\(\int \csc ^2(c+b x) \sin (a+b x) \, dx\) [196]

Optimal result

Integrand size = 15, antiderivative size = 36 \[ \int \csc ^2(c+b x) \sin (a+b x) \, dx=-\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b} \]

[Out]

-arctanh(cos(b*x+c))*cos(a-c)/b-csc(b*x+c)*sin(a-c)/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 36, 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 = {4678, 2686, 8, 3855} \[ \int \csc ^2(c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \text {arctanh}(\cos (b x+c))}{b}-\frac {\sin (a-c) \csc (b x+c)}{b} \]

[In]

Int[Csc[c + b*x]^2*Sin[a + b*x],x]

[Out]

-((ArcTanh[Cos[c + b*x]]*Cos[a - c])/b) - (Csc[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 4678

Int[Csc[w_]^(n_.)*Sin[v_], x_Symbol] :> Dist[Sin[v - w], Int[Cot[w]*Csc[w]^(n - 1), x], x] + Dist[Cos[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 \csc (c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc (c+b x) \, dx \\ & = -\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{b}-\frac {\sin (a-c) \text {Subst}(\int 1 \, dx,x,\csc (c+b x))}{b} \\ & = -\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.50 \[ \int \csc ^2(c+b x) \sin (a+b x) \, dx=-\frac {2 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos (a-c)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b} \]

[In]

Integrate[Csc[c + b*x]^2*Sin[a + b*x],x]

[Out]

((-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])]*Cos[a - c])/b - (Csc[c + b*x]*Sin[a - c])/b

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.38 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.19

method	result	size
risch	\(\frac {{\mathrm e}^{i \left (x b +3 a \right )}-{\mathrm e}^{i \left (x b +a +2 c \right )}}{b \left (-{\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}\)	\(115\)
default	\(\frac {\frac {4 \left (-2 \cos \left (a \right ) \cos \left (c \right )-2 \sin \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+8 \sin \left (a \right ) \cos \left (c \right )-8 \cos \left (a \right ) \sin \left (c \right )}{\left (-4 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \cos \left (a \right )^{2} \cos \left (c \right )^{2}-4 \sin \left (a \right )^{2} \sin \left (c \right )^{2}-4 \cos \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (\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 )\right )}+\frac {4 \left (-2 \cos \left (a \right ) \cos \left (c \right )-2 \sin \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 (-4 \cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \cos \left (a \right )^{2} \cos \left (c \right )^{2}-4 \sin \left (a \right )^{2} \sin \left (c \right )^{2}-4 \cos \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}\)	\(347\)

[In]

int(csc(b*x+c)^2*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/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*c
os(a-c)+ln(exp(I*(b*x+a))-exp(I*(a-c)))/b*cos(a-c)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.97 \[ \int \csc ^2(c+b x) \sin (a+b x) \, dx=-\frac {\cos \left (-a + c\right ) \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - \cos \left (-a + c\right ) \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 2 \, \sin \left (-a + c\right )}{2 \, b \sin \left (b x + c\right )} \]

[In]

integrate(csc(b*x+c)^2*sin(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(cos(-a + c)*log(1/2*cos(b*x + c) + 1/2)*sin(b*x + c) - cos(-a + c)*log(-1/2*cos(b*x + c) + 1/2)*sin(b*x
+ c) - 2*sin(-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. 1690 vs. \(2 (29) = 58\).

Time = 54.94 (sec) , antiderivative size = 3264, normalized size of antiderivative = 90.67 \[ \int \csc ^2(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate(csc(b*x+c)**2*sin(b*x+a),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)*t
an(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*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)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2) + b*ta
n(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)*t
an(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*t
an(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*tan
(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*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)**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*t
an(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)), True))*cos(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*t
an(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)*ta
n(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(ta
n(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))*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)) + tan(c/2)**6*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)**5/(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)**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))*sin(a)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (36) = 72\).

Time = 0.29 (sec) , antiderivative size = 454, normalized size of antiderivative = 12.61 \[ \int \csc ^2(c+b x) \sin (a+b x) \, dx=-\frac {2 \, {\left (\cos \left (b x + 2 \, a\right ) - \cos \left (b x + 2 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) - 2 \, \cos \left (b x + 2 \, a\right ) \cos \left (a\right ) + 2 \, \cos \left (b x + 2 \, c\right ) \cos \left (a\right ) + {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) - 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, \cos \left (-a + c\right ) \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 )} \cos \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} \cos \left (-a + c\right ) - 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, \cos \left (-a + c\right ) \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 )} \cos \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 (\sin \left (b x + 2 \, a\right ) - \sin \left (b x + 2 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) - 2 \, \sin \left (b x + 2 \, a\right ) \sin \left (a\right ) + 2 \, \sin \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(csc(b*x+c)^2*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/2*(2*(cos(b*x + 2*a) - cos(b*x + 2*c))*cos(2*b*x + a + 2*c) - 2*cos(b*x + 2*a)*cos(a) + 2*cos(b*x + 2*c)*co
s(a) + (cos(2*b*x + a + 2*c)^2*cos(-a + c) - 2*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(2*b*x
+ a + 2*c)^2 - 2*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*cos(-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*cos(-a +
c) - 2*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(2*b*x + a + 2*c)^2 - 2*cos(-a + c)*sin(2*b*x
+ a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*cos(-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*(sin(b*x + 2*a) - sin(b*x + 2*c))*sin(2*b*x + a + 2*c) - 2*sin(b*x +
2*a)*sin(a) + 2*sin(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. 349 vs. \(2 (36) = 72\).

Time = 0.32 (sec) , antiderivative size = 349, normalized size of antiderivative = 9.69 \[ \int \csc ^2(c+b x) \sin (a+b x) \, dx=\frac {\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )}}{b} \]

[In]

integrate(csc(b*x+c)^2*sin(b*x+a),x, algorithm="giac")

[Out]

((tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2 + 1)*log(abs(tan(1/2*b*x +
1/2*c)))/(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*c)*tan(1/2*a)^2*t
an(1/2*c) - tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*b*x + 1/2*c)*tan(1/2*a) - tan(1/2*b*x + 1/2
*c)*tan(1/2*c))/(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*c)*tan(1/2*
a)^2*tan(1/2*c)^2 - tan(1/2*b*x + 1/2*c)*tan(1/2*a)^2 + 4*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c) + tan(1/2
*a)^2*tan(1/2*c) - tan(1/2*b*x + 1/2*c)*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*b*x + 1/2*c) + tan(1/
2*a) - tan(1/2*c))/((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*c)))/b

Mupad [B] (verification not implemented)

Time = 25.17 (sec) , antiderivative size = 252, normalized size of antiderivative = 7.00 \[ \int \csc ^2(c+b x) \sin (a+b x) \, dx=-\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \]

[In]

int(sin(a + b*x)/sin(c + b*x)^2,x)

[Out]

(log((exp(a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) + 1)*1i)/(exp(a*2i)*exp(-c*2i))^(1/2) - exp(a*1i)*exp(b*x*1i)
*(exp(a*2i)*exp(-c*2i)*1i + 1i))*(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)*1i + 1i) - (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)
)/(b*(exp(a*2i - c*2i) - exp(a*2i + b*x*2i)))