Optimal. Leaf size=58 \[ \frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^2(c+d x)}{2 d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2836, 12, 75} \[ \frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^2(c+d x)}{2 d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 75
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \sin ^2(c+d x) \tan (c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a (-a-x) (-a+x)^2}{x} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x) (-a+x)^2}{x} \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2-\frac {a^3}{x}+a x-x^2\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac {a \cos (c+d x)}{d}+\frac {a \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{3 d}-\frac {a \log (\cos (c+d x))}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 57, normalized size = 0.98 \[ -\frac {3 a \cos (c+d x)}{4 d}+\frac {a \cos (3 (c+d x))}{12 d}-\frac {a \left (\log (\cos (c+d x))-\frac {1}{2} \cos ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 49, normalized size = 0.84 \[ \frac {2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )^{2} - 6 \, a \cos \left (d x + c\right ) - 6 \, a \log \left (-\cos \left (d x + c\right )\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 66, normalized size = 1.14 \[ -\frac {a \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {2 \, a d^{2} \cos \left (d x + c\right )^{3} + 3 \, a d^{2} \cos \left (d x + c\right )^{2} - 6 \, a d^{2} \cos \left (d x + c\right )}{6 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.61, size = 61, normalized size = 1.05 \[ -\frac {a \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{3 d}-\frac {2 a \cos \left (d x +c \right )}{3 d}-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.62, size = 47, normalized size = 0.81 \[ \frac {2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )^{2} - 6 \, a \cos \left (d x + c\right ) - 6 \, a \log \left (\cos \left (d x + c\right )\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 45, normalized size = 0.78 \[ -\frac {a\,\cos \left (c+d\,x\right )-\frac {a\,{\cos \left (c+d\,x\right )}^2}{2}-\frac {a\,{\cos \left (c+d\,x\right )}^3}{3}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________