\(\int \cos (a+b x) \csc ^2(c+b x) \, dx\) [228]
Optimal result
Integrand size = 15, antiderivative size = 35 \[
\int \cos (a+b x) \csc ^2(c+b x) \, dx=-\frac {\cos (a-c) \csc (c+b x)}{b}+\frac {\text {arctanh}(\cos (c+b x)) \sin (a-c)}{b}
\]
[Out]
-cos(a-c)*csc(b*x+c)/b+arctanh(cos(b*x+c))*sin(a-c)/b
Rubi [A] (verified)
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4677, 2686, 8, 3855}
\[
\int \cos (a+b x) \csc ^2(c+b x) \, dx=\frac {\sin (a-c) \text {arctanh}(\cos (b x+c))}{b}-\frac {\cos (a-c) \csc (b x+c)}{b}
\]
[In]
Int[Cos[a + b*x]*Csc[c + b*x]^2,x]
[Out]
-((Cos[a - c]*Csc[c + b*x])/b) + (ArcTanh[Cos[c + b*x]]*Sin[a - c])/b
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 2686
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Rule 3855
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rule 4677
Int[Cos[v_]*Csc[w_]^(n_.), x_Symbol] :> Dist[Cos[v - w], Int[Cot[w]*Csc[w]^(n - 1), x], x] - Dist[Sin[v - w],
Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Rubi steps \begin{align*}
\text {integral}& = \cos (a-c) \int \cot (c+b x) \csc (c+b x) \, dx-\sin (a-c) \int \csc (c+b x) \, dx \\ & = \frac {\text {arctanh}(\cos (c+b x)) \sin (a-c)}{b}-\frac {\cos (a-c) \text {Subst}(\int 1 \, dx,x,\csc (c+b x))}{b} \\ & = -\frac {\cos (a-c) \csc (c+b x)}{b}+\frac {\text {arctanh}(\cos (c+b x)) \sin (a-c)}{b} \\
\end{align*}
Mathematica [C] (verified)
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.57
\[
\int \cos (a+b x) \csc ^2(c+b x) \, dx=-\frac {\cos (a-c) \csc (c+b x)}{b}+\frac {2 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \sin (a-c)}{b}
\]
[In]
Integrate[Cos[a + b*x]*Csc[c + b*x]^2,x]
[Out]
-((Cos[a - c]*Csc[c + b*x])/b) + ((2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]
))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 1.48 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.29
| | |
| method | result | size |
| | |
| risch |
\(\frac {i \left ({\mathrm e}^{i \left (x b +3 a \right )}+{\mathrm e}^{i \left (x b +a +2 c \right )}\right )}{b \left (-{\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}\) |
\(115\) |
| default |
\(\frac {-\frac {2 \left (-\frac {\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+2 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\cos \left (a \right )^{2} \sin \left (c \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )}+\frac {\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )}{\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\cos \left (a \right )^{2} \sin \left (c \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}\right )}{\cos \left (c \right ) \sin \left (a \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-\sin \left (c \right ) \cos \left (a \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \cos \left (a \right ) \cos \left (c \right )+2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )}-\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2 \cos \left (a \right ) \cos \left (c \right )+2 \sin \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}}}\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\cos \left (a \right )^{2} \sin \left (c \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \sqrt {-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}}}}{b}\) |
\(408\) |
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[In]
int(cos(b*x+a)/sin(b*x+c)^2,x,method=_RETURNVERBOSE)
[Out]
I/b/(-exp(2*I*(b*x+a+c))+exp(2*I*a))*(exp(I*(b*x+3*a))+exp(I*(b*x+a+2*c)))+ln(exp(I*(b*x+a))+exp(I*(a-c)))/b*s
in(a-c)-ln(exp(I*(b*x+a))-exp(I*(a-c)))/b*sin(a-c)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (35) = 70\).
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.03
\[
\int \cos (a+b x) \csc ^2(c+b x) \, dx=-\frac {\log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + 2 \, \cos \left (-a + c\right )}{2 \, b \sin \left (b x + c\right )}
\]
[In]
integrate(cos(b*x+a)/sin(b*x+c)^2,x, algorithm="fricas")
[Out]
-1/2*(log(1/2*cos(b*x + c) + 1/2)*sin(b*x + c)*sin(-a + c) - log(-1/2*cos(b*x + c) + 1/2)*sin(b*x + c)*sin(-a
+ c) + 2*cos(-a + c))/(b*sin(b*x + c))
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1572 vs. \(2 (27) = 54\).
