Optimal. Leaf size=47 \[ -\frac {a^2 \log (\cos (c+d x))}{d}+\frac {2 a b \sec (c+d x)}{d}+\frac {b^2 \sec ^2(c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {3885, 43} \[ -\frac {a^2 \log (\cos (c+d x))}{d}+\frac {2 a b \sec (c+d x)}{d}+\frac {b^2 \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 3885
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^2 \tan (c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^2}{x} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a+\frac {a^2}{x}+x\right ) \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac {a^2 \log (\cos (c+d x))}{d}+\frac {2 a b \sec (c+d x)}{d}+\frac {b^2 \sec ^2(c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 42, normalized size = 0.89 \[ \frac {-2 a^2 \log (\cos (c+d x))+4 a b \sec (c+d x)+b^2 \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 51, normalized size = 1.09 \[ -\frac {2 \, a^{2} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 4 \, a b \cos \left (d x + c\right ) - b^{2}}{2 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.87, size = 191, normalized size = 4.06 \[ \frac {2 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {3 \, a^{2} + 8 \, a b + \frac {6 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {8 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.19, size = 45, normalized size = 0.96 \[ \frac {b^{2} \left (\sec ^{2}\left (d x +c \right )\right )}{2 d}+\frac {2 a b \sec \left (d x +c \right )}{d}+\frac {a^{2} \ln \left (\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.38, size = 42, normalized size = 0.89 \[ -\frac {2 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {4 \, a b \cos \left (d x + c\right ) + b^{2}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.53, size = 81, normalized size = 1.72 \[ \frac {4\,a\,b-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a\,b-2\,b^2\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {2\,a^2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.49, size = 60, normalized size = 1.28 \[ \begin {cases} \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {2 a b \sec {\left (c + d x \right )}}{d} + \frac {b^{2} \sec ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \sec {\relax (c )}\right )^{2} \tan {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________