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 = 0.06, 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 = {2749, 2729,
2727} \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2729
Rule 2749
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 = 0.01, size = 110, normalized size = 1.20 \begin {gather*} \frac {7 a^3 \sec ^5(c+d x)}{15 d}+\frac {a^3 \sec ^4(c+d x) \tan (c+d x)}{d}+\frac {a^3 \sec ^3(c+d x) \tan ^2(c+d x)}{3 d}-\frac {a^3 \sec ^2(c+d x) \tan ^3(c+d x)}{3 d}+\frac {2 a^3 \tan ^5(c+d x)}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 171, normalized size = 1.86
method | result | size |
risch | \(-\frac {4 \left (-a^{3}-5 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+10 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{15 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{5} d}\) | \(55\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )+3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {3 a^{3}}{5 \cos \left (d x +c \right )^{5}}-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(171\) |
default | \(\frac {a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{15 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{15}\right )+3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {3 a^{3}}{5 \cos \left (d x +c \right )^{5}}-a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(171\) |
norman | \(\frac {-\frac {14 a^{3}}{15 d}-\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {34 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {494 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {842 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {178 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {842 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {494 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {34 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {22 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {62 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {130 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {302 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {734 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(324\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.42, size = 103, normalized size = 1.12 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 149, normalized size = 1.62 \begin {gather*} \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 )}} \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 = 7.26, size = 86, normalized size = 0.93 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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