Optimal. Leaf size=89 \[ \frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0998257, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2838, 2592, 302, 206, 2635, 8} \[ \frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x)}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2838
Rule 2592
Rule 302
Rule 206
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \cot (c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cos ^3(c+d x) \cot (c+d x) \, dx+b \int \cos ^4(c+d x) \, dx\\ &=\frac{b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} (3 b) \int \cos ^2(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{3 b \cos (c+d x) \sin (c+d x)}{8 d}+\frac{b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{8} (3 b) \int 1 \, dx-\frac{a \operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{3 b x}{8}+\frac{a \cos (c+d x)}{d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{3 b \cos (c+d x) \sin (c+d x)}{8 d}+\frac{b \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{3 b x}{8}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}+\frac{a \cos (c+d x)}{d}+\frac{a \cos ^3(c+d x)}{3 d}+\frac{3 b \cos (c+d x) \sin (c+d x)}{8 d}+\frac{b \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.122046, size = 109, normalized size = 1.22 \[ \frac{5 a \cos (c+d x)}{4 d}+\frac{a \cos (3 (c+d x))}{12 d}+\frac{a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{d}-\frac{a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{d}+\frac{3 b (c+d x)}{8 d}+\frac{b \sin (2 (c+d x))}{4 d}+\frac{b \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 97, normalized size = 1.1 \begin{align*}{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{\cos \left ( dx+c \right ) a}{d}}+{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}+{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{3\,b\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,bx}{8}}+{\frac{3\,cb}{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11138, size = 109, normalized size = 1.22 \begin{align*} \frac{16 \,{\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a + 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68077, size = 252, normalized size = 2.83 \begin{align*} \frac{8 \, a \cos \left (d x + c\right )^{3} + 9 \, b d x + 24 \, a \cos \left (d x + c\right ) - 12 \, a \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 12 \, a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left (2 \, b \cos \left (d x + c\right )^{3} + 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \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 = 1.35741, size = 196, normalized size = 2.2 \begin{align*} \frac{9 \,{\left (d x + c\right )} b + 24 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \,{\left (15 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 48 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 9 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 96 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 9 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 80 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 32 \, a\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]