\(\int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx\) [21]

Optimal result

Integrand size = 29, antiderivative size = 51 \[ \int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-\frac {3 a A \text {arctanh}(\cos (c+d x))}{2 d}+\frac {2 a A \cot (c+d x)}{d}-\frac {a A \cot (c+d x) \csc (c+d x)}{2 d} \]

[Out]

-3/2*a*A*arctanh(cos(d*x+c))/d+2*a*A*cot(d*x+c)/d-1/2*a*A*cot(d*x+c)*csc(d*x+c)/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {21, 3873, 3852, 8, 4131, 3855} \[ \int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-\frac {3 a A \text {arctanh}(\cos (c+d x))}{2 d}+\frac {2 a A \cot (c+d x)}{d}-\frac {a A \cot (c+d x) \csc (c+d x)}{2 d} \]

[In]

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

[Out]

(-3*a*A*ArcTanh[Cos[c + d*x]])/(2*d) + (2*a*A*Cot[c + d*x])/d - (a*A*Cot[c + d*x]*Csc[c + d*x])/(2*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 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

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 4131

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

Rubi steps \begin{align*} \text {integral}& = \frac {A \int \csc (c+d x) (a-a \csc (c+d x))^2 \, dx}{a} \\ & = \frac {A \int \csc (c+d x) \left (a^2+a^2 \csc ^2(c+d x)\right ) \, dx}{a}-(2 a A) \int \csc ^2(c+d x) \, dx \\ & = -\frac {a A \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} (3 a A) \int \csc (c+d x) \, dx+\frac {(2 a A) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = -\frac {3 a A \text {arctanh}(\cos (c+d x))}{2 d}+\frac {2 a A \cot (c+d x)}{d}-\frac {a A \cot (c+d x) \csc (c+d x)}{2 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(51)=102\).

Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.69 \[ \int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {2 a A \cot (c+d x)}{d}-\frac {a A \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a A \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {a A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a A \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {a A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a A \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d} \]

[In]

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

[Out]

(2*a*A*Cot[c + d*x])/d - (a*A*Csc[(c + d*x)/2]^2)/(8*d) - (a*A*Log[Cos[c/2 + (d*x)/2]])/d - (a*A*Log[Cos[(c +
d*x)/2]])/(2*d) + (a*A*Log[Sin[c/2 + (d*x)/2]])/d + (a*A*Log[Sin[(c + d*x)/2]])/(2*d) + (a*A*Sec[(c + d*x)/2]^
2)/(8*d)

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.31

method	result	size
parallelrisch	\(\frac {A a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+12 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\)	\(67\)
derivativedivides	\(\frac {A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+2 A a \cot \left (d x +c \right )+A a \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d}\)	\(71\)
default	\(\frac {A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+2 A a \cot \left (d x +c \right )+A a \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{d}\)	\(71\)
parts	\(-\frac {A a \ln \left (\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}{d}+\frac {A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}+\frac {2 a A \cot \left (d x +c \right )}{d}\)	\(75\)
norman	\(\frac {\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {A a}{8 d}-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 A a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\)	\(90\)
risch	\(\frac {A a \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+4 i {\mathrm e}^{2 i \left (d x +c \right )}-4 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {3 A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {3 A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\)	\(92\)

[In]

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

[Out]

1/8*A*a*(tan(1/2*d*x+1/2*c)^2-cot(1/2*d*x+1/2*c)^2+12*ln(tan(1/2*d*x+1/2*c))-8*tan(1/2*d*x+1/2*c)+8*cot(1/2*d*
x+1/2*c))/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (47) = 94\).

Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.02 \[ \int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-\frac {8 \, A a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, A a \cos \left (d x + c\right ) + 3 \, {\left (A a \cos \left (d x + c\right )^{2} - A a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (A a \cos \left (d x + c\right )^{2} - A a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]

[In]

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

[Out]

-1/4*(8*A*a*cos(d*x + c)*sin(d*x + c) - 2*A*a*cos(d*x + c) + 3*(A*a*cos(d*x + c)^2 - A*a)*log(1/2*cos(d*x + c)
+ 1/2) - 3*(A*a*cos(d*x + c)^2 - A*a)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^2 - d)

Sympy [F]

\[ \int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=A a \left (\int \csc {\left (c + d x \right )}\, dx + \int \left (- 2 \csc ^{2}{\left (c + d x \right )}\right )\, dx + \int \csc ^{3}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

A*a*(Integral(csc(c + d*x), x) + Integral(-2*csc(c + d*x)**2, x) + Integral(csc(c + d*x)**3, x))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.57 \[ \int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {A a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, A a \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right ) + \frac {8 \, A a}{\tan \left (d x + c\right )}}{4 \, d} \]

[In]

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

[Out]

1/4*(A*a*(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) - 4*A*a*log(cot
(d*x + c) + csc(d*x + c)) + 8*A*a/tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.82 \[ \int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 8 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {18 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]

[In]

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

[Out]

1/8*(A*a*tan(1/2*d*x + 1/2*c)^2 + 12*A*a*log(abs(tan(1/2*d*x + 1/2*c))) - 8*A*a*tan(1/2*d*x + 1/2*c) - (18*A*a
*tan(1/2*d*x + 1/2*c)^2 - 8*A*a*tan(1/2*d*x + 1/2*c) + A*a)/tan(1/2*d*x + 1/2*c)^2)/d

Mupad [B] (verification not implemented)

Time = 18.79 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.69 \[ \int \csc (c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {3\,A\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {A\,a}{8}-A\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {A\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d}+\frac {A\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d} \]

[In]

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

[Out]

(3*A*a*log(tan(c/2 + (d*x)/2)))/(2*d) - (cot(c/2 + (d*x)/2)^2*((A*a)/8 - A*a*tan(c/2 + (d*x)/2)))/d - (A*a*tan
(c/2 + (d*x)/2))/d + (A*a*tan(c/2 + (d*x)/2)^2)/(8*d)