Optimal. Leaf size=34 \[ \frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0902245, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2736, 2671} \[ \frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2736
Rule 2671
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac{a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7}\\ \end{align*}
Mathematica [B] time = 0.776164, size = 93, normalized size = 2.74 \[ \frac{a^3 \left (35 \cos \left (\frac{1}{2} (e+f x)\right )-21 \cos \left (\frac{3}{2} (e+f x)\right )-7 \cos \left (\frac{5}{2} (e+f x)\right )+\cos \left (\frac{7}{2} (e+f x)\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{28 c^4 f (\sin (e+f x)-1)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.099, size = 118, normalized size = 3.5 \begin{align*} 2\,{\frac{{a}^{3}}{f{c}^{4}} \left ( -{\frac{64}{7\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-48\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-5}-6\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-2}-20\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-3}-40\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-4}- \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-1}-32\, \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{-6} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.54743, size = 1411, normalized size = 41.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.36482, size = 528, normalized size = 15.53 \begin{align*} \frac{a^{3} \cos \left (f x + e\right )^{4} - 3 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + 8 \, a^{3} -{\left (a^{3} \cos \left (f x + e\right )^{3} + 4 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} \cos \left (f x + e\right ) - 8 \, a^{3}\right )} \sin \left (f x + e\right )}{7 \,{\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f +{\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 2.04922, size = 104, normalized size = 3.06 \begin{align*} -\frac{2 \,{\left (7 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 35 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 21 \, a^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a^{3}\right )}}{7 \, c^{4} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]