Optimal. Leaf size=44 \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.06, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3090, 3767, 2606, 30} \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 3090
Rule 3767
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\int \left (a \sec ^4(c+d x)+b \sec ^4(c+d x) \tan (c+d x)\right ) \, dx\\ &=a \int \sec ^4(c+d x) \, dx+b \int \sec ^4(c+d x) \tan (c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}+\frac {b \operatorname {Subst}\left (\int x^3 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {b \sec ^4(c+d x)}{4 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 41, normalized size = 0.93 \[ \frac {a \left (\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac {b \sec ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 45, normalized size = 1.02 \[ \frac {4 \, {\left (2 \, a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 3 \, b}{12 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 48, normalized size = 1.09 \[ \frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 6 \, b \tan \left (d x + c\right )^{2} + 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.75, size = 38, normalized size = 0.86 \[ \frac {-a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {b}{4 \cos \left (d x +c \right )^{4}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 41, normalized size = 0.93 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a + \frac {3 \, b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 40, normalized size = 0.91 \[ \frac {\frac {b}{4}+\frac {a\,\sin \left (2\,c+2\,d\,x\right )}{3}+\frac {a\,\sin \left (4\,c+4\,d\,x\right )}{12}}{d\,{\cos \left (c+d\,x\right )}^4} \]
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 \[ \int \left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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