\(\int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx\) [22]

Optimal result

Integrand size = 29, antiderivative size = 33 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx=-2 a A x-\frac {a A \text {arctanh}(\cos (c+d x))}{d}-\frac {a A \cos (c+d x)}{d} \]

[Out]

-2*a*A*x-a*A*arctanh(cos(d*x+c))/d-a*A*cos(d*x+c)/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {21, 3873, 8, 4130, 3855} \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx=-\frac {a A \text {arctanh}(\cos (c+d x))}{d}-\frac {a A \cos (c+d x)}{d}-2 a A x \]

[In]

Int[(a - a*Csc[c + d*x])*(A - A*Csc[c + d*x])*Sin[c + d*x],x]

[Out]

-2*a*A*x - (a*A*ArcTanh[Cos[c + d*x]])/d - (a*A*Cos[c + d*x])/d

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3873

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Dist[2*a*(b/d
), Int[(d*Csc[e + f*x])^(n + 1), x], x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b
, d, e, f, n}, x]

Rule 4130

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e
+ f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {A \int (a-a \csc (c+d x))^2 \sin (c+d x) \, dx}{a} \\ & = \frac {A \int \left (a^2+a^2 \csc ^2(c+d x)\right ) \sin (c+d x) \, dx}{a}-(2 a A) \int 1 \, dx \\ & = -2 a A x-\frac {a A \cos (c+d x)}{d}+(a A) \int \csc (c+d x) \, dx \\ & = -2 a A x-\frac {a A \text {arctanh}(\cos (c+d x))}{d}-\frac {a A \cos (c+d x)}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(33)=66\).

Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx=-2 a A x-\frac {a A \cos (c) \cos (d x)}{d}-\frac {a A \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {a A \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {a A \sin (c) \sin (d x)}{d} \]

[In]

Integrate[(a - a*Csc[c + d*x])*(A - A*Csc[c + d*x])*Sin[c + d*x],x]

[Out]

-2*a*A*x - (a*A*Cos[c]*Cos[d*x])/d - (a*A*Log[Cos[c/2 + (d*x)/2]])/d + (a*A*Log[Sin[c/2 + (d*x)/2]])/d + (a*A*
Sin[c]*Sin[d*x])/d

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97

method	result	size
parallelrisch	\(-\frac {A a \left (2 d x -1-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cos \left (d x +c \right )\right )}{d}\)	\(32\)
parts	\(-\frac {a A \cos \left (d x +c \right )}{d}-\frac {A a \ln \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{d}-2 a A x\)	\(41\)
derivativedivides	\(\frac {A a \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-2 A a \left (d x +c \right )-A a \cos \left (d x +c \right )}{d}\)	\(44\)
default	\(\frac {A a \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-2 A a \left (d x +c \right )-A a \cos \left (d x +c \right )}{d}\)	\(44\)
risch	\(-2 a A x -\frac {A a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {A a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}\)	\(76\)
norman	\(\frac {\frac {2 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}-2 a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 a A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {A a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(94\)

[In]

int((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c),x,method=_RETURNVERBOSE)

[Out]

-A*a*(2*d*x-1-ln(tan(1/2*d*x+1/2*c))+cos(d*x+c))/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx=-\frac {4 \, A a d x + 2 \, A a \cos \left (d x + c\right ) + A a \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - A a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, d} \]

[In]

integrate((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c),x, algorithm="fricas")

[Out]

-1/2*(4*A*a*d*x + 2*A*a*cos(d*x + c) + A*a*log(1/2*cos(d*x + c) + 1/2) - A*a*log(-1/2*cos(d*x + c) + 1/2))/d

Sympy [A] (verification not implemented)

Time = 6.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.48 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx=- 2 A a x + A a \left (\begin {cases} - \frac {\cot {\left (c + d x \right )}}{d \csc {\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x}{\csc {\left (c \right )}} & \text {otherwise} \end {cases}\right ) + A a \left (\begin {cases} \frac {x \cot {\left (c \right )} \csc {\left (c \right )}}{\cot {\left (c \right )} + \csc {\left (c \right )}} + \frac {x \csc ^{2}{\left (c \right )}}{\cot {\left (c \right )} + \csc {\left (c \right )}} & \text {for}\: d = 0 \\- \frac {\log {\left (\cot {\left (c + d x \right )} + \csc {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c),x)

[Out]

-2*A*a*x + A*a*Piecewise((-cot(c + d*x)/(d*csc(c + d*x)), Ne(d, 0)), (x/csc(c), True)) + A*a*Piecewise((x*cot(
c)*csc(c)/(cot(c) + csc(c)) + x*csc(c)**2/(cot(c) + csc(c)), Eq(d, 0)), (-log(cot(c + d*x) + csc(c + d*x))/d,
True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx=-\frac {2 \, {\left (d x + c\right )} A a + A a \cos \left (d x + c\right ) + A a \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{d} \]

[In]

integrate((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c),x, algorithm="maxima")

[Out]

-(2*(d*x + c)*A*a + A*a*cos(d*x + c) + A*a*log(cot(d*x + c) + csc(d*x + c)))/d

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx=-\frac {2 \, {\left (d x + c\right )} A a - A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {2 \, A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}}{d} \]

[In]

integrate((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c),x, algorithm="giac")

[Out]

-(2*(d*x + c)*A*a - A*a*log(abs(tan(1/2*d*x + 1/2*c))) + 2*A*a/(tan(1/2*d*x + 1/2*c)^2 + 1))/d

Mupad [B] (verification not implemented)

Time = 18.96 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.91 \[ \int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin (c+d x) \, dx=\frac {A\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,A\,a}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {4\,A\,a\,\mathrm {atan}\left (\frac {16\,A^2\,a^2}{8\,A^2\,a^2+16\,A^2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {8\,A^2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,A^2\,a^2+16\,A^2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]

[In]

int(sin(c + d*x)*(A - A/sin(c + d*x))*(a - a/sin(c + d*x)),x)

[Out]

(A*a*log(tan(c/2 + (d*x)/2)))/d - (2*A*a)/(d*(tan(c/2 + (d*x)/2)^2 + 1)) + (4*A*a*atan((16*A^2*a^2)/(8*A^2*a^2
+ 16*A^2*a^2*tan(c/2 + (d*x)/2)) - (8*A^2*a^2*tan(c/2 + (d*x)/2))/(8*A^2*a^2 + 16*A^2*a^2*tan(c/2 + (d*x)/2))
))/d