Optimal. Leaf size=64 \[ a^3 x-\frac {2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+\frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d} \]
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Rubi [A] time = 0.14, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2855, 2670, 2680, 8} \[ -\frac {2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+a^3 x+\frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2670
Rule 2680
Rule 2855
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx &=\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-a \int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-a^5 \int \frac {\cos ^2(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\frac {2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}+a^3 \int 1 \, dx\\ &=a^3 x+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\frac {2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 107, normalized size = 1.67 \[ -\frac {a^3 \left (-9 (c+d x+2) \cos \left (\frac {1}{2} (c+d x)\right )+(3 c+3 d x+14) \cos \left (\frac {3}{2} (c+d x)\right )+6 \sin \left (\frac {1}{2} (c+d x)\right ) (2 (c+d x+2)+(c+d x) \cos (c+d x))\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 143, normalized size = 2.23 \[ -\frac {6 \, a^{3} d x + 2 \, a^{3} - {\left (3 \, a^{3} d x + 7 \, a^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{3} d x - 5 \, a^{3}\right )} \cos \left (d x + c\right ) - {\left (6 \, a^{3} d x - 2 \, a^{3} + {\left (3 \, a^{3} d x - 7 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 67, normalized size = 1.05 \[ \frac {3 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 126, normalized size = 1.97 \[ \frac {a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+3 a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {a^{3}}{3 \cos \left (d x +c \right )^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 84, normalized size = 1.31 \[ \frac {3 \, a^{3} \tan \left (d x + c\right )^{3} + {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} - \frac {3 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{3}}{\cos \left (d x + c\right )^{3}} + \frac {a^{3}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.26, size = 102, normalized size = 1.59 \[ a^3\,x+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3\,\left (9\,d\,x-24\right )}{3}-3\,a^3\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (9\,d\,x-6\right )}{3}-3\,a^3\,d\,x\right )-\frac {a^3\,\left (3\,d\,x-10\right )}{3}+a^3\,d\,x}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3} \]
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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