Optimal. Leaf size=22 \[ -\frac {(a \cos (c+d x)+b)^2}{2 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4377, 12, 2638, 2564, 30} \[ \frac {a \sin ^2(c+d x)}{2 d}-\frac {b \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 2564
Rule 2638
Rule 4377
Rubi steps
\begin {align*} \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \cos (c+d x) \sin (c+d x) \, dx+\int b \sin (c+d x) \, dx\\ &=b \int \sin (c+d x) \, dx+\frac {a \operatorname {Subst}(\int x \, dx,x,\sin (c+d x))}{d}\\ &=-\frac {b \cos (c+d x)}{d}+\frac {a \sin ^2(c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 40, normalized size = 1.82 \[ -\frac {a \cos ^2(c+d x)}{2 d}+\frac {b \sin (c) \sin (d x)}{d}-\frac {b \cos (c) \cos (d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 25, normalized size = 1.14 \[ -\frac {a \cos \left (d x + c\right )^{2} + 2 \, b \cos \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.27, size = 102, normalized size = 4.64 \[ -\frac {a \cos \left (2 \, d x + 2 \, c\right )}{4 \, d} - \frac {b \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, c\right )^{2} + b}{d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d \tan \left (\frac {1}{2} \, d x\right )^{2} + d \tan \left (\frac {1}{2} \, c\right )^{2} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 26, normalized size = 1.18 \[ -\frac {\frac {\left (\cos ^{2}\left (d x +c \right )\right ) a}{2}+b \cos \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 25, normalized size = 1.14 \[ -\frac {a \cos \left (d x + c\right )^{2} + 2 \, b \cos \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.63, size = 28, normalized size = 1.27 \[ -\frac {\left (\cos \left (c+d\,x\right )+1\right )\,\left (2\,b-a+a\,\cos \left (c+d\,x\right )\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________