3.8 \(\int -\sin (e+f x) \, dx\)
Optimal. Leaf size=10 \[ \frac{\cos (e+f x)}{f} \]
[Out]
Cos[e + f*x]/f
________________________________________________________________________________________
Rubi [A] time = 0.0038965, antiderivative size = 10, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{2638} \[ \frac{\cos (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
Int[-Sin[e + f*x],x]
[Out]
Cos[e + f*x]/f
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int -\sin (e+f x) \, dx &=\frac{\cos (e+f x)}{f}\\ \end{align*}
Mathematica [B] time = 0.0066503, size = 22, normalized size = 2.2 \[ \frac{\cos (e) \cos (f x)}{f}-\frac{\sin (e) \sin (f x)}{f} \]
Antiderivative was successfully verified.
[In]
Integrate[-Sin[e + f*x],x]
[Out]
(Cos[e]*Cos[f*x])/f - (Sin[e]*Sin[f*x])/f
________________________________________________________________________________________
Maple [A] time = 0.03, size = 11, normalized size = 1.1 \begin{align*}{\frac{\cos \left ( fx+e \right ) }{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-csc(f*x+e)*sin(f*x+e)^2,x)
[Out]
cos(f*x+e)/f
________________________________________________________________________________________
Maxima [A] time = 0.935987, size = 14, normalized size = 1.4 \begin{align*} \frac{\cos \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-csc(f*x+e)*sin(f*x+e)^2,x, algorithm="maxima")
[Out]
cos(f*x + e)/f
________________________________________________________________________________________
Fricas [A] time = 1.5676, size = 22, normalized size = 2.2 \begin{align*} \frac{\cos \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-csc(f*x+e)*sin(f*x+e)^2,x, algorithm="fricas")
[Out]
cos(f*x + e)/f
________________________________________________________________________________________
Sympy [B] time = 106.913, size = 4124, normalized size = 412.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-csc(f*x+e)*sin(f*x+e)**2,x)
[Out]
-Piecewise((-log(cot(e + f*x) + csc(e + f*x))/f, Ne(f, 0)), (x*(cot(e)*csc(e) + csc(e)**2)/(cot(e) + csc(e)),
True))/2 - 2*Piecewise((x, Eq(e, 0) & Eq(f, 0)), (0, Eq(f, 0)), (sin(f*x)/f, Eq(e, 0)), (2*log(tan(e/2) + tan(
f*x/2))*tan(e/2)**3*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2
+ 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2*log(tan(e/2) + tan(f*x/2))*tan(e/2)**3/(f*tan(e/2)**4*tan(f*x/2)
**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*log(tan(e/2)
+ tan(f*x/2))*tan(e/2)*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)
**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*log(tan(e/2) + tan(f*x/2))*tan(e/2)/(f*tan(e/2)**4*tan(f*x/2)
**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*log(tan(f*x/2
) - 1/tan(e/2))*tan(e/2)**3*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f
*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*log(tan(f*x/2) - 1/tan(e/2))*tan(e/2)**3/(f*tan(e/2)**4*
tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2*log
(tan(f*x/2) - 1/tan(e/2))*tan(e/2)*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**
2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2*log(tan(f*x/2) - 1/tan(e/2))*tan(e/2)/(f*tan(e/2)
**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2
*tan(e/2)**4*tan(f*x/2)/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan
(e/2)**2 + f*tan(f*x/2)**2 + f) - 4*tan(e/2)**3/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2
*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 4*tan(e/2)/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)
**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2*tan(f*x/2)/(f*tan(e/2)**4*tan
(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f), True))*si
n(e)*cos(e) - Piecewise((zoo*x, Eq(e, 0) & Eq(f, 0)), (x/sin(e), Eq(f, 0)), (log(tan(f*x/2))/f, Eq(e, 0)), (lo
g(tan(e/2) + tan(f*x/2))/f - log(tan(f*x/2) - 1/tan(e/2))/f, True))*cos(e)**2 + Piecewise((zoo*x, Eq(e, 0) & E
q(f, 0)), (x/sin(e), Eq(f, 0)), (log(tan(f*x/2))/f, Eq(e, 0)), (log(tan(e/2) + tan(f*x/2))/f - log(tan(f*x/2)
- 1/tan(e/2))/f, True))/2 + 2*Piecewise((zoo*x, Eq(e, 0) & Eq(f, 0)), (x/sin(e), Eq(f, 0)), (log(tan(f*x/2))*t
an(f*x/2)**2/(f*tan(f*x/2)**2 + f) + log(tan(f*x/2))/(f*tan(f*x/2)**2 + f) + 2/(f*tan(f*x/2)**2 + f), Eq(e, 0)
), (log(tan(e/2) + tan(f*x/2))*tan(e/2)**4*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*ta
n(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + log(tan(e/2) + tan(f*x/2))*tan(e/2)**4/(f*t
an(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 +
