Optimal. Leaf size=83 \[ \frac{(6 a+b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{(6 a+b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x (6 a+b)-\frac{b \sin (e+f x) \cos ^5(e+f x)}{6 f} \]
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Rubi [A] time = 0.0511575, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3191, 385, 199, 203} \[ \frac{(6 a+b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{(6 a+b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x (6 a+b)-\frac{b \sin (e+f x) \cos ^5(e+f x)}{6 f} \]
Antiderivative was successfully verified.
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Rule 3191
Rule 385
Rule 199
Rule 203
Rubi steps
\begin{align*} \int \cos ^4(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+(a+b) x^2}{\left (1+x^2\right )^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{b \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{(6 a+b) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac{(6 a+b) \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{b \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{(6 a+b) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac{(6 a+b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac{(6 a+b) \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{b \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{(6 a+b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 f}\\ &=\frac{1}{16} (6 a+b) x+\frac{(6 a+b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac{(6 a+b) \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{b \cos ^5(e+f x) \sin (e+f x)}{6 f}\\ \end{align*}
Mathematica [A] time = 0.14942, size = 64, normalized size = 0.77 \[ \frac{3 (16 a+b) \sin (2 (e+f x))+(6 a-3 b) \sin (4 (e+f x))+72 a e+72 a f x-b \sin (6 (e+f x))+12 b f x}{192 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 92, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ( b \left ( -{\frac{\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{5}}{6}}+{\frac{\sin \left ( fx+e \right ) }{24} \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\cos \left ( fx+e \right ) }{2}} \right ) }+{\frac{fx}{16}}+{\frac{e}{16}} \right ) +a \left ({\frac{\sin \left ( fx+e \right ) }{4} \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\cos \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46822, size = 131, normalized size = 1.58 \begin{align*} \frac{3 \,{\left (f x + e\right )}{\left (6 \, a + b\right )} + \frac{3 \,{\left (6 \, a + b\right )} \tan \left (f x + e\right )^{5} + 8 \,{\left (6 \, a + b\right )} \tan \left (f x + e\right )^{3} + 3 \,{\left (10 \, a - b\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{6} + 3 \, \tan \left (f x + e\right )^{4} + 3 \, \tan \left (f x + e\right )^{2} + 1}}{48 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93832, size = 159, normalized size = 1.92 \begin{align*} \frac{3 \,{\left (6 \, a + b\right )} f x -{\left (8 \, b \cos \left (f x + e\right )^{5} - 2 \,{\left (6 \, a + b\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left (6 \, a + b\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.95717, size = 250, normalized size = 3.01 \begin{align*} \begin{cases} \frac{3 a x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{3 a x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{3 a x \cos ^{4}{\left (e + f x \right )}}{8} + \frac{3 a \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} + \frac{5 a \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac{b x \sin ^{6}{\left (e + f x \right )}}{16} + \frac{3 b x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac{3 b x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac{b x \cos ^{6}{\left (e + f x \right )}}{16} + \frac{b \sin ^{5}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{16 f} + \frac{b \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac{b \sin{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} & \text{for}\: f \neq 0 \\x \left (a + b \sin ^{2}{\left (e \right )}\right ) \cos ^{4}{\left (e \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13837, size = 90, normalized size = 1.08 \begin{align*} \frac{1}{16} \,{\left (6 \, a + b\right )} x - \frac{b \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{{\left (2 \, a - b\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac{{\left (16 \, a + b\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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