Time = 61.01 (sec) , antiderivative size = 3264, normalized size of antiderivative = 93.26
\[
\int \cos (a+b x) \csc ^2(c+b x) \, dx=\text {Too large to display}
\]
[In]
integrate(cos(b*x+a)/sin(b*x+c)**2,x)
[Out]
-Piecewise((0, Eq(b, 0) & Eq(c, 0)), (log(tan(b*x/2))/b, Eq(c, 0)), (0, Eq(b, 0)), (-log(tan(c/2) + tan(b*x/2)
)*tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*
tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - log(tan(c/2) + tan(b*x/2))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)*
*4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*
x/2)) + log(tan(c/2) + tan(b*x/2))*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan
(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) + 2*log(tan(c/2) + tan(b*x/2))*tan(c/2)**2*ta
n(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 -
b*tan(c/2) - b*tan(b*x/2)) + log(tan(c/2) + tan(b*x/2))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2) + b*t
an(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - log(tan(c/2
) + tan(b*x/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*
tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - log(tan(c/2) + tan(b*x/2))*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) +
b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) + log(tan
(b*x/2) - 1/tan(c/2))*tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c
/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) + log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)**3*tan(
b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 -
b*tan(c/2) - b*tan(b*x/2)) - log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)*
*3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - 2*log(tan(b*x/2) -
1/tan(c/2))*tan(c/2)**2*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b
*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)*tan(b*x/2)**2/(b*
tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) -
b*tan(b*x/2)) + log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2
- b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) + log(tan(b*x/2) - 1/tan(c/2))*tan(b*
x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*ta
n(c/2) - b*tan(b*x/2)) + tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*ta
n(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - 2*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2) +
b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - 2*tan(c/
2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(
c/2) - b*tan(b*x/2)) - tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*
tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)), True))*sin(a) + Piecewise((zoo*x, Eq(b, 0) & Eq(c, 0)), (
-1/(b*sin(b*x)), Eq(c, 0)), (x/sin(c)**2, Eq(b, 0)), (4*log(tan(c/2) + tan(b*x/2))*tan(c/2)**4*tan(b*x/2)/(2*b
*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*
b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) + 4*log(tan(c/2) + tan(b*x/2))*tan(c/2)**3*tan(b*x/2)**2/(2*b*tan(c/2
)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/
2)**2 - 2*b*tan(c/2)*tan(b*x/2)) - 4*log(tan(c/2) + tan(b*x/2))*tan(c/2)**3/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*
tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*t
an(b*x/2)) - 4*log(tan(c/2) + tan(b*x/2))*tan(c/2)**2*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4
*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2))
- 4*log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)**4*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/
2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) - 4*log(t
an(b*x/2) - 1/tan(c/2))*tan(c/2)**3*tan(b*x/2)**2/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2
- 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) + 4*log(tan(b*x
/2) - 1/tan(c/2))*tan(c/2)**3/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 +
2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) + 4*log(tan(b*x/2) - 1/tan(c/2))*ta
n(c/2)**2*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c
/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) + tan(c/2)**6*tan(b*x/2)/(2*b*tan(c/2)**5*ta
n(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 -
2*b*tan(c/2)*tan(b*x/2)) - 2*tan(c/2)**5/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*ta
n(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) - tan(c/2)**4*tan(b*x/2
)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**
2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) - tan(c/2)**2*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(
c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b
*x/2)) + 2*tan(c/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/
2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) + tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*
b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)
*tan(b*x/2)), True))*cos(a)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (35) = 70\).
Time = 0.25 (sec) , antiderivative size = 450, normalized size of antiderivative = 12.86
\[
\int \cos (a+b x) \csc ^2(c+b x) \, dx=\frac {2 \, {\left (\sin \left (b x + 2 \, a\right ) + \sin \left (b x + 2 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) - {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \sin \left (-a + c\right ) + \sin \left (2 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) \sin \left (-a + c\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) + {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \sin \left (-a + c\right ) + \sin \left (2 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) \sin \left (-a + c\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - 2 \, {\left (\cos \left (b x + 2 \, a\right ) + \cos \left (b x + 2 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) - 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) - 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, c\right ) + 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (a\right )}{2 \, {\left (b \cos \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) + b \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} b\right )}}
\]
[In]
integrate(cos(b*x+a)/sin(b*x+c)^2,x, algorithm="maxima")
[Out]
1/2*(2*(sin(b*x + 2*a) + sin(b*x + 2*c))*cos(2*b*x + a + 2*c) - (cos(2*b*x + a + 2*c)^2*sin(-a + c) - 2*cos(2*
b*x + a + 2*c)*cos(a)*sin(-a + c) + sin(2*b*x + a + 2*c)^2*sin(-a + c) - 2*sin(2*b*x + a + 2*c)*sin(a)*sin(-a
+ c) + (cos(a)^2 + sin(a)^2)*sin(-a + c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b
*x)*sin(c) + sin(c)^2) + (cos(2*b*x + a + 2*c)^2*sin(-a + c) - 2*cos(2*b*x + a + 2*c)*cos(a)*sin(-a + c) + sin
(2*b*x + a + 2*c)^2*sin(-a + c) - 2*sin(2*b*x + a + 2*c)*sin(a)*sin(-a + c) + (cos(a)^2 + sin(a)^2)*sin(-a + c
))*log(cos(b*x)^2 - 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2) - 2*(cos(b*x + 2
*a) + cos(b*x + 2*c))*sin(2*b*x + a + 2*c) - 2*cos(a)*sin(b*x + 2*a) - 2*cos(a)*sin(b*x + 2*c) + 2*cos(b*x + 2
*a)*sin(a) + 2*cos(b*x + 2*c)*sin(a))/(b*cos(2*b*x + a + 2*c)^2 - 2*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(2*b*
x + a + 2*c)^2 - 2*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*b)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 893 vs. \(2 (35) = 70\).