f) - 2*log(tan(e/2) + tan(f*x/2))*tan(e/2)**2*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*
f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*log(tan(e/2) + tan(f*x/2))*tan(e/2)**
2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2
)**2 + f) + log(tan(e/2) + tan(f*x/2))*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/
2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + log(tan(e/2) + tan(f*x/2))/(f*tan(e/2)**4*tan(f
*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - log(tan(f*
x/2) - 1/tan(e/2))*tan(e/2)**4*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*ta
n(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - log(tan(f*x/2) - 1/tan(e/2))*tan(e/2)**4/(f*tan(e/2)**4
*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2*lo
g(tan(f*x/2) - 1/tan(e/2))*tan(e/2)**2*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/
2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2*log(tan(f*x/2) - 1/tan(e/2))*tan(e/2)**2/(f*t
an(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 +
f) - log(tan(f*x/2) - 1/tan(e/2))*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**
2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - log(tan(f*x/2) - 1/tan(e/2))/(f*tan(e/2)**4*tan(f*x
/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*tan(e/2)**
4/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2
)**2 + f) + 4*tan(e/2)**3*tan(f*x/2)/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)
**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 4*tan(e/2)*tan(f*x/2)/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)
**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2/(f*tan(e/2)**4*tan(f*x/2)**2
+ f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f), True))*cos(e)**2 - P
iecewise((zoo*x, Eq(e, 0) & Eq(f, 0)), (x/sin(e), Eq(f, 0)), (log(tan(f*x/2))*tan(f*x/2)**2/(f*tan(f*x/2)**2 +
f) + log(tan(f*x/2))/(f*tan(f*x/2)**2 + f) + 2/(f*tan(f*x/2)**2 + f), Eq(e, 0)), (log(tan(e/2) + tan(f*x/2))*
tan(e/2)**4*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*t
an(e/2)**2 + f*tan(f*x/2)**2 + f) + log(tan(e/2) + tan(f*x/2))*tan(e/2)**4/(f*tan(e/2)**4*tan(f*x/2)**2 + f*ta
n(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*log(tan(e/2) + tan(f*x/
2))*tan(e/2)**2*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2
*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*log(tan(e/2) + tan(f*x/2))*tan(e/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2
+ f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + log(tan(e/2) + tan(
f*x/2))*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e
/2)**2 + f*tan(f*x/2)**2 + f) + log(tan(e/2) + tan(f*x/2))/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*
tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - log(tan(f*x/2) - 1/tan(e/2))*tan(e/2)**4*
tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 +
f*tan(f*x/2)**2 + f) - log(tan(f*x/2) - 1/tan(e/2))*tan(e/2)**4/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4
+ 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2*log(tan(f*x/2) - 1/tan(e/2))*tan(
e/2)**2*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e
/2)**2 + f*tan(f*x/2)**2 + f) + 2*log(tan(f*x/2) - 1/tan(e/2))*tan(e/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*ta
n(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - log(tan(f*x/2) - 1/tan(e/
2))*tan(f*x/2)**2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)*
*2 + f*tan(f*x/2)**2 + f) - log(tan(f*x/2) - 1/tan(e/2))/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*ta
n(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) - 2*tan(e/2)**4/(f*tan(e/2)**4*tan(f*x/2)**2
+ f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 4*tan(e/2)**3*tan(f
*x/2)/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f
*x/2)**2 + f) + 4*tan(e/2)*tan(f*x/2)/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)**2*tan(f*x/2
)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f) + 2/(f*tan(e/2)**4*tan(f*x/2)**2 + f*tan(e/2)**4 + 2*f*tan(e/2)*
*2*tan(f*x/2)**2 + 2*f*tan(e/2)**2 + f*tan(f*x/2)**2 + f), True))
________________________________________________________________________________________
Giac [B] time = 1.14573, size = 41, normalized size = 4.1 \begin{align*} -\frac{2}{f{\left (\frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-csc(f*x+e)*sin(f*x+e)^2,x, algorithm="giac")
[Out]
-2/(f*((cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 1))