Optimal. Leaf size=66 \[ \frac {2 \cos (c+d x)}{a^2 d}-\frac {\cos ^2(c+d x)}{a^2 d}+\frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {2 \log (1+\cos (c+d x))}{a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12,
78} \begin {gather*} \frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {\cos ^2(c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \log (\cos (c+d x)+1)}{a^2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 78
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x) x^2}{a^2 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x) x^2}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \left (-2 a^2+\frac {2 a^3}{a-x}-2 a x-x^2\right ) \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {2 \cos (c+d x)}{a^2 d}-\frac {\cos ^2(c+d x)}{a^2 d}+\frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {2 \log (1+\cos (c+d x))}{a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 51, normalized size = 0.77 \begin {gather*} \frac {-22+27 \cos (c+d x)-6 \cos (2 (c+d x))+\cos (3 (c+d x))-48 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{12 a^2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 58, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {-\frac {1}{3 \sec \left (d x +c \right )^{3}}+\frac {1}{\sec \left (d x +c \right )^{2}}-\frac {2}{\sec \left (d x +c \right )}-2 \ln \left (\sec \left (d x +c \right )\right )+2 \ln \left (1+\sec \left (d x +c \right )\right )}{d \,a^{2}}\) | \(58\) |
default | \(-\frac {-\frac {1}{3 \sec \left (d x +c \right )^{3}}+\frac {1}{\sec \left (d x +c \right )^{2}}-\frac {2}{\sec \left (d x +c \right )}-2 \ln \left (\sec \left (d x +c \right )\right )+2 \ln \left (1+\sec \left (d x +c \right )\right )}{d \,a^{2}}\) | \(58\) |
norman | \(\frac {\frac {10 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {14}{3 a d}+\frac {12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} a}+\frac {2 \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d}\) | \(90\) |
risch | \(\frac {2 i x}{a^{2}}+\frac {9 \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{2} d}+\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{2} d}+\frac {4 i c}{a^{2} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d}+\frac {\cos \left (3 d x +3 c \right )}{12 d \,a^{2}}-\frac {\cos \left (2 d x +2 c \right )}{2 d \,a^{2}}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 51, normalized size = 0.77 \begin {gather*} \frac {\frac {\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right )}{a^{2}} - \frac {6 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.60, size = 48, normalized size = 0.73 \begin {gather*} \frac {\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right ) - 6 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{3 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.50, size = 75, normalized size = 1.14 \begin {gather*} -\frac {2 \, \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a^{2} d} + \frac {a^{4} d^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{4} d^{2} \cos \left (d x + c\right )^{2} + 6 \, a^{4} d^{2} \cos \left (d x + c\right )}{3 \, a^{6} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 56, normalized size = 0.85 \begin {gather*} -\frac {\frac {2\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{a^2}-\frac {2\,\cos \left (c+d\,x\right )}{a^2}+\frac {{\cos \left (c+d\,x\right )}^2}{a^2}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________