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

Optimal. Leaf size=78 \[ \frac {a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {6 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc (c+d x)}{d}+\frac {4 a^4 \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ \frac {a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {6 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc (c+d x)}{d}+\frac {4 a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^4*Csc[c + d*x])/d) + (4*a^4*Log[Sin[c + d*x]])/d + (6*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (a
^4*Sin[c + d*x]^3)/(3*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 78, normalized size = 1.00 \[ \frac {a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {6 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc (c+d x)}{d}+\frac {4 a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^4*Csc[c + d*x])/d) + (4*a^4*Log[Sin[c + d*x]])/d + (6*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (a
^4*Sin[c + d*x]^3)/(3*d)

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fricas [A]  time = 0.50, size = 91, normalized size = 1.17 \[ \frac {a^{4} \cos \left (d x + c\right )^{4} - 20 \, a^{4} \cos \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 16 \, a^{4} - 3 \, {\left (2 \, a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \sin \left (d x + c\right )}{3 \, d \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/3*(a^4*cos(d*x + c)^4 - 20*a^4*cos(d*x + c)^2 + 12*a^4*log(1/2*sin(d*x + c))*sin(d*x + c) + 16*a^4 - 3*(2*a^
4*cos(d*x + c)^2 - a^4)*sin(d*x + c))/(d*sin(d*x + c))

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giac [A]  time = 0.20, size = 68, normalized size = 0.87 \[ \frac {a^{4} \sin \left (d x + c\right )^{3} + 6 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 18 \, a^{4} \sin \left (d x + c\right ) - \frac {3 \, a^{4}}{\sin \left (d x + c\right )}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/3*(a^4*sin(d*x + c)^3 + 6*a^4*sin(d*x + c)^2 + 12*a^4*log(abs(sin(d*x + c))) + 18*a^4*sin(d*x + c) - 3*a^4/s
in(d*x + c))/d

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maple [A]  time = 0.15, size = 79, normalized size = 1.01 \[ \frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}+\frac {2 a^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {6 a^{4} \sin \left (d x +c \right )}{d}-\frac {a^{4}}{d \sin \left (d x +c \right )}+\frac {4 a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^4,x)

[Out]

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

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maxima [A]  time = 0.40, size = 67, normalized size = 0.86 \[ \frac {a^{4} \sin \left (d x + c\right )^{3} + 6 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 18 \, a^{4} \sin \left (d x + c\right ) - \frac {3 \, a^{4}}{\sin \left (d x + c\right )}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/3*(a^4*sin(d*x + c)^3 + 6*a^4*sin(d*x + c)^2 + 12*a^4*log(sin(d*x + c)) + 18*a^4*sin(d*x + c) - 3*a^4/sin(d*
x + c))/d

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mupad [B]  time = 8.65, size = 235, normalized size = 3.01 \[ \frac {8\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}-\frac {8\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {4\,a^4\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {4\,a^4\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {28\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {16\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {8\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {23\,a^4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]

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)^2,x)

[Out]

(8*a^4*cos(c/2 + (d*x)/2)^2)/d - (8*a^4*cos(c/2 + (d*x)/2)^4)/d - (4*a^4*log(1/cos(c/2 + (d*x)/2)^2))/d + (4*a
^4*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d - (28*a^4*cos(c/2 + (d*x)/2)^3)/(3*d*sin(c/2 + (d*x)/2)) - (1
6*a^4*cos(c/2 + (d*x)/2)^5)/(3*d*sin(c/2 + (d*x)/2)) + (8*a^4*cos(c/2 + (d*x)/2)^7)/(3*d*sin(c/2 + (d*x)/2)) +
(23*a^4*cos(c/2 + (d*x)/2))/(2*d*sin(c/2 + (d*x)/2)) - (a^4*sin(c/2 + (d*x)/2))/(2*d*cos(c/2 + (d*x)/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*csc(d*x+c)**2*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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