\(\int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx\) [19]

Optimal result

Integrand size = 31, antiderivative size = 91 \[ \int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-\frac {7 a A \text {arctanh}(\cos (c+d x))}{8 d}+\frac {2 a A \cot (c+d x)}{d}+\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d} \]

[Out]

-7/8*a*A*arctanh(cos(d*x+c))/d+2*a*A*cot(d*x+c)/d+2/3*a*A*cot(d*x+c)^3/d-7/8*a*A*cot(d*x+c)*csc(d*x+c)/d-1/4*a
*A*cot(d*x+c)*csc(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {21, 3873, 3852, 4131, 3853, 3855} \[ \int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=-\frac {7 a A \text {arctanh}(\cos (c+d x))}{8 d}+\frac {2 a A \cot ^3(c+d x)}{3 d}+\frac {2 a A \cot (c+d x)}{d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d} \]

[In]

Int[Csc[c + d*x]^3*(a - a*Csc[c + d*x])*(A - A*Csc[c + d*x]),x]

[Out]

(-7*a*A*ArcTanh[Cos[c + d*x]])/(8*d) + (2*a*A*Cot[c + d*x])/d + (2*a*A*Cot[c + d*x]^3)/(3*d) - (7*a*A*Cot[c +
d*x]*Csc[c + d*x])/(8*d) - (a*A*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)

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 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

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 ^3(c+d x) (a-a \csc (c+d x))^2 \, dx}{a} \\ & = \frac {A \int \csc ^3(c+d x) \left (a^2+a^2 \csc ^2(c+d x)\right ) \, dx}{a}-(2 a A) \int \csc ^4(c+d x) \, dx \\ & = -\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} (7 a A) \int \csc ^3(c+d x) \, dx+\frac {(2 a A) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {2 a A \cot (c+d x)}{d}+\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} (7 a A) \int \csc (c+d x) \, dx \\ & = -\frac {7 a A \text {arctanh}(\cos (c+d x))}{8 d}+\frac {2 a A \cot (c+d x)}{d}+\frac {2 a A \cot ^3(c+d x)}{3 d}-\frac {7 a A \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a A \cot (c+d x) \csc ^3(c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.79 \[ \int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {4 a A \cot (c+d x)}{3 d}-\frac {7 a A \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a A \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {2 a A \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac {7 a A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {7 a A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {7 a A \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a A \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d} \]

[In]

Integrate[Csc[c + d*x]^3*(a - a*Csc[c + d*x])*(A - A*Csc[c + d*x]),x]

[Out]

(4*a*A*Cot[c + d*x])/(3*d) - (7*a*A*Csc[(c + d*x)/2]^2)/(32*d) - (a*A*Csc[(c + d*x)/2]^4)/(64*d) + (2*a*A*Cot[
c + d*x]*Csc[c + d*x]^2)/(3*d) - (7*a*A*Log[Cos[(c + d*x)/2]])/(8*d) + (7*a*A*Log[Sin[(c + d*x)/2]])/(8*d) + (
7*a*A*Sec[(c + d*x)/2]^2)/(32*d) + (a*A*Sec[(c + d*x)/2]^4)/(64*d)

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(\frac {A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )-2 A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )+A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\)	\(112\)
default	\(\frac {A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )-2 A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )+A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\)	\(112\)
parts	\(\frac {A a \left (-\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}+\frac {A a \left (\left (-\frac {\csc \left (d x +c \right )^{3}}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{8}\right )}{d}-\frac {2 A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )}{d}\)	\(117\)
parallelrisch	\(-\frac {a \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {16 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+16 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-56 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) A}{64 d}\)	\(119\)
risch	\(\frac {A a \left (21 \,{\mathrm e}^{7 i \left (d x +c \right )}-45 \,{\mathrm e}^{5 i \left (d x +c \right )}-45 \,{\mathrm e}^{3 i \left (d x +c \right )}-96 i {\mathrm e}^{4 i \left (d x +c \right )}+21 \,{\mathrm e}^{i \left (d x +c \right )}+128 i {\mathrm e}^{2 i \left (d x +c \right )}-32 i\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {7 A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {7 A a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\)	\(131\)
norman	\(\frac {-\frac {A a}{64 d}+\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d}-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 d}+\frac {3 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}-\frac {3 A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}+\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 d}-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{12 d}+\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {7 A a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\)	\(163\)

