Optimal. Leaf size=66 \[ \frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {\cos ^2(c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \log (\cos (c+d x)+1)}{a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, 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 = {3872, 2836, 12, 77} \[ \frac {\cos ^3(c+d x)}{3 a^2 d}-\frac {\cos ^2(c+d x)}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {2 \log (\cos (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 77
Rule 2836
Rule 3872
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 {\operatorname {Subst}\left (\int \frac {(-a-x) x^2}{a^2 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x) x^2}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {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 = 0.24, size = 51, normalized size = 0.77 \[ \frac {27 \cos (c+d x)-6 \cos (2 (c+d x))+\cos (3 (c+d x))-48 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-22}{12 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 48, normalized size = 0.73 \[ \frac {\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right ) - 6 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.42, size = 75, normalized size = 1.14 \[ -\frac {2 \, \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a^{2} d} + \frac {a^{4} d^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{4} d^{2} \cos \left (d x + c\right )^{2} + 6 \, a^{4} d^{2} \cos \left (d x + c\right )}{3 \, a^{6} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.65, size = 82, normalized size = 1.24 \[ \frac {1}{3 d \,a^{2} \sec \left (d x +c \right )^{3}}-\frac {1}{d \,a^{2} \sec \left (d x +c \right )^{2}}+\frac {2}{d \,a^{2} \sec \left (d x +c \right )}+\frac {2 \ln \left (\sec \left (d x +c \right )\right )}{d \,a^{2}}-\frac {2 \ln \left (1+\sec \left (d x +c \right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 51, normalized size = 0.77 \[ \frac {\frac {\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right )}{a^{2}} - \frac {6 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 56, normalized size = 0.85 \[ -\frac {\frac {2\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{a^2}-\frac {2\,\cos \left (c+d\,x\right )}{a^2}+\frac {{\cos \left (c+d\,x\right )}^2}{a^2}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}}{d} \]
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]
________________________________________________________________________________________