Optimal. Leaf size=44 \[ \frac{\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{b}+\frac{\cos (a-c) \sec (b x+c)}{b}+\frac{\cos (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.035963, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4576, 4579, 2638, 3770, 2606, 8} \[ \frac{\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{b}+\frac{\cos (a-c) \sec (b x+c)}{b}+\frac{\cos (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4576
Rule 4579
Rule 2638
Rule 3770
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \sin (a+b x) \tan ^2(c+b x) \, dx &=\cos (a-c) \int \sec (c+b x) \tan (c+b x) \, dx-\int \cos (a+b x) \tan (c+b x) \, dx\\ &=\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{\tanh ^{-1}(\sin (c+b x)) \sin (a-c)}{b}\\ \end{align*}
Mathematica [C] time = 0.10031, size = 109, normalized size = 2.48 \[ \frac{\cos (a-c) \sec (b x+c)}{b}-\frac{2 i \sin (a-c) \tan ^{-1}\left (\frac{(\sin (c)+i \cos (c)) \left (\sin (c) \cos \left (\frac{b x}{2}\right )+\cos (c) \sin \left (\frac{b x}{2}\right )\right )}{\cos (c) \cos \left (\frac{b x}{2}\right )-i \sin (c) \cos \left (\frac{b x}{2}\right )}\right )}{b}-\frac{\sin (a) \sin (b x)}{b}+\frac{\cos (a) \cos (b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.083, size = 143, normalized size = 3.3 \begin{align*}{\frac{{{\rm e}^{i \left ( bx+a \right ) }}}{2\,b}}+{\frac{{{\rm e}^{-i \left ( bx+a \right ) }}}{2\,b}}+{\frac{{{\rm e}^{i \left ( bx+3\,a \right ) }}+{{\rm e}^{i \left ( bx+a+2\,c \right ) }}}{b \left ({{\rm e}^{2\,i \left ( bx+a+c \right ) }}+{{\rm e}^{2\,ia}} \right ) }}-{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-i{{\rm e}^{i \left ( a-c \right ) }} \right ) \sin \left ( a-c \right ) }{b}}+{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+i{{\rm e}^{i \left ( a-c \right ) }} \right ) \sin \left ( a-c \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.99207, size = 702, normalized size = 15.95 \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 = 0.538988, size = 857, normalized size = 19.48 \begin{align*} -\frac{4 \,{\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right )^{2} - 4 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac{\sqrt{2}{\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) +{\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \sin \left (b x + a\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}} + 4 \, \cos \left (-2 \, a + 2 \, c\right ) + 4}{4 \,{\left (b \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) -{\left (b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \cos \left (b x + a\right )\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*} \int \sin{\left (a + b x \right )} \tan ^{2}{\left (b x + c \right )}\, dx \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 \sin \left (b x + a\right ) \tan \left (b x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]