3.1.86 \(\int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [86]

Optimal. Leaf size=135 \[ \frac {8 \csc (c+d x)}{a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {8 \csc ^3(c+d x)}{3 a^4 d}-\frac {7 \csc ^4(c+d x)}{4 a^4 d}+\frac {4 \csc ^5(c+d x)}{5 a^4 d}-\frac {\csc ^6(c+d x)}{6 a^4 d}+\frac {8 \log (\sin (c+d x))}{a^4 d}-\frac {8 \log (1+\sin (c+d x))}{a^4 d} \]

[Out]

8*csc(d*x+c)/a^4/d-4*csc(d*x+c)^2/a^4/d+8/3*csc(d*x+c)^3/a^4/d-7/4*csc(d*x+c)^4/a^4/d+4/5*csc(d*x+c)^5/a^4/d-1
/6*csc(d*x+c)^6/a^4/d+8*ln(sin(d*x+c))/a^4/d-8*ln(1+sin(d*x+c))/a^4/d

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Rubi [A]
time = 0.06, antiderivative size = 135, 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, 90} \begin {gather*} -\frac {\csc ^6(c+d x)}{6 a^4 d}+\frac {4 \csc ^5(c+d x)}{5 a^4 d}-\frac {7 \csc ^4(c+d x)}{4 a^4 d}+\frac {8 \csc ^3(c+d x)}{3 a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {8 \csc (c+d x)}{a^4 d}+\frac {8 \log (\sin (c+d x))}{a^4 d}-\frac {8 \log (\sin (c+d x)+1)}{a^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4,x]

[Out]

(8*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (8*Csc[c + d*x]^3)/(3*a^4*d) - (7*Csc[c + d*x]^4)/(4*a
^4*d) + (4*Csc[c + d*x]^5)/(5*a^4*d) - Csc[c + d*x]^6/(6*a^4*d) + (8*Log[Sin[c + d*x]])/(a^4*d) - (8*Log[1 + S
in[c + d*x]])/(a^4*d)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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))^4} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^3}{x^7 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2}{x^7}-\frac {4 a}{x^6}+\frac {7}{x^5}-\frac {8}{a x^4}+\frac {8}{a^2 x^3}-\frac {8}{a^3 x^2}+\frac {8}{a^4 x}-\frac {8}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {8 \csc (c+d x)}{a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {8 \csc ^3(c+d x)}{3 a^4 d}-\frac {7 \csc ^4(c+d x)}{4 a^4 d}+\frac {4 \csc ^5(c+d x)}{5 a^4 d}-\frac {\csc ^6(c+d x)}{6 a^4 d}+\frac {8 \log (\sin (c+d x))}{a^4 d}-\frac {8 \log (1+\sin (c+d x))}{a^4 d}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 89, normalized size = 0.66 \begin {gather*} \frac {480 \csc (c+d x)-240 \csc ^2(c+d x)+160 \csc ^3(c+d x)-105 \csc ^4(c+d x)+48 \csc ^5(c+d x)-10 \csc ^6(c+d x)+480 \log (\sin (c+d x))-480 \log (1+\sin (c+d x))}{60 a^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4,x]

[Out]

(480*Csc[c + d*x] - 240*Csc[c + d*x]^2 + 160*Csc[c + d*x]^3 - 105*Csc[c + d*x]^4 + 48*Csc[c + d*x]^5 - 10*Csc[
c + d*x]^6 + 480*Log[Sin[c + d*x]] - 480*Log[1 + Sin[c + d*x]])/(60*a^4*d)

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Maple [A]
time = 0.38, size = 89, normalized size = 0.66

method	result	size
derivativedivides	\(\frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {4}{5 \sin \left (d x +c \right )^{5}}-\frac {7}{4 \sin \left (d x +c \right )^{4}}+\frac {8}{3 \sin \left (d x +c \right )^{3}}-\frac {4}{\sin \left (d x +c \right )^{2}}+\frac {8}{\sin \left (d x +c \right )}+8 \ln \left (\sin \left (d x +c \right )\right )-8 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{4}}\)	\(89\)
default	\(\frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {4}{5 \sin \left (d x +c \right )^{5}}-\frac {7}{4 \sin \left (d x +c \right )^{4}}+\frac {8}{3 \sin \left (d x +c \right )^{3}}-\frac {4}{\sin \left (d x +c \right )^{2}}+\frac {8}{\sin \left (d x +c \right )}+8 \ln \left (\sin \left (d x +c \right )\right )-8 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{4}}\)	\(89\)
risch	\(\frac {4 i \left (-60 i {\mathrm e}^{10 i \left (d x +c \right )}+60 \,{\mathrm e}^{11 i \left (d x +c \right )}+345 i {\mathrm e}^{8 i \left (d x +c \right )}-380 \,{\mathrm e}^{9 i \left (d x +c \right )}-610 i {\mathrm e}^{6 i \left (d x +c \right )}+936 \,{\mathrm e}^{7 i \left (d x +c \right )}+345 i {\mathrm e}^{4 i \left (d x +c \right )}-936 \,{\mathrm e}^{5 i \left (d x +c \right )}-60 i {\mathrm e}^{2 i \left (d x +c \right )}+380 \,{\mathrm e}^{3 i \left (d x +c \right )}-60 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {16 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{4}}+\frac {8 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{4}}\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d/a^4*(-1/6/sin(d*x+c)^6+4/5/sin(d*x+c)^5-7/4/sin(d*x+c)^4+8/3/sin(d*x+c)^3-4/sin(d*x+c)^2+8/sin(d*x+c)+8*ln
(sin(d*x+c))-8*ln(1+sin(d*x+c)))

