Optimal. Leaf size=44 \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.04, antiderivative size = 44, 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 = {2669, 3767} \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 3767
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+b \sin (c+d x)) \, dx &=\frac {b \sec ^3(c+d x)}{3 d}+a \int \sec ^4(c+d x) \, dx\\ &=\frac {b \sec ^3(c+d x)}{3 d}-\frac {a \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {b \sec ^3(c+d x)}{3 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 = 0.07, size = 41, normalized size = 0.93 \[ \frac {a \left (\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac {b \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 35, normalized size = 0.80 \[ \frac {{\left (2 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + b}{3 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 76, normalized size = 1.73 \[ -\frac {2 \, {\left (3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b\right )}}{3 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 38, normalized size = 0.86 \[ \frac {-a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {b}{3 \cos \left (d x +c \right )^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 35, normalized size = 0.80 \[ \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a + \frac {b}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.27, size = 42, normalized size = 0.95 \[ \frac {\frac {2\,a\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^2}{3}+\frac {b}{3}+\frac {a\,\sin \left (c+d\,x\right )}{3}}{d\,{\cos \left (c+d\,x\right )}^3} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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