Optimal. Leaf size=92 \[ \frac {a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac {2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac {2 a^6 \cos (c+d x)}{15 d \left (a^3-a^3 \sin (c+d x)\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2670, 2650, 2648} \[ \frac {2 a^6 \cos (c+d x)}{15 d \left (a^3-a^3 \sin (c+d x)\right )}+\frac {a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac {2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2670
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=a^6 \int \frac {1}{(a-a \sin (c+d x))^3} \, dx\\ &=\frac {a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac {1}{5} \left (2 a^5\right ) \int \frac {1}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac {a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac {2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac {1}{15} \left (2 a^4\right ) \int \frac {1}{a-a \sin (c+d x)} \, dx\\ &=\frac {a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac {2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac {2 a^4 \cos (c+d x)}{15 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 110, normalized size = 1.20 \[ \frac {2 a^3 \tan ^5(c+d x)}{15 d}+\frac {7 a^3 \sec ^5(c+d x)}{15 d}+\frac {a^3 \tan (c+d x) \sec ^4(c+d x)}{d}+\frac {a^3 \tan ^2(c+d x) \sec ^3(c+d x)}{3 d}-\frac {a^3 \tan ^3(c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 149, normalized size = 1.62 \[ \frac {2 \, a^{3} \cos \left (d x + c\right )^{3} - 4 \, a^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{3} \cos \left (d x + c\right ) - 3 \, a^{3} + {\left (2 \, a^{3} \cos \left (d x + c\right )^{2} + 6 \, a^{3} \cos \left (d x + c\right ) - 3 \, a^{3}\right )} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right )^{3} + 3 \, d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) - 4 \, d\right )} \sin \left (d x + c\right ) - 4 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 86, normalized size = 0.93 \[ -\frac {2 \, {\left (15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 20 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{3}\right )}}{15 \, d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 171, normalized size = 1.86 \[ \frac {a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )+3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {3 a^{3}}{5 \cos \left (d x +c \right )^{5}}-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 103, normalized size = 1.12 \[ \frac {{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{3} + 3 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} a^{3} - \frac {{\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} a^{3}}{\cos \left (d x + c\right )^{5}} + \frac {9 \, a^{3}}{\cos \left (d x + c\right )^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 135, normalized size = 1.47 \[ \frac {2\,a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-20\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+40\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-30\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}{15\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^5} \]
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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