^{2}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{2} \cos \left (d x + c\right )^{3} - 42 \, a^{2} d x + 9 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} - {\left (21 \, a^{2} d x - 40 \, a^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 135, normalized size = 1.34 \[ \frac {21 \, {\left (d x + c\right )} a^{2} + \frac {6 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {4 \, {\left (9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 186, normalized size = 1.84 \[ \frac {a^{2} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+2 a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a^{2} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 120, normalized size = 1.19 \[ \frac {{\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{2} + 2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{2} - 4 \, a^{2} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.69, size = 287, normalized size = 2.84 \[ \frac {7\,a^2\,x}{2}+\frac {\frac {7\,a^2\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {21\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (63\,c+63\,d\,x-150\right )}{6}\right )-\frac {a^2\,\left (21\,c+21\,d\,x-64\right )}{6}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {21\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (63\,c+63\,d\,x-42\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {35\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (105\,c+105\,d\,x-126\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {35\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (105\,c+105\,d\,x-194\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {49\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (147\,c+147\,d\,x-196\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {49\,a^2\,\left (c+d\,x\right )}{2}-\frac {a^2\,\left (147\,c+147\,d\,x-252\right )}{6}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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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