Optimal. Leaf size=64 \[ \frac {2 \sec ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{5 d}+\frac {3 a^2 \tan (c+d x)}{5 d}+\frac {a^2 \tan ^3(c+d x)}{5 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2755, 3852}
\begin {gather*} \frac {a^2 \tan ^3(c+d x)}{5 d}+\frac {3 a^2 \tan (c+d x)}{5 d}+\frac {2 \sec ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2755
Rule 3852
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {2 \sec ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{5 d}+\frac {1}{5} \left (3 a^2\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac {2 \sec ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{5 d}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac {2 \sec ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{5 d}+\frac {3 a^2 \tan (c+d x)}{5 d}+\frac {a^2 \tan ^3(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 82, normalized size = 1.28 \begin {gather*} \frac {2 a^2 \sec ^5(c+d x)}{5 d}+\frac {a^2 \sec ^4(c+d x) \tan (c+d x)}{d}-\frac {a^2 \sec ^2(c+d x) \tan ^3(c+d x)}{d}+\frac {2 a^2 \tan ^5(c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 93, normalized size = 1.45
method | result | size |
risch | \(-\frac {4 i a^{2} \left (-4 i {\mathrm e}^{i \left (d x +c \right )}+5 \,{\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{5 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) d}\) | \(63\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {2 a^{2}}{5 \cos \left (d x +c \right )^{5}}-a^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(93\) |
default | \(\frac {a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {2 a^{2}}{5 \cos \left (d x +c \right )^{5}}-a^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(93\) |
norman | \(\frac {-\frac {4 a^{2}}{5 d}-\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {54 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {88 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {54 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {4 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {8 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {12 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {16 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {44 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(286\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 77, normalized size = 1.20 \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^{2} + {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} a^{2} + \frac {6 \, a^{2}}{\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 = 0.34, size = 85, normalized size = 1.33 \begin {gather*} -\frac {4 \, a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2}\right )} \sin \left (d x + c\right )}{5 \, {\left (d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.98, size = 106, normalized size = 1.66 \begin {gather*} -\frac {\frac {5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1} + \frac {35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 90 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, a^{2}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{5}}}{20 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.63, size = 156, normalized size = 2.44 \begin {gather*} \frac {2\,a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+10\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-10\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}{5\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^5\,\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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