Optimal. Leaf size=66 \[ \frac{a^3 \sec ^3(c+d x)}{3 d}+\frac{3 a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0388226, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3879, 43} \[ \frac{a^3 \sec ^3(c+d x)}{3 d}+\frac{3 a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3879
Rule 43
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^3 \tan (c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^3}{x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^4}+\frac{3 a^3}{x^3}+\frac{3 a^3}{x^2}+\frac{a^3}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^3 \log (\cos (c+d x))}{d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{3 a^3 \sec ^2(c+d x)}{2 d}+\frac{a^3 \sec ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.156663, size = 64, normalized size = 0.97 \[ -\frac{a^3 \sec ^3(c+d x) (-18 \cos (2 (c+d x))+9 \cos (c+d x) (\log (\cos (c+d x))-2)+3 \cos (3 (c+d x)) \log (\cos (c+d x))-22)}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 62, normalized size = 0.9 \begin{align*}{\frac{{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{3\,{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{{a}^{3}\sec \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.48249, size = 78, normalized size = 1.18 \begin{align*} -\frac{6 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac{18 \, a^{3} \cos \left (d x + c\right )^{2} + 9 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.945276, size = 162, normalized size = 2.45 \begin{align*} -\frac{6 \, a^{3} \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) - 18 \, a^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{3} \cos \left (d x + c\right ) - 2 \, a^{3}}{6 \, d \cos \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.91752, size = 76, normalized size = 1.15 \begin{align*} \begin{cases} \frac{a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{3} \sec ^{3}{\left (c + d x \right )}}{3 d} + \frac{3 a^{3} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac{3 a^{3} \sec{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sec{\left (c \right )} + a\right )^{3} \tan{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.43613, size = 225, normalized size = 3.41 \begin{align*} \frac{6 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 6 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{51 \, a^{3} + \frac{69 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{45 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{11 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]