3.3.13 \(\int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\) [213]

Optimal. Leaf size=30 \[ -\frac {\csc ^4(c+d x) (a+a \sin (c+d x))^4}{4 a d} \]

[Out]

-1/4*csc(d*x+c)^4*(a+a*sin(d*x+c))^4/a/d

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Rubi [A]
time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 37} \begin {gather*} -\frac {\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/4*(Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4)/(a*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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 2912

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/b)*x)^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 ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\text {Subst}\left (\int \frac {a^5 (a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^4 \text {Subst}\left (\int \frac {(a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^4(c+d x) (a+a \sin (c+d x))^4}{4 a d}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 20, normalized size = 0.67 \begin {gather*} -\frac {a^3 (1+\csc (c+d x))^4}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

-1/4*(a^3*(1 + Csc[c + d*x])^4)/d

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Maple [A]
time = 0.13, size = 49, normalized size = 1.63

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{\sin \left (d x +c \right )}-\frac {3}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{\sin \left (d x +c \right )^{3}}\right )}{d}\)	\(49\)
default	\(\frac {a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{\sin \left (d x +c \right )}-\frac {3}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{\sin \left (d x +c \right )^{3}}\right )}{d}\)	\(49\)
risch	\(-\frac {2 i a^{3} \left (3 i {\mathrm e}^{6 i \left (d x +c \right )}+{\mathrm e}^{7 i \left (d x +c \right )}-8 i {\mathrm e}^{4 i \left (d x +c \right )}-7 \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i {\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}\)	\(102\)
norman	\(\frac {-\frac {a^{3}}{64 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {31 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {5 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {31 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {11 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {31 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {5 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {31 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {37 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {37 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\)	\(263\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 54, normalized size = 1.80 \begin {gather*} -\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*(4*a^3*sin(d*x + c)^3 + 6*a^3*sin(d*x + c)^2 + 4*a^3*sin(d*x + c) + a^3)/(d*sin(d*x + c)^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (28) = 56\).
time = 0.33, size = 72, normalized size = 2.40 \begin {gather*} \frac {6 \, a^{3} \cos \left (d x + c\right )^{2} - 7 \, a^{3} + 4 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/4*(6*a^3*cos(d*x + c)^2 - 7*a^3 + 4*(a^3*cos(d*x + c)^2 - 2*a^3)*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d
*x + c)^2 + d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)**5*(a+a*sin(d*x+c))**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3003 deep

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Giac [A]
time = 0.43, size = 54, normalized size = 1.80 \begin {gather*} -\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/4*(4*a^3*sin(d*x + c)^3 + 6*a^3*sin(d*x + c)^2 + 4*a^3*sin(d*x + c) + a^3)/(d*sin(d*x + c)^4)

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Mupad [B]
time = 8.61, size = 54, normalized size = 1.80 \begin {gather*} -\frac {4\,a^3\,{\sin \left (c+d\,x\right )}^3+6\,a^3\,{\sin \left (c+d\,x\right )}^2+4\,a^3\,\sin \left (c+d\,x\right )+a^3}{4\,d\,{\sin \left (c+d\,x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)*(a + a*sin(c + d*x))^3)/sin(c + d*x)^5,x)

[Out]

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

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