Optimal. Leaf size=111 \[ \frac {\sec ^3(e+f x)}{7 a^2 f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {4 \tan (e+f x)}{7 a^2 c^4 f}+\frac {4 \tan ^3(e+f x)}{21 a^2 c^4 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751,
3852} \begin {gather*} \frac {4 \tan ^3(e+f x)}{21 a^2 c^4 f}+\frac {4 \tan (e+f x)}{7 a^2 c^4 f}+\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^2-c^2 \sin (e+f x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2751
Rule 2815
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^4} \, dx &=\frac {\int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{a^2 c^2}\\ &=\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {5 \int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)} \, dx}{7 a^2 c^3}\\ &=\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {4 \int \sec ^4(e+f x) \, dx}{7 a^2 c^4}\\ &=\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^4-c^4 \sin (e+f x)\right )}-\frac {4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{7 a^2 c^4 f}\\ &=\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {\sec ^3(e+f x)}{7 a^2 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {4 \tan (e+f x)}{7 a^2 c^4 f}+\frac {4 \tan ^3(e+f x)}{21 a^2 c^4 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.64, size = 151, normalized size = 1.36 \begin {gather*} \frac {-210+512 \cos (e+f x)-255 \cos (2 (e+f x))+768 \cos (3 (e+f x))-30 \cos (4 (e+f x))+256 \cos (5 (e+f x))+15 \cos (6 (e+f x))+120 \sin (e+f x)+1088 \sin (2 (e+f x))+180 \sin (3 (e+f x))+128 \sin (4 (e+f x))+60 \sin (5 (e+f x))-64 \sin (6 (e+f x))}{5376 a^2 c^4 f (-1+\sin (e+f x))^4 (1+\sin (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.38, size = 163, normalized size = 1.47
method | result | size |
risch | \(-\frac {16 i \left (-8 i {\mathrm e}^{3 i \left (f x +e \right )}+14 \,{\mathrm e}^{4 i \left (f x +e \right )}-4 i {\mathrm e}^{i \left (f x +e \right )}+3 \,{\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{21 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,c^{4} a^{2}}\) | \(89\) |
derivativedivides | \(\frac {-\frac {4}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {55}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {23}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {13}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{2} f \,c^{4}}\) | \(163\) |
default | \(\frac {-\frac {4}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {55}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {23}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {13}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{2} f \,c^{4}}\) | \(163\) |
norman | \(\frac {\frac {4 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {1}{14 a c f}-\frac {\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a c f}+\frac {5 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a c f}-\frac {20 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {12 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7 a c f}-\frac {68 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 a c f}+\frac {5 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {13 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}+\frac {53 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{14 a c f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 463 vs.
\(2 (111) = 222\).
time = 0.33, size = 463, normalized size = 4.17 \begin {gather*} -\frac {2 \, {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {24 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {76 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {28 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {42 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {56 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {28 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {42 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {21 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - 6\right )}}{21 \, {\left (a^{2} c^{4} - \frac {4 \, a^{2} c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {8 \, a^{2} c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {14 \, a^{2} c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {14 \, a^{2} c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {8 \, a^{2} c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {3 \, a^{2} c^{4} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {4 \, a^{2} c^{4} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - \frac {a^{2} c^{4} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.31, size = 122, normalized size = 1.10 \begin {gather*} -\frac {16 \, \cos \left (f x + e\right )^{4} - 8 \, \cos \left (f x + e\right )^{2} - {\left (8 \, \cos \left (f x + e\right )^{4} - 12 \, \cos \left (f x + e\right )^{2} - 5\right )} \sin \left (f x + e\right ) - 2}{21 \, {\left (a^{2} c^{4} f \cos \left (f x + e\right )^{5} + 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2213 vs.
\(2 (97) = 194\).
time = 12.96, size = 2213, normalized size = 19.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.47, size = 161, normalized size = 1.45 \begin {gather*} -\frac {\frac {7 \, {\left (9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8\right )}}{a^{2} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {273 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1155 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 2450 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2870 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2037 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 791 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 152}{a^{2} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{168 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.98, size = 119, normalized size = 1.07 \begin {gather*} -\frac {\frac {\sin \left (e+f\,x\right )}{3}+\frac {4\,\cos \left (2\,e+2\,f\,x\right )}{21}+\frac {2\,\cos \left (4\,e+4\,f\,x\right )}{21}+\frac {\sin \left (3\,e+3\,f\,x\right )}{14}-\frac {\sin \left (5\,e+5\,f\,x\right )}{42}}{a^2\,c^4\,f\,\left (\frac {\cos \left (5\,e+5\,f\,x\right )}{16}-\frac {3\,\cos \left (3\,e+3\,f\,x\right )}{16}-\frac {7\,\cos \left (e+f\,x\right )}{8}+\frac {\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {\sin \left (4\,e+4\,f\,x\right )}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________