Optimal. Leaf size=52 \[ -\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0469152, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 43} \[ -\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{a \operatorname{Subst}\left (\int \frac{(a+x)^2}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \operatorname{Subst}\left (\int \left (-3 a+\frac{4 a^2}{a-x}-x\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \log (1-\sin (c+d x))}{d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{a^3 \sin ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0290341, size = 41, normalized size = 0.79 \[ \frac{a^3 \left (-\frac{1}{2} \sin ^2(c+d x)-3 \sin (c+d x)-4 \log (1-\sin (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 69, normalized size = 1.3 \begin{align*} -{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-4\,{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.96589, size = 58, normalized size = 1.12 \begin{align*} -\frac{a^{3} \sin \left (d x + c\right )^{2} + 8 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65545, size = 108, normalized size = 2.08 \begin{align*} \frac{a^{3} \cos \left (d x + c\right )^{2} - 8 \, a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int 3 \sin{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \sec{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.16096, size = 173, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (2 \, a^{3} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 4 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 7 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{3}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]