Optimal. Leaf size=104 \[ -\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c^2 f}+\frac{16 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c f}-\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.269207, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {2736, 2674, 2673} \[ -\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c^2 f}+\frac{16 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c f}-\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2736
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int \frac{(c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^3} \, dx &=\frac{\int \sec ^6(e+f x) (c-c \sin (e+f x))^{11/2} \, dx}{a^3 c^3}\\ &=-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c^2 f}-\frac{8 \int \sec ^6(e+f x) (c-c \sin (e+f x))^{9/2} \, dx}{a^3 c^2}\\ &=\frac{16 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c f}-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c^2 f}+\frac{32 \int \sec ^6(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{3 a^3 c}\\ &=-\frac{64 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 a^3 f}+\frac{16 \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 a^3 c f}-\frac{2 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{a^3 c^2 f}\\ \end{align*}
Mathematica [A] time = 0.841201, size = 104, normalized size = 1. \[ \frac{c^2 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (-20 \sin (e+f x)+15 \cos (2 (e+f x))-29)}{15 a^3 f (\sin (e+f x)+1)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.504, size = 71, normalized size = 0.7 \begin{align*}{\frac{2\,{c}^{3} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 15\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}+10\,\sin \left ( fx+e \right ) +7 \right ) }{15\,{a}^{3} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) f}{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.82795, size = 513, normalized size = 4.93 \begin{align*} \frac{2 \,{\left (7 \, c^{\frac{5}{2}} + \frac{20 \, c^{\frac{5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{95 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{80 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{250 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{120 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac{250 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac{80 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac{95 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac{20 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac{7 \, c^{\frac{5}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}}\right )}}{15 \,{\left (a^{3} + \frac{5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} f{\left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.12084, size = 230, normalized size = 2.21 \begin{align*} -\frac{2 \,{\left (15 \, c^{2} \cos \left (f x + e\right )^{2} - 10 \, c^{2} \sin \left (f x + e\right ) - 22 \, c^{2}\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{15 \,{\left (a^{3} f \cos \left (f x + e\right )^{3} - 2 \, a^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} f \cos \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 = 1.94364, size = 878, normalized size = 8.44 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]