3.1.68 \(\int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [68]

Optimal. Leaf size=73 \[ \frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{6 a^2 d} \]

[Out]

1/2*csc(d*x+c)^2/a^2/d-2/3*csc(d*x+c)^3/a^2/d+2/5*csc(d*x+c)^5/a^2/d-1/6*csc(d*x+c)^6/a^2/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, 76} \begin {gather*} -\frac {\csc ^6(c+d x)}{6 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {\csc ^2(c+d x)}{2 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^2,x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 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))^2} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^3 (a+x)}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^4}{x^7}-\frac {2 a^3}{x^6}+\frac {2 a}{x^4}-\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{6 a^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 73, normalized size = 1.00 \begin {gather*} \frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{6 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^2,x]

[Out]

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

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Maple [A]
time = 0.32, size = 49, normalized size = 0.67

method	result	size
derivativedivides	\(\frac {\frac {2}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {2}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}}{d \,a^{2}}\)	\(49\)
default	\(\frac {\frac {2}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {2}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}}{d \,a^{2}}\)	\(49\)
risch	\(-\frac {2 \left (15 \,{\mathrm e}^{10 i \left (d x +c \right )}-60 \,{\mathrm e}^{8 i \left (d x +c \right )}-40 i {\mathrm e}^{9 i \left (d x +c \right )}+10 \,{\mathrm e}^{6 i \left (d x +c \right )}+24 i {\mathrm e}^{7 i \left (d x +c \right )}-60 \,{\mathrm e}^{4 i \left (d x +c \right )}-24 i {\mathrm e}^{5 i \left (d x +c \right )}+15 \,{\mathrm e}^{2 i \left (d x +c \right )}+40 i {\mathrm e}^{3 i \left (d x +c \right )}\right )}{15 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}\)	\(126\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 46, normalized size = 0.63 \begin {gather*} \frac {15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 12 \, \sin \left (d x + c\right ) - 5}{30 \, a^{2} 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))^2,x, algorithm="maxima")

[Out]

1/30*(15*sin(d*x + c)^4 - 20*sin(d*x + c)^3 + 12*sin(d*x + c) - 5)/(a^2*d*sin(d*x + c)^6)

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Fricas [A]
time = 0.36, size = 94, normalized size = 1.29 \begin {gather*} -\frac {15 \, \cos \left (d x + c\right )^{4} - 30 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 10}{30 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} 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))^2,x, algorithm="fricas")

[Out]

-1/30*(15*cos(d*x + c)^4 - 30*cos(d*x + c)^2 + 4*(5*cos(d*x + c)^2 - 2)*sin(d*x + c) + 10)/(a^2*d*cos(d*x + c)
^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*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 ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \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))**2,x)

[Out]

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

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Giac [A]
time = 26.12, size = 46, normalized size = 0.63 \begin {gather*} \frac {15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 12 \, \sin \left (d x + c\right ) - 5}{30 \, a^{2} 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))^2,x, algorithm="giac")

[Out]

1/30*(15*sin(d*x + c)^4 - 20*sin(d*x + c)^3 + 12*sin(d*x + c) - 5)/(a^2*d*sin(d*x + c)^6)

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Mupad [B]
time = 6.37, size = 46, normalized size = 0.63 \begin {gather*} \frac {15\,{\sin \left (c+d\,x\right )}^4-20\,{\sin \left (c+d\,x\right )}^3+12\,\sin \left (c+d\,x\right )-5}{30\,a^2\,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))^2,x)

[Out]

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

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