Optimal. Leaf size=72 \[ \frac{3 \sin (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \sec (b x+c)}{b}-\frac{\sin (a-c) \tan (b x+c) \sec (b x+c)}{2 b}+\frac{\cos (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0779723, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {4579, 4576, 2638, 3770, 2606, 8, 2611} \[ \frac{3 \sin (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \sec (b x+c)}{b}-\frac{\sin (a-c) \tan (b x+c) \sec (b x+c)}{2 b}+\frac{\cos (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4579
Rule 4576
Rule 2638
Rule 3770
Rule 2606
Rule 8
Rule 2611
Rubi steps
\begin{align*} \int \cos (a+b x) \tan ^3(c+b x) \, dx &=-\left (\sin (a-c) \int \sec (c+b x) \tan ^2(c+b x) \, dx\right )+\int \sin (a+b x) \tan ^2(c+b x) \, dx\\ &=-\frac{\sec (c+b x) \sin (a-c) \tan (c+b x)}{2 b}+\cos (a-c) \int \sec (c+b x) \tan (c+b x) \, dx+\frac{1}{2} \sin (a-c) \int \sec (c+b x) \, dx-\int \cos (a+b x) \tan (c+b x) \, dx\\ &=\frac{\tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{2 b}-\frac{\sec (c+b x) \sin (a-c) \tan (c+b x)}{2 b}+\frac{\cos (a-c) \operatorname{Subst}(\int 1 \, dx,x,\sec (c+b x))}{b}+\sin (a-c) \int \sec (c+b x) \, dx-\int \sin (a+b x) \, dx\\ &=\frac{\cos (a+b x)}{b}+\frac{\cos (a-c) \sec (c+b x)}{b}+\frac{3 \tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{2 b}-\frac{\sec (c+b x) \sin (a-c) \tan (c+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.36262, size = 70, normalized size = 0.97 \[ \frac{\sec ^2(b x+c) (2 \cos (a-b x-2 c)+\cos (a+3 b x+2 c)+5 \cos (a+b x))+12 \sin (a-c) \tanh ^{-1}\left (\cos (c) \tan \left (\frac{b x}{2}\right )+\sin (c)\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.099, size = 181, normalized size = 2.5 \begin{align*}{\frac{{{\rm e}^{i \left ( bx+a \right ) }}}{2\,b}}+{\frac{{{\rm e}^{-i \left ( bx+a \right ) }}}{2\,b}}+{\frac{3\,{{\rm e}^{i \left ( 3\,bx+5\,a+2\,c \right ) }}+{{\rm e}^{i \left ( 3\,bx+3\,a+4\,c \right ) }}+{{\rm e}^{i \left ( bx+5\,a \right ) }}+3\,{{\rm e}^{i \left ( bx+3\,a+2\,c \right ) }}}{2\,b \left ({{\rm e}^{2\,i \left ( bx+a+c \right ) }}+{{\rm e}^{2\,ia}} \right ) ^{2}}}+{\frac{3\,\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+i{{\rm e}^{i \left ( a-c \right ) }} \right ) \sin \left ( a-c \right ) }{2\,b}}-{\frac{3\,\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-i{{\rm e}^{i \left ( a-c \right ) }} \right ) \sin \left ( a-c \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.03128, size = 1386, normalized size = 19.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.570641, size = 986, normalized size = 13.69 \begin{align*} \frac{16 \, \cos \left (b x + a\right )^{3} \cos \left (-2 \, a + 2 \, c\right ) - 4 \,{\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - 4 \,{\left (\cos \left (-2 \, a + 2 \, c\right ) - 5\right )} \cos \left (b x + a\right ) + \frac{3 \, \sqrt{2}{\left (2 \,{\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) +{\left (2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (-2 \, a + 2 \, c\right )\right )} \log \left (-\frac{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac{2 \, \sqrt{2}{\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt{\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) - 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1}\right )}{\sqrt{\cos \left (-2 \, a + 2 \, c\right ) + 1}}}{8 \,{\left (2 \, b \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, b \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - b \cos \left (-2 \, a + 2 \, c\right ) + b\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x + a\right ) \tan \left (b x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]