Time = 0.41 (sec) , antiderivative size = 893, normalized size of antiderivative = 25.51
\[
\int \cos (a+b x) \csc ^2(c+b x) \, dx=\text {Too large to display}
\]
[In]
integrate(cos(b*x+a)/sin(b*x+c)^2,x, algorithm="giac")
[Out]
-1/2*(4*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(2*tan(1/2*b*x +
1/2*a)*tan(1/2*a)^2*tan(1/2*c) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(1/2*b*x + 1/2*a)*tan(1
/2*a) - 2*tan(1/2*a)^2 - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*c) + 4*tan(1/2*a)*tan(1/2*c) - 2*tan(1/2*c)^2)/abs(2*t
an(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(1/2*a)^2*
tan(1/2*c)^2 + 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*c) + 4*tan(1/2*a)*tan(1/2*c)
+ 2))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) - (tan(1/2*b*x + 1/2*a)*tan(1/2*a)^4*tan(
1/2*c)^4 - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^4*tan(1/2*c)^2 + 8*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^3*tan(1/2*c)^3
- 2*tan(1/2*a)^4*tan(1/2*c)^3 - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c)^4 + 2*tan(1/2*a)^3*tan(1/2*c)^
4 + tan(1/2*b*x + 1/2*a)*tan(1/2*a)^4 - 8*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^3*tan(1/2*c) + 2*tan(1/2*a)^4*tan(1/
2*c) + 20*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c)^2 - 12*tan(1/2*a)^3*tan(1/2*c)^2 - 8*tan(1/2*b*x + 1/2*
a)*tan(1/2*a)*tan(1/2*c)^3 + 12*tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*b*x + 1/2*a)*tan(1/2*c)^4 - 2*tan(1/2*a)*t
an(1/2*c)^4 - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2 + 2*tan(1/2*a)^3 + 8*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2
*c) - 12*tan(1/2*a)^2*tan(1/2*c) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*c)^2 + 12*tan(1/2*a)*tan(1/2*c)^2 - 2*tan(1/
2*c)^3 + tan(1/2*b*x + 1/2*a) - 2*tan(1/2*a) + 2*tan(1/2*c))/((tan(1/2*b*x + 1/2*a)^2*tan(1/2*a)^2*tan(1/2*c)
- tan(1/2*b*x + 1/2*a)^2*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*b*
x + 1/2*a)^2*tan(1/2*a) - tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2 - tan(1/2*b*x + 1/2*a)^2*tan(1/2*c) + 4*tan(1/2*b*
x + 1/2*a)*tan(1/2*a)*tan(1/2*c) - tan(1/2*a)^2*tan(1/2*c) - tan(1/2*b*x + 1/2*a)*tan(1/2*c)^2 + tan(1/2*a)*ta
n(1/2*c)^2 + tan(1/2*b*x + 1/2*a) - tan(1/2*a) + tan(1/2*c))*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^
2 + tan(1/2*a) - tan(1/2*c))))/b
Mupad [B] (verification not implemented)
Time = 26.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 7.20
\[
\int \cos (a+b x) \csc ^2(c+b x) \, dx=-\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}
\]
[In]
int(cos(a + b*x)/sin(c + b*x)^2,x)
[Out]
(log(exp(a*1i)*exp(b*x*1i)*(exp(a*2i)*exp(-c*2i) - 1) + (exp(a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) - 1)*1i)/(
-exp(a*2i)*exp(-c*2i))^(1/2))*(exp(a*2i - c*2i) - 1))/(2*b*(-exp(a*2i - c*2i))^(1/2)) - (log(exp(a*1i)*exp(b*x
*1i)*(exp(a*2i)*exp(-c*2i) - 1) - (exp(a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) - 1)*1i)/(-exp(a*2i)*exp(-c*2i))
^(1/2))*(exp(a*2i - c*2i) - 1))/(2*b*(-exp(a*2i - c*2i))^(1/2)) + (exp(a*1i + b*x*1i)*(exp(a*2i - c*2i) + 1)*1
i)/(b*(exp(a*2i - c*2i) - exp(a*2i + b*x*2i)))