Optimal. Leaf size=77 \[ -\frac{a}{8 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{1}{4 d (a \sin (c+d x)+a)}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0798467, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2667, 44, 206} \[ -\frac{a}{8 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{1}{4 d (a \sin (c+d x)+a)}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{8 a^3 (a-x)^2}+\frac{1}{4 a^2 (a+x)^3}+\frac{1}{4 a^3 (a+x)^2}+\frac{3}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{1}{8 d (a-a \sin (c+d x))}-\frac{a}{8 d (a+a \sin (c+d x))^2}-\frac{1}{4 d (a+a \sin (c+d x))}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{a}{8 d (a+a \sin (c+d x))^2}-\frac{1}{4 d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.102742, size = 75, normalized size = 0.97 \[ -\frac{\sec ^2(c+d x) \left (-3 \sin ^2(c+d x)-3 \sin (c+d x)+3 (\sin (c+d x)-1) (\sin (c+d x)+1)^2 \tanh ^{-1}(\sin (c+d x))+2\right )}{8 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.058, size = 90, normalized size = 1.2 \begin{align*} -{\frac{1}{8\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{16\,da}}-{\frac{1}{8\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{4\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{16\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.966444, size = 123, normalized size = 1.6 \begin{align*} -\frac{\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 2\right )}}{a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac{3 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{3 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7482, size = 336, normalized size = 4.36 \begin{align*} -\frac{6 \, \cos \left (d x + c\right )^{2} - 3 \,{\left (\cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (\cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \, \sin \left (d x + c\right ) - 2}{16 \,{\left (a d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{2}\right )}} \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*} \frac{\int \frac{\sec ^{3}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23488, size = 130, normalized size = 1.69 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{2 \,{\left (3 \, \sin \left (d x + c\right ) - 5\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}} - \frac{9 \, \sin \left (d x + c\right )^{2} + 26 \, \sin \left (d x + c\right ) + 21}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]