Optimal. Leaf size=115 \[ -\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{9 a^2 x}{8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.268433, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3872, 2872, 2637, 2635, 8, 2633, 3770, 3767} \[ -\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{9 a^2 x}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2872
Rule 2637
Rule 2635
Rule 8
Rule 2633
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx &=\int (-a-a \cos (c+d x))^2 \sin ^2(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{\int \left (-a^6-4 a^6 \cos (c+d x)-a^6 \cos ^2(c+d x)+2 a^6 \cos ^3(c+d x)+a^6 \cos ^4(c+d x)+2 a^6 \sec (c+d x)+a^6 \sec ^2(c+d x)\right ) \, dx}{a^4}\\ &=-a^2 x-a^2 \int \cos ^2(c+d x) \, dx+a^2 \int \cos ^4(c+d x) \, dx+a^2 \int \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^3(c+d x) \, dx+\left (2 a^2\right ) \int \sec (c+d x) \, dx-\left (4 a^2\right ) \int \cos (c+d x) \, dx\\ &=-a^2 x+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{4 a^2 \sin (c+d x)}{d}-\frac{a^2 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{1}{2} a^2 \int 1 \, dx+\frac{1}{4} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac{a^2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac{3 a^2 x}{2}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{1}{8} \left (3 a^2\right ) \int 1 \, dx\\ &=-\frac{9 a^2 x}{8}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a^2 \sin (c+d x)}{d}-\frac{a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.27868, size = 94, normalized size = 0.82 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (64 \sin ^3(c+d x)+192 \sin (c+d x)-3 \sin (4 (c+d x))+60 \tan ^{-1}(\tan (c+d x))-96 \tan (c+d x)-192 \tanh ^{-1}(\sin (c+d x))+48 c+48 d x\right )}{384 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 134, normalized size = 1.2 \begin{align*}{\frac{3\,{a}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{9\,{a}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}-{\frac{9\,{a}^{2}x}{8}}-{\frac{9\,{a}^{2}c}{8\,d}}-{\frac{2\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d}}+2\,{\frac{{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.506, size = 170, normalized size = 1.48 \begin{align*} -\frac{32 \,{\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{2} - 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 )} a^{2} + 48 \,{\left (3 \, d x + 3 \, c - \frac{\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{2}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.88363, size = 344, normalized size = 2.99 \begin{align*} -\frac{27 \, a^{2} d x \cos \left (d x + c\right ) - 24 \, a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + 24 \, a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (6 \, a^{2} \cos \left (d x + c\right )^{4} + 16 \, a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )^{2} - 64 \, a^{2} \cos \left (d x + c\right ) + 24 \, a^{2}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\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 = 1.37841, size = 217, normalized size = 1.89 \begin{align*} -\frac{27 \,{\left (d x + c\right )} a^{2} - 48 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 48 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{48 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1} + \frac{2 \,{\left (51 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 187 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 229 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 45 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\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]