Optimal. Leaf size=63 \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0457791, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3787, 3768, 3770, 3767} \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3787
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sec (c+d x)) \, dx &=a \int \sec ^3(c+d x) \, dx+a \int \sec ^4(c+d x) \, dx\\ &=\frac{a \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} a \int \sec (c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x) \tan (c+d x)}{2 d}+\frac{a \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.133453, size = 60, normalized size = 0.95 \[ \frac{a \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.029, size = 72, normalized size = 1.1 \begin{align*}{\frac{a\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,a\tan \left ( dx+c \right ) }{3\,d}}+{\frac{a\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04754, size = 95, normalized size = 1.51 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a - 3 \, a{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.63606, size = 236, normalized size = 3.75 \begin{align*} \frac{3 \, a \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (4 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + 2 \, a\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \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*} a \left (\int \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.29445, size = 130, normalized size = 2.06 \begin{align*} \frac{3 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, a \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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]