3.221 \(\int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=81 \[ \frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {3 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \log (\sin (c+d x))}{d} \]

[Out]

a^4*ln(sin(d*x+c))/d+4*a^4*sin(d*x+c)/d+3*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 = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2707, 43} \[ \frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {3 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (3*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) +
(a^4*Sin[c + d*x]^4)/(4*d)

Rule 43

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 2707

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 (c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^3+\frac {a^4}{x}+6 a^2 x+4 a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^4 \log (\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {3 a^4 \sin ^2(c+d x)}{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 = 0.04, size = 81, normalized size = 1.00 \[ \frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {3 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (3*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) +
(a^4*Sin[c + d*x]^4)/(4*d)

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fricas [A]  time = 0.53, size = 72, normalized size = 0.89 \[ \frac {3 \, a^{4} \cos \left (d x + c\right )^{4} - 42 \, a^{4} \cos \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 16 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 4 \, a^{4}\right )} \sin \left (d x + c\right )}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*a^4*cos(d*x + c)^4 - 42*a^4*cos(d*x + c)^2 + 12*a^4*log(1/2*sin(d*x + c)) - 16*(a^4*cos(d*x + c)^2 - 4
*a^4)*sin(d*x + c))/d

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giac [A]  time = 0.20, size = 69, normalized size = 0.85 \[ \frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 36 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 48 \, a^{4} \sin \left (d x + c\right )}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(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 + 36*a^4*sin(d*x + c)^2 + 12*a^4*log(abs(sin(d*x + c))) + 4
8*a^4*sin(d*x + c))/d

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maple [A]  time = 0.18, size = 78, normalized size = 0.96 \[ \frac {a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {4 a^{4} \sin \left (d x +c \right )}{d}+\frac {3 a^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {4 a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)*(a+a*sin(d*x+c))^4,x)

[Out]

a^4*ln(sin(d*x+c))/d+4*a^4*sin(d*x+c)/d+3*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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maxima [A]  time = 0.43, size = 68, normalized size = 0.84 \[ \frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 36 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 48 \, a^{4} \sin \left (d x + c\right )}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(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 + 36*a^4*sin(d*x + c)^2 + 12*a^4*log(sin(d*x + c)) + 48*a^4
*sin(d*x + c))/d

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mupad [B]  time = 8.62, size = 118, normalized size = 1.46 \[ \frac {16\,a^4\,\sin \left (c+d\,x\right )}{3\,d}-\frac {a^4\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a^4\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {7\,a^4\,{\cos \left (c+d\,x\right )}^2}{2\,d}+\frac {a^4\,{\cos \left (c+d\,x\right )}^4}{4\,d}-\frac {4\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)*(a + a*sin(c + d*x))^4)/sin(c + d*x),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{4} \left (\int \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int 4 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(a+a*sin(d*x+c))**4,x)

[Out]

a**4*(Integral(cos(c + d*x)*csc(c + d*x), x) + Integral(4*sin(c + d*x)*cos(c + d*x)*csc(c + d*x), x) + Integra
l(6*sin(c + d*x)**2*cos(c + d*x)*csc(c + d*x), x) + Integral(4*sin(c + d*x)**3*cos(c + d*x)*csc(c + d*x), x) +
Integral(sin(c + d*x)**4*cos(c + d*x)*csc(c + d*x), x))

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