Optimal. Leaf size=31 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0426453, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3885, 63, 207} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3885
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+x}} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{-a+x^2} \, dx,x,\sqrt{a+b \sec (c+d x)}\right )}{d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}\\ \end{align*}
Mathematica [B] time = 0.192134, size = 108, normalized size = 3.48 \[ \frac{\sqrt{a \cos (c+d x)+b} \left (\log \left (1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right )-\log \left (\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right )\right )}{d \sqrt{a \cos (c+d x)} \sqrt{a+b \sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 26, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{d\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b\sec \left ( dx+c \right ) }}{\sqrt{a}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.88349, size = 373, normalized size = 12.03 \begin{align*} \left [\frac{\log \left (-8 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a b \cos \left (d x + c\right ) - b^{2} + 4 \,{\left (2 \, a \cos \left (d x + c\right )^{2} + b \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}\right )}{2 \, \sqrt{a} d}, \frac{\sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + b}\right )}{a d}\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 \frac{\tan{\left (c + d x \right )}}{\sqrt{a + b \sec{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.55222, size = 147, normalized size = 4.74 \begin{align*} -\frac{2 \, \arctan \left (-\frac{\sqrt{a - b} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a + b} + \sqrt{a - b}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} d \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]