3.230 \(\int \csc ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx\)

Optimal. Leaf size=78 \[ \frac{a^3 A \cos (c+d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x \]

[Out]

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

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Rubi [A]  time = 0.120552, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2966, 3767, 8, 3768, 3770, 2638} \[ \frac{a^3 A \cos (c+d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 A \cot (c+d x) \csc (c+d x)}{2 d}-2 a^3 A x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2966

Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; Fr
eeQ[{a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IntegerQ[n]

Rule 3767

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 8

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

Rule 3768

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 3770

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

Rule 2638

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

Rubi steps

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

Mathematica [A]  time = 0.034722, size = 142, normalized size = 1.82 \[ -\frac{a^3 A \sin (c) \sin (d x)}{d}+\frac{a^3 A \cos (c) \cos (d x)}{d}-\frac{2 a^3 A \cot (c+d x)}{d}-\frac{a^3 A \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a^3 A \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a^3 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{a^3 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-2 a^3 A x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.058, size = 94, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}A\cos \left ( dx+c \right ) }{d}}-2\,{a}^{3}Ax-2\,{\frac{A{a}^{3}c}{d}}-2\,{\frac{{a}^{3}A\cot \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}A\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}A\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.97063, size = 122, normalized size = 1.56 \begin{align*} -\frac{8 \,{\left (d x + c\right )} A a^{3} - A a^{3}{\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^{3} \cos \left (d x + c\right ) + \frac{8 \, A a^{3}}{\tan \left (d x + c\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.028, size = 377, normalized size = 4.83 \begin{align*} -\frac{8 \, A a^{3} d x \cos \left (d x + c\right )^{2} - 4 \, A a^{3} \cos \left (d x + c\right )^{3} - 8 \, A a^{3} d x - 8 \, A a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 \, A a^{3} \cos \left (d x + c\right ) +{\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (A a^{3} \cos \left (d x + c\right )^{2} - A a^{3}\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.22309, size = 185, normalized size = 2.37 \begin{align*} \frac{A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 16 \,{\left (d x + c\right )} A a^{3} + 4 \, A a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 8 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{16 \, A a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} - \frac{6 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(A*a^3*tan(1/2*d*x + 1/2*c)^2 - 16*(d*x + c)*A*a^3 + 4*A*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 8*A*a^3*tan(
1/2*d*x + 1/2*c) + 16*A*a^3/(tan(1/2*d*x + 1/2*c)^2 + 1) - (6*A*a^3*tan(1/2*d*x + 1/2*c)^2 + 8*A*a^3*tan(1/2*d
*x + 1/2*c) + A*a^3)/tan(1/2*d*x + 1/2*c)^2)/d