3.1.78 \(\int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [78]

Optimal. Leaf size=73 \[ \frac {\csc ^3(c+d x)}{3 a^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {3 \csc ^5(c+d x)}{5 a^3 d}-\frac {\csc ^6(c+d x)}{6 a^3 d} \]

[Out]

1/3*csc(d*x+c)^3/a^3/d-3/4*csc(d*x+c)^4/a^3/d+3/5*csc(d*x+c)^5/a^3/d-1/6*csc(d*x+c)^6/a^3/d

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Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2786, 45} \begin {gather*} -\frac {\csc ^6(c+d x)}{6 a^3 d}+\frac {3 \csc ^5(c+d x)}{5 a^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {\csc ^3(c+d x)}{3 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^3,x]

[Out]

Csc[c + d*x]^3/(3*a^3*d) - (3*Csc[c + d*x]^4)/(4*a^3*d) + (3*Csc[c + d*x]^5)/(5*a^3*d) - Csc[c + d*x]^6/(6*a^3
*d)

Rule 45

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 2786

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

\begin {align*} \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^3}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^3}{x^7}-\frac {3 a^2}{x^6}+\frac {3 a}{x^5}-\frac {1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc ^3(c+d x)}{3 a^3 d}-\frac {3 \csc ^4(c+d x)}{4 a^3 d}+\frac {3 \csc ^5(c+d x)}{5 a^3 d}-\frac {\csc ^6(c+d x)}{6 a^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 48, normalized size = 0.66 \begin {gather*} \frac {\csc ^3(c+d x) \left (20-45 \csc (c+d x)+36 \csc ^2(c+d x)-10 \csc ^3(c+d x)\right )}{60 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^3,x]

[Out]

(Csc[c + d*x]^3*(20 - 45*Csc[c + d*x] + 36*Csc[c + d*x]^2 - 10*Csc[c + d*x]^3))/(60*a^3*d)

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Maple [A]
time = 0.30, size = 49, normalized size = 0.67

method	result	size
derivativedivides	\(\frac {\frac {3}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}}{d \,a^{3}}\)	\(49\)
default	\(\frac {\frac {3}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {3}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}}{d \,a^{3}}\)	\(49\)
risch	\(-\frac {4 i \left (-45 i {\mathrm e}^{8 i \left (d x +c \right )}+10 \,{\mathrm e}^{9 i \left (d x +c \right )}+130 i {\mathrm e}^{6 i \left (d x +c \right )}-102 \,{\mathrm e}^{7 i \left (d x +c \right )}-45 i {\mathrm e}^{4 i \left (d x +c \right )}+102 \,{\mathrm e}^{5 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{15 d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d/a^3*(3/5/sin(d*x+c)^5+1/3/sin(d*x+c)^3-3/4/sin(d*x+c)^4-1/6/sin(d*x+c)^6)

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Maxima [A]
time = 0.29, size = 46, normalized size = 0.63 \begin {gather*} \frac {20 \, \sin \left (d x + c\right )^{3} - 45 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right ) - 10}{60 \, a^{3} d \sin \left (d x + c\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/60*(20*sin(d*x + c)^3 - 45*sin(d*x + c)^2 + 36*sin(d*x + c) - 10)/(a^3*d*sin(d*x + c)^6)

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Fricas [A]
time = 0.34, size = 84, normalized size = 1.15 \begin {gather*} -\frac {45 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 14\right )} \sin \left (d x + c\right ) - 55}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/60*(45*cos(d*x + c)^2 - 4*(5*cos(d*x + c)^2 - 14)*sin(d*x + c) - 55)/(a^3*d*cos(d*x + c)^6 - 3*a^3*d*cos(d*
x + c)^4 + 3*a^3*d*cos(d*x + c)^2 - a^3*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\cot ^{7}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**7/(a+a*sin(d*x+c))**3,x)

[Out]

Integral(cot(c + d*x)**7/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3

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Giac [A]
time = 7.55, size = 46, normalized size = 0.63 \begin {gather*} \frac {20 \, \sin \left (d x + c\right )^{3} - 45 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right ) - 10}{60 \, a^{3} d \sin \left (d x + c\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/60*(20*sin(d*x + c)^3 - 45*sin(d*x + c)^2 + 36*sin(d*x + c) - 10)/(a^3*d*sin(d*x + c)^6)

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Mupad [B]
time = 6.69, size = 46, normalized size = 0.63 \begin {gather*} \frac {20\,{\sin \left (c+d\,x\right )}^3-45\,{\sin \left (c+d\,x\right )}^2+36\,\sin \left (c+d\,x\right )-10}{60\,a^3\,d\,{\sin \left (c+d\,x\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^7/(a + a*sin(c + d*x))^3,x)

[Out]

(36*sin(c + d*x) - 45*sin(c + d*x)^2 + 20*sin(c + d*x)^3 - 10)/(60*a^3*d*sin(c + d*x)^6)

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