Optimal. Leaf size=60 \[ \frac {b \sec ^5(c+d x)}{5 d}+\frac {a \tan (c+d x)}{d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2748, 3852}
\begin {gather*} \frac {a \tan ^5(c+d x)}{5 d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec ^5(c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2748
Rule 3852
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+b \sin (c+d x)) \, dx &=\frac {b \sec ^5(c+d x)}{5 d}+a \int \sec ^6(c+d x) \, dx\\ &=\frac {b \sec ^5(c+d x)}{5 d}-\frac {a \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {b \sec ^5(c+d x)}{5 d}+\frac {a \tan (c+d x)}{d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 53, normalized size = 0.88 \begin {gather*} \frac {b \sec ^5(c+d x)}{5 d}+\frac {a \left (\tan (c+d x)+\frac {2}{3} \tan ^3(c+d x)+\frac {1}{5} \tan ^5(c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 48, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {b}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(48\) |
default | \(\frac {-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {b}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(48\) |
risch | \(\frac {\frac {32 i a \,{\mathrm e}^{4 i \left (d x +c \right )}}{3}+\frac {32 b \,{\mathrm e}^{5 i \left (d x +c \right )}}{5}+\frac {16 i {\mathrm e}^{2 i \left (d x +c \right )} a}{3}+\frac {16 i a}{15}}{d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{5}}\) | \(62\) |
norman | \(\frac {-\frac {2 b}{5 d}-\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {76 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {76 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {2 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {2 b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 b \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(224\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 48, normalized size = 0.80 \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 + \frac {3 \, b}{\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 = 50, normalized size = 0.83 \begin {gather*} \frac {{\left (8 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 3 \, a\right )} \sin \left (d x + c\right ) + 3 \, b}{15 \, d \cos \left (d x + c\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right ) \sec ^{6}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (54) = 108\).
time = 6.15, size = 120, normalized size = 2.00 \begin {gather*} -\frac {2 \, {\left (15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 20 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 58 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 20 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, b\right )}}{15 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.30, size = 75, normalized size = 1.25 \begin {gather*} \frac {b}{5\,d\,{\cos \left (c+d\,x\right )}^5}+\frac {8\,a\,\sin \left (c+d\,x\right )}{15\,d\,\cos \left (c+d\,x\right )}+\frac {4\,a\,\sin \left (c+d\,x\right )}{15\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {a\,\sin \left (c+d\,x\right )}{5\,d\,{\cos \left (c+d\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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