Optimal. Leaf size=54 \[ \frac {(a-b) \sin ^5(c+d x)}{5 d}-\frac {(2 a-b) \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 54, 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 = {3676, 373} \[ \frac {(a-b) \sin ^5(c+d x)}{5 d}-\frac {(2 a-b) \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3676
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a-(a-b) x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a-(2 a-b) x^2+(a-b) x^4\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a \sin (c+d x)}{d}-\frac {(2 a-b) \sin ^3(c+d x)}{3 d}+\frac {(a-b) \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 52, normalized size = 0.96 \[ \frac {\sin (c+d x) (4 (7 a-2 b) \cos (2 (c+d x))+3 (a-b) \cos (4 (c+d x))+89 a+11 b)}{120 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 47, normalized size = 0.87 \[ \frac {{\left (3 \, {\left (a - b\right )} \cos \left (d x + c\right )^{4} + {\left (4 \, a + b\right )} \cos \left (d x + c\right )^{2} + 8 \, a + 2 \, b\right )} \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.80, size = 72, normalized size = 1.33 \[ \frac {\frac {a \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 47, normalized size = 0.87 \[ \frac {3 \, {\left (a - b\right )} \sin \left (d x + c\right )^{5} - 5 \, {\left (2 \, a - b\right )} \sin \left (d x + c\right )^{3} + 15 \, a \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.07, size = 71, normalized size = 1.31 \[ \frac {\frac {5\,a\,\sin \left (c+d\,x\right )}{8}+\frac {b\,\sin \left (c+d\,x\right )}{8}+\frac {5\,a\,\sin \left (3\,c+3\,d\,x\right )}{48}+\frac {a\,\sin \left (5\,c+5\,d\,x\right )}{80}-\frac {b\,\sin \left (3\,c+3\,d\,x\right )}{48}-\frac {b\,\sin \left (5\,c+5\,d\,x\right )}{80}}{d} \]
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 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right ) \cos ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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