3.195 \(\int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=33 \[ -\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d} \]

[Out]

-1/2*a*csc(d*x+c)^2/d-1/3*a*csc(d*x+c)^3/d

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Rubi [A]  time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ -\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^4 (a+x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {a+x}{x^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.00 \[ -\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3)/(3*d)

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fricas [A]  time = 0.44, size = 39, normalized size = 1.18 \[ \frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/6*(3*a*sin(d*x + c) + 2*a)/((d*cos(d*x + c)^2 - d)*sin(d*x + c))

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giac [A]  time = 0.15, size = 26, normalized size = 0.79 \[ -\frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/6*(3*a*sin(d*x + c) + 2*a)/(d*sin(d*x + c)^3)

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maple [A]  time = 0.09, size = 27, normalized size = 0.82 \[ \frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^4*(a+a*sin(d*x+c)),x)

[Out]

a/d*(-1/2/sin(d*x+c)^2-1/3/sin(d*x+c)^3)

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maxima [A]  time = 0.30, size = 26, normalized size = 0.79 \[ -\frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/6*(3*a*sin(d*x + c) + 2*a)/(d*sin(d*x + c)^3)

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mupad [B]  time = 8.60, size = 39, normalized size = 1.18 \[ -\frac {\frac {5\,a\,\sin \left (c+d\,x\right )}{16}+\frac {a\,\left (\frac {3\,\sin \left (3\,c+3\,d\,x\right )}{16}+1\right )}{3}}{d\,{\sin \left (c+d\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)*(a + a*sin(c + d*x)))/sin(c + d*x)^4,x)

[Out]

-((5*a*sin(c + d*x))/16 + (a*((3*sin(3*c + 3*d*x))/16 + 1))/3)/(d*sin(c + d*x)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \cos {\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)**4*(a+a*sin(d*x+c)),x)

[Out]

a*(Integral(cos(c + d*x)*csc(c + d*x)**4, x) + Integral(sin(c + d*x)*cos(c + d*x)*csc(c + d*x)**4, x))

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