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Maxima [A]
time = 0.28, size = 95, normalized size = 0.70 \begin {gather*} -\frac {\frac {480 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} - \frac {480 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}} - \frac {480 \, \sin \left (d x + c\right )^{5} - 240 \, \sin \left (d x + c\right )^{4} + 160 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 48 \, \sin \left (d x + c\right ) - 10}{a^{4} \sin \left (d x + c\right )^{6}}}{60 \, d} \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))^4,x, algorithm="maxima")

[Out]

-1/60*(480*log(sin(d*x + c) + 1)/a^4 - 480*log(sin(d*x + c))/a^4 - (480*sin(d*x + c)^5 - 240*sin(d*x + c)^4 +
160*sin(d*x + c)^3 - 105*sin(d*x + c)^2 + 48*sin(d*x + c) - 10)/(a^4*sin(d*x + c)^6))/d

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Fricas [A]
time = 0.36, size = 186, normalized size = 1.38 \begin {gather*} \frac {240 \, \cos \left (d x + c\right )^{4} - 585 \, \cos \left (d x + c\right )^{2} + 480 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 480 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 16 \, {\left (30 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{2} + 43\right )} \sin \left (d x + c\right ) + 355}{60 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} 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))^4,x, algorithm="fricas")

[Out]

1/60*(240*cos(d*x + c)^4 - 585*cos(d*x + c)^2 + 480*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)
*log(1/2*sin(d*x + c)) - 480*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(sin(d*x + c) + 1)
- 16*(30*cos(d*x + c)^4 - 70*cos(d*x + c)^2 + 43)*sin(d*x + c) + 355)/(a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x
+ c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*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 ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \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))**4,x)

[Out]

Integral(cot(c + d*x)**7/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c + d*x) + 1), x)/a*
*4

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Giac [A]
time = 9.92, size = 232, normalized size = 1.72 \begin {gather*} -\frac {\frac {30720 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {15360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac {37632 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2835 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} + \frac {5 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 240 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 880 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2835 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10080 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{24}}}{1920 \, d} \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))^4,x, algorithm="giac")

[Out]

-1/1920*(30720*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 15360*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 + (37632*tan(
1/2*d*x + 1/2*c)^6 - 10080*tan(1/2*d*x + 1/2*c)^5 + 2835*tan(1/2*d*x + 1/2*c)^4 - 880*tan(1/2*d*x + 1/2*c)^3 +
240*tan(1/2*d*x + 1/2*c)^2 - 48*tan(1/2*d*x + 1/2*c) + 5)/(a^4*tan(1/2*d*x + 1/2*c)^6) + (5*a^20*tan(1/2*d*x
+ 1/2*c)^6 - 48*a^20*tan(1/2*d*x + 1/2*c)^5 + 240*a^20*tan(1/2*d*x + 1/2*c)^4 - 880*a^20*tan(1/2*d*x + 1/2*c)^
3 + 2835*a^20*tan(1/2*d*x + 1/2*c)^2 - 10080*a^20*tan(1/2*d*x + 1/2*c))/a^24)/d

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Mupad [B]
time = 6.81, size = 235, normalized size = 1.74 \begin {gather*} \frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^4\,d}-\frac {189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{8\,a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,a^4\,d}+\frac {8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {16\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^4\,d}+\frac {21\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^4\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (336\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {189\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+\frac {88\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}-\frac {1}{6}\right )}{64\,a^4\,d} \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))^4,x)

[Out]

(11*tan(c/2 + (d*x)/2)^3)/(24*a^4*d) - (189*tan(c/2 + (d*x)/2)^2)/(128*a^4*d) - tan(c/2 + (d*x)/2)^4/(8*a^4*d)
+ tan(c/2 + (d*x)/2)^5/(40*a^4*d) - tan(c/2 + (d*x)/2)^6/(384*a^4*d) + (8*log(tan(c/2 + (d*x)/2)))/(a^4*d) -
(16*log(tan(c/2 + (d*x)/2) + 1))/(a^4*d) + (21*tan(c/2 + (d*x)/2))/(4*a^4*d) + (cot(c/2 + (d*x)/2)^6*((8*tan(c
/2 + (d*x)/2))/5 - 8*tan(c/2 + (d*x)/2)^2 + (88*tan(c/2 + (d*x)/2)^3)/3 - (189*tan(c/2 + (d*x)/2)^4)/2 + 336*t
an(c/2 + (d*x)/2)^5 - 1/6))/(64*a^4*d)

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