Optimal. Leaf size=27 \[ \frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d}-b x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2838, 2606, 8, 3473} \[ \frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d}-b x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2606
Rule 2838
Rule 3473
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x)) \tan (c+d x) \, dx &=a \int \sec (c+d x) \tan (c+d x) \, dx+b \int \tan ^2(c+d x) \, dx\\ &=\frac {b \tan (c+d x)}{d}-b \int 1 \, dx+\frac {a \operatorname {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=-b x+\frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 36, normalized size = 1.33 \[ \frac {a \sec (c+d x)}{d}-\frac {b \tan ^{-1}(\tan (c+d x))}{d}+\frac {b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 36, normalized size = 1.33 \[ -\frac {b d x \cos \left (d x + c\right ) - b \sin \left (d x + c\right ) - a}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 43, normalized size = 1.59 \[ -\frac {{\left (d x + c\right )} b + \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.17, size = 32, normalized size = 1.19 \[ \frac {\frac {a}{\cos \left (d x +c \right )}+b \left (\tan \left (d x +c \right )-d x -c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 32, normalized size = 1.19 \[ -\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} b - \frac {a}{\cos \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 11.94, size = 41, normalized size = 1.52 \[ -b\,x-\frac {2\,a+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + d x \right )}\right ) \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________