\(\int -\sin (e+f x) \, dx\) [8]
Optimal result
Integrand size = 8, antiderivative size = 10 \[
\int -\sin (e+f x) \, dx=\frac {\cos (e+f x)}{f}
\]
[Out]
cos(f*x+e)/f
Rubi [A] (verified)
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2718}
\[
\int -\sin (e+f x) \, dx=\frac {\cos (e+f x)}{f}
\]
[In]
Int[-Sin[e + f*x],x]
[Out]
Cos[e + f*x]/f
Rule 2718
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rubi steps \begin{align*}
\text {integral}& = \frac {\cos (e+f x)}{f} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(22\) vs. \(2(10)=20\).
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 2.20
\[
\int -\sin (e+f x) \, dx=\frac {\cos (e) \cos (f x)}{f}-\frac {\sin (e) \sin (f x)}{f}
\]
[In]
Integrate[-Sin[e + f*x],x]
[Out]
(Cos[e]*Cos[f*x])/f - (Sin[e]*Sin[f*x])/f
Maple [A] (verified)
Time = 0.34 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {\cos \left (f x +e \right )}{f}\) |
\(11\) |
| default |
\(\frac {\cos \left (f x +e \right )}{f}\) |
\(11\) |
| risch |
\(\frac {\cos \left (f x +e \right )}{f}\) |
\(11\) |
| parallelrisch |
\(-\frac {-\cos \left (f x +e \right )-1}{f}\) |
\(16\) |
| norman |
\(-\frac {2 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\) |
\(32\) |
| | |
|
|
|
|
[In]
int(-sin(f*x+e)^2*csc(f*x+e),x,method=_RETURNVERBOSE)
[Out]
cos(f*x+e)/f
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
\[
\int -\sin (e+f x) \, dx=\frac {\cos \left (f x + e\right )}{f}
\]
[In]
integrate(-csc(f*x+e)*sin(f*x+e)^2,x, algorithm="fricas")
[Out]
cos(f*x + e)/f
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (7) = 14\).
Time = 10.22 (sec) , antiderivative size = 4124, normalized size of antiderivative = 412.40
\[
\int -\sin (e+f x) \, dx=\text {Too large to display}
\]
[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)), (sin(f*x)/f, Eq(e, 0)), (0, Eq(f, 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)), (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)), (x/sin(e), Eq(f, 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)), (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)), (x/sin(e), Eq(f, 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))
Maxima [A] (verification not implemented)
none
Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
\[
\int -\sin (e+f x) \, dx=\frac {\cos \left (f x + e\right )}{f}
\]
[In]
integrate(-csc(f*x+e)*sin(f*x+e)^2,x, algorithm="maxima")
[Out]
cos(f*x + e)/f
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (10) = 20\).
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.80
\[
\int -\sin (e+f x) \, dx=-\frac {2}{f {\left (\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 1\right )}}
\]
[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))
Mupad [B] (verification not implemented)
Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
\[
\int -\sin (e+f x) \, dx=\frac {\cos \left (e+f\,x\right )}{f}
\]
[In]
int(-sin(e + f*x),x)
[Out]
cos(e + f*x)/f