Optimal. Leaf size=86 \[ -\frac{a^3 \cot (c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.169914, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2872, 3770, 3767, 8, 2650, 2648} \[ -\frac{a^3 \cot (c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2872
Rule 3770
Rule 3767
Rule 8
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \csc ^2(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=a^4 \int \left (\frac{3 \csc (c+d x)}{a}+\frac{\csc ^2(c+d x)}{a}+\frac{2}{a (-1+\sin (c+d x))^2}-\frac{3}{a (-1+\sin (c+d x))}\right ) \, dx\\ &=a^3 \int \csc ^2(c+d x) \, dx+\left (2 a^3\right ) \int \frac{1}{(-1+\sin (c+d x))^2} \, dx+\left (3 a^3\right ) \int \csc (c+d x) \, dx-\left (3 a^3\right ) \int \frac{1}{-1+\sin (c+d x)} \, dx\\ &=-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{3 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{1}{3} \left (2 a^3\right ) \int \frac{1}{-1+\sin (c+d x)} \, dx-\frac{a^3 \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac{11 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.954951, size = 135, normalized size = 1.57 \[ \frac{a^3 \left (3 \tan \left (\frac{1}{2} (c+d x)\right )-3 \cot \left (\frac{1}{2} (c+d x)\right )+18 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-18 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{4 \sin \left (\frac{1}{2} (c+d x)\right ) (11 \sin (c+d x)-13)}{\left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{4}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}\right )}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.136, size = 155, normalized size = 1.8 \begin{align*}{\frac{4\,{a}^{3}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+2\,{\frac{{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+3\,{\frac{{a}^{3}}{d\cos \left ( dx+c \right ) }}+3\,{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{3}}{3\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{4\,{a}^{3}}{3\,d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-{\frac{8\,{a}^{3}\cot \left ( dx+c \right ) }{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10084, size = 166, normalized size = 1.93 \begin{align*} \frac{2 \,{\left (\tan \left (d x + c\right )^{3} - \frac{3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a^{3} + 6 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} + 3 \, a^{3}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{2 \, a^{3}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.56735, size = 819, normalized size = 9.52 \begin{align*} \frac{28 \, a^{3} \cos \left (d x + c\right )^{3} - 10 \, a^{3} \cos \left (d x + c\right )^{2} - 34 \, a^{3} \cos \left (d x + c\right ) + 4 \, a^{3} - 9 \,{\left (a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} -{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 9 \,{\left (a^{3} \cos \left (d x + c\right )^{3} + 2 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} -{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (14 \, a^{3} \cos \left (d x + c\right )^{2} + 19 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{6 \,{\left (d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) -{\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.323, size = 159, normalized size = 1.85 \begin{align*} \frac{18 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{3 \,{\left (6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \frac{4 \,{\left (15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 13 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]