Optimal. Leaf size=87 \[ -\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\frac {10 a^2 \tan (c+d x)}{3 d}+\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3872, 2869, 2766, 2978, 2748, 3767, 8, 3770} \[ \frac {10 a^2 \tan (c+d x)}{3 d}+\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2748
Rule 2766
Rule 2869
Rule 2978
Rule 3767
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \csc ^4(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc ^4(c+d x) \sec ^2(c+d x) \, dx\\ &=a^4 \int \frac {\sec ^2(c+d x)}{(-a+a \cos (c+d x))^2} \, dx\\ &=-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\frac {1}{3} a^2 \int \frac {(-4 a-2 a \cos (c+d x)) \sec ^2(c+d x)}{-a+a \cos (c+d x)} \, dx\\ &=-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\frac {1}{3} \int \left (10 a^2+6 a^2 \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\left (2 a^2\right ) \int \sec (c+d x) \, dx+\frac {1}{3} \left (10 a^2\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}-\frac {\left (10 a^2\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {10 a^2 \tan (c+d x)}{3 d}-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.85, size = 228, normalized size = 2.62 \[ \frac {a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (-\cot \left (\frac {c}{2}\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )+6 \left (\frac {\sin (d x)}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-\left (\csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) (7 \cos (c+d x)-8) \csc ^3\left (\frac {1}{2} (c+d x)\right )\right )\right )}{24 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 159, normalized size = 1.83 \[ -\frac {10 \, a^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{2} \cos \left (d x + c\right )^{2} - 11 \, a^{2} \cos \left (d x + c\right ) - 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 3 \, a^{2}}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 104, normalized size = 1.20 \[ \frac {12 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 12 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.88, size = 140, normalized size = 1.61 \[ -\frac {10 a^{2} \cot \left (d x +c \right )}{3 d}-\frac {a^{2} \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{3 d}-\frac {2 a^{2}}{3 d \sin \left (d x +c \right )^{3}}-\frac {2 a^{2}}{d \sin \left (d x +c \right )}+\frac {2 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {a^{2}}{3 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4 a^{2}}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 113, normalized size = 1.30 \[ -\frac {a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + a^{2} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )} + \frac {{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} a^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.48, size = 91, normalized size = 1.05 \[ \frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {-9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {14\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {a^2}{3}}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int 2 \csc ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________