3.1209 \(\int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx\)

Optimal. Leaf size=61 \[ -\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc ^5(c+d x)}{5 d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc (c+d x)}{d} \]

[Out]

-1/6*a*cot(d*x+c)^6/d-b*csc(d*x+c)/d+2/3*b*csc(d*x+c)^3/d-1/5*b*csc(d*x+c)^5/d

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Rubi [A]  time = 0.11, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2834, 2607, 30, 2606, 194} \[ -\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc ^5(c+d x)}{5 d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x]),x]

[Out]

-(a*Cot[c + d*x]^6)/(6*d) - (b*Csc[c + d*x])/d + (2*b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2834

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e +
f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0]
&& NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])

Rubi steps

\begin {align*} \int \cot ^5(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^2(c+d x) \, dx+b \int \cot ^5(c+d x) \csc (c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{d}-\frac {b \operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc (c+d x)}{d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 61, normalized size = 1.00 \[ -\frac {a \cot ^6(c+d x)}{6 d}-\frac {b \csc ^5(c+d x)}{5 d}+\frac {2 b \csc ^3(c+d x)}{3 d}-\frac {b \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x]),x]

[Out]

-1/6*(a*Cot[c + d*x]^6)/d - (b*Csc[c + d*x])/d + (2*b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d)

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fricas [A]  time = 0.73, size = 100, normalized size = 1.64 \[ \frac {15 \, a \cos \left (d x + c\right )^{4} - 15 \, a \cos \left (d x + c\right )^{2} + 2 \, {\left (15 \, b \cos \left (d x + c\right )^{4} - 20 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right ) + 5 \, a}{30 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/30*(15*a*cos(d*x + c)^4 - 15*a*cos(d*x + c)^2 + 2*(15*b*cos(d*x + c)^4 - 20*b*cos(d*x + c)^2 + 8*b)*sin(d*x
+ c) + 5*a)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)

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giac [A]  time = 0.24, size = 70, normalized size = 1.15 \[ -\frac {30 \, b \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, b \sin \left (d x + c\right )^{3} - 15 \, a \sin \left (d x + c\right )^{2} + 6 \, b \sin \left (d x + c\right ) + 5 \, a}{30 \, d \sin \left (d x + c\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/30*(30*b*sin(d*x + c)^5 + 15*a*sin(d*x + c)^4 - 20*b*sin(d*x + c)^3 - 15*a*sin(d*x + c)^2 + 6*b*sin(d*x + c
) + 5*a)/(d*sin(d*x + c)^6)

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maple [A]  time = 0.43, size = 110, normalized size = 1.80 \[ \frac {-\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{6 \sin \left (d x +c \right )^{6}}+b \left (-\frac {\cos ^{6}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{15 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{5 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^7*(a+b*sin(d*x+c)),x)

[Out]

1/d*(-1/6*a/sin(d*x+c)^6*cos(d*x+c)^6+b*(-1/5/sin(d*x+c)^5*cos(d*x+c)^6+1/15/sin(d*x+c)^3*cos(d*x+c)^6-1/5/sin
(d*x+c)*cos(d*x+c)^6-1/5*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))

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maxima [A]  time = 0.32, size = 70, normalized size = 1.15 \[ -\frac {30 \, b \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, b \sin \left (d x + c\right )^{3} - 15 \, a \sin \left (d x + c\right )^{2} + 6 \, b \sin \left (d x + c\right ) + 5 \, a}{30 \, d \sin \left (d x + c\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/30*(30*b*sin(d*x + c)^5 + 15*a*sin(d*x + c)^4 - 20*b*sin(d*x + c)^3 - 15*a*sin(d*x + c)^2 + 6*b*sin(d*x + c
) + 5*a)/(d*sin(d*x + c)^6)

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mupad [B]  time = 11.79, size = 69, normalized size = 1.13 \[ -\frac {b\,{\sin \left (c+d\,x\right )}^5+\frac {a\,{\sin \left (c+d\,x\right )}^4}{2}-\frac {2\,b\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {b\,\sin \left (c+d\,x\right )}{5}+\frac {a}{6}}{d\,{\sin \left (c+d\,x\right )}^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^5*(a + b*sin(c + d*x)))/sin(c + d*x)^7,x)

[Out]

-(a/6 + (b*sin(c + d*x))/5 - (a*sin(c + d*x)^2)/2 + (a*sin(c + d*x)^4)/2 - (2*b*sin(c + d*x)^3)/3 + b*sin(c +
d*x)^5)/(d*sin(c + d*x)^6)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**7*(a+b*sin(d*x+c)),x)

[Out]

Timed out

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