Optimal. Leaf size=101 \[ -\frac {4 a^4 \cos (c+d x)}{d}-\frac {a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {17 a^4 x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2872, 2650, 2648, 2638, 2635, 8} \[ -\frac {4 a^4 \cos (c+d x)}{d}-\frac {a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {17 a^4 x}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 2638
Rule 2648
Rule 2650
Rule 2872
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx &=a^4 \int \left (8+\frac {4}{(-1+\sin (c+d x))^2}+\frac {12}{-1+\sin (c+d x)}+4 \sin (c+d x)+\sin ^2(c+d x)\right ) \, dx\\ &=8 a^4 x+a^4 \int \sin ^2(c+d x) \, dx+\left (4 a^4\right ) \int \frac {1}{(-1+\sin (c+d x))^2} \, dx+\left (4 a^4\right ) \int \sin (c+d x) \, dx+\left (12 a^4\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx\\ &=8 a^4 x-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {12 a^4 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} a^4 \int 1 \, dx-\frac {1}{3} \left (4 a^4\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx\\ &=\frac {17 a^4 x}{2}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.80, size = 158, normalized size = 1.56 \[ -\frac {a^4 \left (-3 (204 c+204 d x+161) \cos \left (\frac {1}{2} (c+d x)\right )+(204 c+204 d x+647) \cos \left (\frac {3}{2} (c+d x)\right )-39 \cos \left (\frac {5}{2} (c+d x)\right )+3 \cos \left (\frac {7}{2} (c+d x)\right )+6 \sin \left (\frac {1}{2} (c+d x)\right ) ((68 c+68 d x-59) \cos (c+d x)-14 \cos (2 (c+d x))-\cos (3 (c+d x))+136 c+136 d x+146)\right )}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 197, normalized size = 1.95 \[ \frac {3 \, a^{4} \cos \left (d x + c\right )^{4} - 18 \, a^{4} \cos \left (d x + c\right )^{3} - 102 \, a^{4} d x - 8 \, a^{4} + 17 \, {\left (3 \, a^{4} d x + 5 \, a^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (51 \, a^{4} d x - 98 \, a^{4}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{4} \cos \left (d x + c\right )^{3} - 102 \, a^{4} d x + 21 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - {\left (51 \, a^{4} d x - 106 \, a^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 135, normalized size = 1.34 \[ \frac {51 \, {\left (d x + c\right )} a^{4} + \frac {6 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {16 \, {\left (6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.57, size = 268, normalized size = 2.65 \[ \frac {a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+6 a^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+4 a^{4} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 158, normalized size = 1.56 \[ \frac {2 \, a^{4} \tan \left (d x + c\right )^{3} + {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{4} + 12 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} - 8 \, a^{4} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {8 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{4}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 14.85, size = 287, normalized size = 2.84 \[ \frac {17\,a^4\,x}{2}+\frac {\frac {17\,a^4\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {51\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (153\,c+153\,d\,x-378\right )}{6}\right )-\frac {a^4\,\left (51\,c+51\,d\,x-160\right )}{6}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {51\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (153\,c+153\,d\,x-102\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {85\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (255\,c+255\,d\,x-306\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {85\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (255\,c+255\,d\,x-494\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {119\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (357\,c+357\,d\,x-460\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {119\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (357\,c+357\,d\,x-660\right )}{6}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________