Optimal. Leaf size=71 \[ \frac {2 a^2 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {5 a^2 x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2709, 2648, 2638, 2635, 8} \[ \frac {2 a^2 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {5 a^2 x}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 2638
Rule 2648
Rule 2709
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx &=a^2 \int \left (-2-\frac {2}{-1+\sin (c+d x)}-2 \sin (c+d x)-\sin ^2(c+d x)\right ) \, dx\\ &=-2 a^2 x-a^2 \int \sin ^2(c+d x) \, dx-\left (2 a^2\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx-\left (2 a^2\right ) \int \sin (c+d x) \, dx\\ &=-2 a^2 x+\frac {2 a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} a^2 \int 1 \, dx\\ &=-\frac {5 a^2 x}{2}+\frac {2 a^2 \cos (c+d x)}{d}+\frac {2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.39, size = 145, normalized size = 2.04 \[ -\frac {a^2 (\sin (c+d x)+1)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right ) (10 (c+d x)-\sin (2 (c+d x))-8 \cos (c+d x))+\sin \left (\frac {1}{2} (c+d x)\right ) (-2 (5 c+5 d x+8)+\sin (2 (c+d x))+8 \cos (c+d x))\right )}{4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 125, normalized size = 1.76 \[ \frac {a^{2} \cos \left (d x + c\right )^{3} - 5 \, a^{2} d x + 4 \, a^{2} \cos \left (d x + c\right )^{2} + 4 \, a^{2} - {\left (5 \, a^{2} d x - 7 \, a^{2}\right )} \cos \left (d x + c\right ) + {\left (5 \, a^{2} d x + a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \cos \left (d x + c\right ) + 4 \, a^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 102, normalized size = 1.44 \[ -\frac {5 \, {\left (d x + c\right )} a^{2} + \frac {8 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} + \frac {2 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.42, size = 117, normalized size = 1.65 \[ \frac {a^{2} \left (\frac {\sin ^{5}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+2 a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+a^{2} \left (\tan \left (d x +c \right )-d x -c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 84, normalized size = 1.18 \[ -\frac {{\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{2} + 2 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{2} - 4 \, a^{2} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 11.39, size = 183, normalized size = 2.58 \[ -\frac {5\,a^2\,x}{2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^2\,\left (5\,d\,x-6\right )}{2}-\frac {5\,a^2\,d\,x}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2\,\left (5\,d\,x-10\right )}{2}-\frac {5\,a^2\,d\,x}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^2\,\left (10\,d\,x-10\right )}{2}-5\,a^2\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2\,\left (10\,d\,x-22\right )}{2}-5\,a^2\,d\,x\right )-\frac {a^2\,\left (5\,d\,x-16\right )}{2}+\frac {5\,a^2\,d\,x}{2}}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________