3.1.37 \(\int \cot ^3(c+d x) (a+a \sin (c+d x))^4 \, dx\) [37]

Optimal. Leaf size=102 \[ -\frac {4 a^4 \csc (c+d x)}{d}-\frac {a^4 \csc ^2(c+d x)}{2 d}+\frac {5 a^4 \log (\sin (c+d x))}{d}-\frac {5 a^4 \sin ^2(c+d x)}{2 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {a^4 \sin ^4(c+d x)}{4 d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 102, 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 {a^4 \sin ^4(c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {5 a^4 \sin ^2(c+d x)}{2 d}-\frac {a^4 \csc ^2(c+d x)}{2 d}-\frac {4 a^4 \csc (c+d x)}{d}+\frac {5 a^4 \log (\sin (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^4*Csc[c + d*x])/d - (a^4*Csc[c + d*x]^2)/(2*d) + (5*a^4*Log[Sin[c + d*x]])/d - (5*a^4*Sin[c + d*x]^2)/(2
*d) - (4*a^4*Sin[c + d*x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*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 \cot ^3(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x) (a+x)^5}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^6}{x^3}+\frac {4 a^5}{x^2}+\frac {5 a^4}{x}-5 a^2 x-4 a x^2-x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {4 a^4 \csc (c+d x)}{d}-\frac {a^4 \csc ^2(c+d x)}{2 d}+\frac {5 a^4 \log (\sin (c+d x))}{d}-\frac {5 a^4 \sin ^2(c+d x)}{2 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {a^4 \sin ^4(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 78, normalized size = 0.76 \begin {gather*} -\frac {a^4 \left (3+16 \csc (c+d x)+30 \csc ^2(c+d x)+48 \csc ^5(c+d x)+6 \csc ^6(c+d x)+\csc ^4(c+d x) (90-60 \log (\sin (c+d x)))\right ) \sin ^4(c+d x)}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(a^4*(3 + 16*Csc[c + d*x] + 30*Csc[c + d*x]^2 + 48*Csc[c + d*x]^5 + 6*Csc[c + d*x]^6 + Csc[c + d*x]^4*(9
0 - 60*Log[Sin[c + d*x]]))*Sin[c + d*x]^4)/d

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Maple [A]
time = 0.21, size = 129, normalized size = 1.26

method	result	size
derivativedivides	\(\frac {-\frac {a^{4} \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {4 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} \left (\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{4} \left (-\frac {\cos ^{4}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )+a^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(129\)
default	\(\frac {-\frac {a^{4} \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {4 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 a^{4} \left (\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+4 a^{4} \left (-\frac {\cos ^{4}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )+a^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(129\)
risch	\(-5 i a^{4} x -\frac {i a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{6 d}+\frac {11 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {11 a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}+\frac {i a^{4} {\mathrm e}^{-3 i \left (d x +c \right )}}{6 d}-\frac {10 i a^{4} c}{d}-\frac {2 i a^{4} \left (i {\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{3 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {5 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {a^{4} \cos \left (4 d x +4 c \right )}{32 d}\)	\(219\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 82, normalized size = 0.80 \begin {gather*} -\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + \frac {6 \, {\left (8 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{2}}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 30*a^4*sin(d*x + c)^2 - 60*a^4*log(sin(d*x + c)) + 6*(8*
a^4*sin(d*x + c) + a^4)/sin(d*x + c)^2)/d

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Fricas [A]
time = 0.38, size = 131, normalized size = 1.28 \begin {gather*} -\frac {24 \, a^{4} \cos \left (d x + c\right )^{6} - 312 \, a^{4} \cos \left (d x + c\right )^{4} + 423 \, a^{4} \cos \left (d x + c\right )^{2} - 183 \, a^{4} - 480 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 128 \, {\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + 4 \, a^{4}\right )} \sin \left (d x + c\right )}{96 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/96*(24*a^4*cos(d*x + c)^6 - 312*a^4*cos(d*x + c)^4 + 423*a^4*cos(d*x + c)^2 - 183*a^4 - 480*(a^4*cos(d*x +
c)^2 - a^4)*log(1/2*sin(d*x + c)) - 128*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + 4*a^4)*sin(d*x + c))/(d*c
os(d*x + c)^2 - d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{4} \left (\int 4 \sin {\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \cot ^{3}{\left (c + d x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3*(a+a*sin(d*x+c))**4,x)

[Out]

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

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Giac [A]
time = 15.68, size = 96, normalized size = 0.94 \begin {gather*} -\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac {6 \, {\left (15 \, a^{4} \sin \left (d x + c\right )^{2} + 8 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{2}}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 30*a^4*sin(d*x + c)^2 - 60*a^4*log(abs(sin(d*x + c))) +
6*(15*a^4*sin(d*x + c)^2 + 8*a^4*sin(d*x + c) + a^4)/sin(d*x + c)^2)/d

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Mupad [B]
time = 6.40, size = 298, normalized size = 2.92 \begin {gather*} \frac {5\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {81\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+\frac {224\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+98\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {272\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+43\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+32\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d}-\frac {5\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^3*(a + a*sin(c + d*x))^4,x)

[Out]

(5*a^4*log(tan(c/2 + (d*x)/2)))/d - (a^4*tan(c/2 + (d*x)/2)^2)/(8*d) - (2*a^4*tan(c/2 + (d*x)/2)^2 + 32*a^4*ta
n(c/2 + (d*x)/2)^3 + 43*a^4*tan(c/2 + (d*x)/2)^4 + (272*a^4*tan(c/2 + (d*x)/2)^5)/3 + 98*a^4*tan(c/2 + (d*x)/2
)^6 + (224*a^4*tan(c/2 + (d*x)/2)^7)/3 + (81*a^4*tan(c/2 + (d*x)/2)^8)/2 + 8*a^4*tan(c/2 + (d*x)/2)^9 + a^4/2
+ 8*a^4*tan(c/2 + (d*x)/2))/(d*(4*tan(c/2 + (d*x)/2)^2 + 16*tan(c/2 + (d*x)/2)^4 + 24*tan(c/2 + (d*x)/2)^6 + 1
6*tan(c/2 + (d*x)/2)^8 + 4*tan(c/2 + (d*x)/2)^10)) - (2*a^4*tan(c/2 + (d*x)/2))/d - (5*a^4*log(tan(c/2 + (d*x)
/2)^2 + 1))/d

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