[In]

int(csc(d*x+c)^3*(a-a*csc(d*x+c))*(A-A*csc(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(A*a*((-1/4*csc(d*x+c)^3-3/8*csc(d*x+c))*cot(d*x+c)+3/8*ln(-cot(d*x+c)+csc(d*x+c)))-2*A*a*(-2/3-1/3*csc(d*
x+c)^2)*cot(d*x+c)+A*a*(-1/2*csc(d*x+c)*cot(d*x+c)+1/2*ln(-cot(d*x+c)+csc(d*x+c))))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.77 \[ \int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {42 \, A a \cos \left (d x + c\right )^{3} - 54 \, A a \cos \left (d x + c\right ) - 21 \, {\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 21 \, {\left (A a \cos \left (d x + c\right )^{4} - 2 \, A a \cos \left (d x + c\right )^{2} + A a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (2 \, A a \cos \left (d x + c\right )^{3} - 3 \, A a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]

[In]

integrate(csc(d*x+c)^3*(a-a*csc(d*x+c))*(A-A*csc(d*x+c)),x, algorithm="fricas")

[Out]

1/48*(42*A*a*cos(d*x + c)^3 - 54*A*a*cos(d*x + c) - 21*(A*a*cos(d*x + c)^4 - 2*A*a*cos(d*x + c)^2 + A*a)*log(1
/2*cos(d*x + c) + 1/2) + 21*(A*a*cos(d*x + c)^4 - 2*A*a*cos(d*x + c)^2 + A*a)*log(-1/2*cos(d*x + c) + 1/2) - 3
2*(2*A*a*cos(d*x + c)^3 - 3*A*a*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)

Sympy [F]

\[ \int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=A a \left (\int \csc ^{3}{\left (c + d x \right )}\, dx + \int \left (- 2 \csc ^{4}{\left (c + d x \right )}\right )\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

A*a*(Integral(csc(c + d*x)**3, x) + Integral(-2*csc(c + d*x)**4, x) + Integral(csc(c + d*x)**5, x))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.59 \[ \int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {3 \, A a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, A a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{48 \, d} \]

[In]

integrate(csc(d*x+c)^3*(a-a*csc(d*x+c))*(A-A*csc(d*x+c)),x, algorithm="maxima")

[Out]

1/48*(3*A*a*(2*(3*cos(d*x + c)^3 - 5*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 3*log(cos(d*x + c
) + 1) + 3*log(cos(d*x + c) - 1)) + 12*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)) + 32*(3*tan(d*x + c)^2 + 1)*A*a/tan(d*x + c)^3)/d

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.70 \[ \int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 168 \, A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 144 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {350 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 144 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]

[In]

integrate(csc(d*x+c)^3*(a-a*csc(d*x+c))*(A-A*csc(d*x+c)),x, algorithm="giac")

[Out]

1/192*(3*A*a*tan(1/2*d*x + 1/2*c)^4 - 16*A*a*tan(1/2*d*x + 1/2*c)^3 + 48*A*a*tan(1/2*d*x + 1/2*c)^2 + 168*A*a*
log(abs(tan(1/2*d*x + 1/2*c))) - 144*A*a*tan(1/2*d*x + 1/2*c) - (350*A*a*tan(1/2*d*x + 1/2*c)^4 - 144*A*a*tan(
1/2*d*x + 1/2*c)^3 + 48*A*a*tan(1/2*d*x + 1/2*c)^2 - 16*A*a*tan(1/2*d*x + 1/2*c) + 3*A*a)/tan(1/2*d*x + 1/2*c)
^4)/d

Mupad [B] (verification not implemented)

Time = 19.28 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.66 \[ \int \csc ^3(c+d x) (a-a \csc (c+d x)) (A-A \csc (c+d x)) \, dx=\frac {A\,a\,\left (3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-144\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+144\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+168\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}{192\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \]

[In]

int(((A - A/sin(c + d*x))*(a - a/sin(c + d*x)))/sin(c + d*x)^3,x)

[Out]

(A*a*(3*sin(c/2 + (d*x)/2)^8 - 3*cos(c/2 + (d*x)/2)^8 - 16*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^7 + 16*cos(c/
2 + (d*x)/2)^7*sin(c/2 + (d*x)/2) + 48*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^6 - 144*cos(c/2 + (d*x)/2)^3*si
n(c/2 + (d*x)/2)^5 + 144*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^3 - 48*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2
)^2 + 168*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^4))/(192*d*cos(c/
2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^4)