Optimal. Leaf size=23 \[ \frac{a \tan (c+d x)}{d}+\frac{b \sec (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0311392, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2669, 3767, 8} \[ \frac{a \tan (c+d x)}{d}+\frac{b \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx &=\frac{b \sec (c+d x)}{d}+a \int \sec ^2(c+d x) \, dx\\ &=\frac{b \sec (c+d x)}{d}-\frac{a \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac{b \sec (c+d x)}{d}+\frac{a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0110951, size = 23, normalized size = 1. \[ \frac{a \tan (c+d x)}{d}+\frac{b \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.028, size = 24, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( a\tan \left ( dx+c \right ) +{\frac{b}{\cos \left ( dx+c \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.955777, size = 31, normalized size = 1.35 \begin{align*} \frac{a \tan \left (d x + c\right ) + \frac{b}{\cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.05694, size = 53, normalized size = 2.3 \begin{align*} \frac{a \sin \left (d x + c\right ) + b}{d \cos \left (d x + c\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 \left (a + b \sin{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09731, size = 45, normalized size = 1.96 \begin{align*} -\frac{2 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]