Optimal. Leaf size=318 \[ \frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \sin (e+f x) \cos (e+f x)}{240 f}+\frac {1}{16} a^2 x \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right )+\frac {a^2 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.46, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2763, 2753, 2734} \[ \frac {a^2 \left (-311 c^3 d^2-448 c^2 d^3-48 c^4 d+4 c^5-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}+\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 \left (-48 c^2 d+4 c^3-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (-438 c^2 d^2-96 c^3 d+8 c^4-464 c d^3-165 d^4\right ) \sin (e+f x) \cos (e+f x)}{240 f}+\frac {1}{16} a^2 x \left (84 c^2 d^2+64 c^3 d+24 c^4+48 c d^3+11 d^4\right )-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2734
Rule 2753
Rule 2763
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4 \, dx &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int \left (11 a^2 d-a^2 (c-12 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^4 \, dx}{6 d}\\ &=\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (3 a^2 d (17 c+16 d)-a^2 \left (4 c^2-48 c d-55 d^2\right ) \sin (e+f x)\right ) \, dx}{30 d}\\ &=\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 a^2 d \left (64 c^2+112 c d+55 d^2\right )-3 a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \sin (e+f x)\right ) \, dx}{120 d}\\ &=\frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}+\frac {\int (c+d \sin (e+f x)) \left (3 a^2 d \left (184 c^3+432 c^2 d+411 c d^2+128 d^3\right )-3 a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \sin (e+f x)\right ) \, dx}{360 d}\\ &=\frac {1}{16} a^2 \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) x+\frac {a^2 \left (4 c^5-48 c^4 d-311 c^3 d^2-448 c^2 d^3-288 c d^4-64 d^5\right ) \cos (e+f x)}{60 d f}+\frac {a^2 \left (8 c^4-96 c^3 d-438 c^2 d^2-464 c d^3-165 d^4\right ) \cos (e+f x) \sin (e+f x)}{240 f}+\frac {a^2 \left (4 c^3-48 c^2 d-123 c d^2-64 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d f}+\frac {a^2 \left (4 c^2-48 c d-55 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d f}+\frac {a^2 (c-12 d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^5}{6 d f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.40, size = 262, normalized size = 0.82 \[ -\frac {a^2 \cos (e+f x) \left (30 \left (24 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (10 d^2 \left (36 c^2+48 c d+11 d^2\right ) \sin ^3(e+f x)+64 d \left (5 c^3+15 c^2 d+9 c d^2+2 d^3\right ) \sin ^2(e+f x)+15 \left (8 c^4+64 c^3 d+84 c^2 d^2+48 c d^3+11 d^4\right ) \sin (e+f x)+32 \left (15 c^4+50 c^3 d+60 c^2 d^2+36 c d^3+8 d^4\right )+96 d^3 (2 c+d) \sin ^4(e+f x)+40 d^4 \sin ^5(e+f x)\right )\right )}{240 f \sqrt {\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 299, normalized size = 0.94 \[ -\frac {96 \, {\left (2 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right )^{5} - 320 \, {\left (a^{2} c^{3} d + 3 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (24 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 84 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 11 \, a^{2} d^{4}\right )} f x + 480 \, {\left (a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, a^{2} d^{4} \cos \left (f x + e\right )^{5} - 2 \, {\left (36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 108 \, a^{2} c^{2} d^{2} + 80 \, a^{2} c d^{3} + 21 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 458, normalized size = 1.44 \[ \frac {a^{2} c d^{3} \cos \left (3 \, f x + 3 \, e\right )}{3 \, f} - \frac {a^{2} d^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {a^{2} d^{4} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {1}{16} \, {\left (8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} x + \frac {1}{8} \, {\left (8 \, a^{2} c^{4} + 24 \, a^{2} c^{2} d^{2} + 3 \, a^{2} d^{4}\right )} x - \frac {{\left (2 \, a^{2} c d^{3} + a^{2} d^{4}\right )} \cos \left (5 \, f x + 5 \, e\right )}{40 \, f} + \frac {{\left (8 \, a^{2} c^{3} d + 24 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} \cos \left (3 \, f x + 3 \, e\right )}{24 \, f} - \frac {{\left (8 \, a^{2} c^{4} + 12 \, a^{2} c^{3} d + 36 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 5 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (4 \, a^{2} c^{3} d + 3 \, a^{2} c d^{3}\right )} \cos \left (f x + e\right )}{f} + \frac {{\left (12 \, a^{2} c^{2} d^{2} + 16 \, a^{2} c d^{3} + 3 \, a^{2} d^{4}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {{\left (16 \, a^{2} c^{4} + 128 \, a^{2} c^{3} d + 96 \, a^{2} c^{2} d^{2} + 128 \, a^{2} c d^{3} + 15 \, a^{2} d^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} - \frac {{\left (6 \, a^{2} c^{2} d^{2} + a^{2} d^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.39, size = 462, normalized size = 1.45 \[ \frac {a^{2} c^{4} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+6 a^{2} c^{2} d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {4 a^{2} c \,d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} d^{4} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-2 a^{2} c^{4} \cos \left (f x +e \right )+8 a^{2} c^{3} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-4 a^{2} c^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+8 a^{2} c \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 a^{2} d^{4} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} c^{4} \left (f x +e \right )-4 a^{2} c^{3} d \cos \left (f x +e \right )+6 a^{2} c^{2} d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} c \,d^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} d^{4} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 451, normalized size = 1.42 \[ \frac {240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{4} + 960 \, {\left (f x + e\right )} a^{2} c^{4} + 1280 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{3} d + 1920 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{3} d + 3840 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c^{2} d^{2} + 180 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d^{2} + 1440 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} d^{2} - 256 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c d^{3} + 1280 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c d^{3} + 240 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d^{3} - 128 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} d^{4} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{4} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{4} - 1920 \, a^{2} c^{4} \cos \left (f x + e\right ) - 3840 \, a^{2} c^{3} d \cos \left (f x + e\right )}{960 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.92, size = 865, normalized size = 2.72 \[ \frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (24\,c^4+64\,c^3\,d+84\,c^2\,d^2+48\,c\,d^3+11\,d^4\right )}{8\,\left (3\,a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )}\right )\,\left (24\,c^4+64\,c^3\,d+84\,c^2\,d^2+48\,c\,d^3+11\,d^4\right )}{8\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (20\,a^2\,c^4+56\,a^2\,c^3\,d+48\,a^2\,c^2\,d^2+16\,a^2\,c\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (4\,a^2\,c^4+8\,d\,a^2\,c^3\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )+4\,a^2\,c^4+\frac {32\,a^2\,d^4}{15}-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (a^2\,c^4+8\,a^2\,c^3\,d+\frac {21\,a^2\,c^2\,d^2}{2}+6\,a^2\,c\,d^3+\frac {11\,a^2\,d^4}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (2\,a^2\,c^4+16\,a^2\,c^3\,d+33\,a^2\,c^2\,d^2+28\,a^2\,c\,d^3+\frac {47\,a^2\,d^4}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,a^2\,c^4+16\,a^2\,c^3\,d+33\,a^2\,c^2\,d^2+28\,a^2\,c\,d^3+\frac {47\,a^2\,d^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,a^2\,c^4+24\,a^2\,c^3\,d+\frac {87\,a^2\,c^2\,d^2}{2}+34\,a^2\,c\,d^3+\frac {187\,a^2\,d^4}{24}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (3\,a^2\,c^4+24\,a^2\,c^3\,d+\frac {87\,a^2\,c^2\,d^2}{2}+34\,a^2\,c\,d^3+\frac {187\,a^2\,d^4}{24}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (40\,a^2\,c^4+144\,a^2\,c^3\,d+192\,a^2\,c^2\,d^2+128\,a^2\,c\,d^3+32\,a^2\,d^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (20\,a^2\,c^4+72\,a^2\,c^3\,d+96\,a^2\,c^2\,d^2+\frac {288\,a^2\,c\,d^3}{5}+\frac {64\,a^2\,d^4}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (40\,a^2\,c^4+\frac {400\,a^2\,c^3\,d}{3}+160\,a^2\,c^2\,d^2+96\,a^2\,c\,d^3+\frac {64\,a^2\,d^4}{3}\right )+\frac {48\,a^2\,c\,d^3}{5}+\frac {40\,a^2\,c^3\,d}{3}+16\,a^2\,c^2\,d^2}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.42, size = 1136, normalized size = 3.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________