Optimal. Leaf size=50 \[ \frac {(A+C) \sin (c+d x)}{d}-\frac {(2 A+C) \sin ^3(c+d x)}{3 d}+\frac {A \sin ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 50, 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 = {4129, 3092,
380} \begin {gather*} -\frac {(2 A+C) \sin ^3(c+d x)}{3 d}+\frac {(A+C) \sin (c+d x)}{d}+\frac {A \sin ^5(c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 380
Rule 3092
Rule 4129
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^3(c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx\\ &=-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (A+C-A x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (A \left (1+\frac {C}{A}\right )-(2 A+C) x^2+A x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {(A+C) \sin (c+d x)}{d}-\frac {(2 A+C) \sin ^3(c+d x)}{3 d}+\frac {A \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 74, normalized size = 1.48 \begin {gather*} \frac {5 A \sin (c+d x)}{8 d}+\frac {C \sin (c+d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d}+\frac {5 A \sin (3 (c+d x))}{48 d}+\frac {A \sin (5 (c+d x))}{80 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 54, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(54\) |
default | \(\frac {\frac {A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(54\) |
risch | \(\frac {5 A \sin \left (d x +c \right )}{8 d}+\frac {3 C \sin \left (d x +c \right )}{4 d}+\frac {A \sin \left (5 d x +5 c \right )}{80 d}+\frac {5 A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) C}{12 d}\) | \(71\) |
norman | \(\frac {-\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (A +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (A +5 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (A +5 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 \left (19 A +5 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {4 \left (19 A +5 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 43, normalized size = 0.86 \begin {gather*} \frac {3 \, A \sin \left (d x + c\right )^{5} - 5 \, {\left (2 \, A + C\right )} \sin \left (d x + c\right )^{3} + 15 \, {\left (A + C\right )} \sin \left (d x + c\right )}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.24, size = 45, normalized size = 0.90 \begin {gather*} \frac {{\left (3 \, A \cos \left (d x + c\right )^{4} + {\left (4 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, A + 10 \, C\right )} \sin \left (d x + c\right )}{15 \, d} \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 + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{5}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 57, normalized size = 1.14 \begin {gather*} \frac {3 \, A \sin \left (d x + c\right )^{5} - 10 \, A \sin \left (d x + c\right )^{3} - 5 \, C \sin \left (d x + c\right )^{3} + 15 \, A \sin \left (d x + c\right ) + 15 \, C \sin \left (d x + c\right )}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.37, size = 43, normalized size = 0.86 \begin {gather*} \frac {\frac {A\,{\sin \left (c+d\,x\right )}^5}{5}+\left (-\frac {2\,A}{3}-\frac {C}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (A+C\right )\,\sin \left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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