Optimal. Leaf size=210 \[ \frac{256 a^2 c^6 (13 A-3 B) \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 c^5 (13 A-3 B) \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 (13 A-3 B) \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^3 (13 A-3 B) \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.554384, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2967, 2856, 2674, 2673} \[ \frac{256 a^2 c^6 (13 A-3 B) \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 c^5 (13 A-3 B) \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 (13 A-3 B) \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^3 (13 A-3 B) \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2967
Rule 2856
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac{1}{13} \left (a^2 (13 A-3 B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac{2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac{1}{143} \left (12 a^2 (13 A-3 B) c^3\right ) \int \cos ^4(e+f x) \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac{1}{429} \left (32 a^2 (13 A-3 B) c^4\right ) \int \frac{\cos ^4(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac{\left (128 a^2 (13 A-3 B) c^5\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx}{3003}\\ &=\frac{256 a^2 (13 A-3 B) c^6 \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}\\ \end{align*}
Mathematica [B] time = 6.70347, size = 1355, normalized size = 6.45 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.12, size = 121, normalized size = 0.6 \begin{align*}{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ){c}^{4} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{3}{a}^{2} \left ( \left ( -1365\,A+4935\,B \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+ \left ( 11180\,A-11820\,B \right ) \sin \left ( fx+e \right ) +1155\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( 5915\,A-10605\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}-12844\,A+12204\,B \right ) }{15015\,f\cos \left ( fx+e \right ) }{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54687, size = 895, normalized size = 4.26 \begin{align*} \frac{2 \,{\left (1155 \, B a^{2} c^{3} \cos \left (f x + e\right )^{7} + 105 \,{\left (13 \, A - 14 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{6} + 35 \,{\left (91 \, A - 87 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} - 20 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 32 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} - 64 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right ) + 512 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} +{\left (1155 \, B a^{2} c^{3} \cos \left (f x + e\right )^{6} - 105 \,{\left (13 \, A - 25 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} + 140 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 160 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} + 192 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right ) + 512 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{15015 \,{\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\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 [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]