Optimal. Leaf size=90 \[ -\frac{2 b \left (4 a^2+b^2\right ) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left (2 a^2+3 b^2\right )-\frac{5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0687045, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2656, 2734} \[ -\frac{2 b \left (4 a^2+b^2\right ) \cos (e+f x)}{3 f}+\frac{1}{2} a x \left (2 a^2+3 b^2\right )-\frac{5 a b^2 \sin (e+f x) \cos (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2656
Rule 2734
Rubi steps
\begin{align*} \int (a+b \sin (e+f x))^3 \, dx &=-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}+\frac{1}{3} \int (a+b \sin (e+f x)) \left (3 a^2+2 b^2+5 a b \sin (e+f x)\right ) \, dx\\ &=\frac{1}{2} a \left (2 a^2+3 b^2\right ) x-\frac{2 b \left (4 a^2+b^2\right ) \cos (e+f x)}{3 f}-\frac{5 a b^2 \cos (e+f x) \sin (e+f x)}{6 f}-\frac{b \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}\\ \end{align*}
Mathematica [A] time = 0.169682, size = 71, normalized size = 0.79 \[ \frac{6 a \left (2 a^2+3 b^2\right ) (e+f x)-9 b \left (4 a^2+b^2\right ) \cos (e+f x)-9 a b^2 \sin (2 (e+f x))+b^3 \cos (3 (e+f x))}{12 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 76, normalized size = 0.8 \begin{align*}{\frac{1}{f} \left ( -{\frac{{b}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+3\,a{b}^{2} \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -3\,{a}^{2}b\cos \left ( fx+e \right ) +{a}^{3} \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.038, size = 100, normalized size = 1.11 \begin{align*} a^{3} x + \frac{3 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2}}{4 \, f} + \frac{{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{3}}{3 \, f} - \frac{3 \, a^{2} b \cos \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56898, size = 169, normalized size = 1.88 \begin{align*} \frac{2 \, b^{3} \cos \left (f x + e\right )^{3} - 9 \, a b^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} f x - 6 \,{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.654677, size = 128, normalized size = 1.42 \begin{align*} \begin{cases} a^{3} x - \frac{3 a^{2} b \cos{\left (e + f x \right )}}{f} + \frac{3 a b^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{3 a b^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{3 a b^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{b^{3} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{2 b^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} & \text{for}\: f \neq 0 \\x \left (a + b \sin{\left (e \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.43832, size = 101, normalized size = 1.12 \begin{align*} \frac{b^{3} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac{3 \, a b^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac{1}{2} \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} x - \frac{3 \,{\left (4 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]