3.538 \(\int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=60 \[ \frac {\sin (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 75} \[ \frac {\sin (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - Log[Sin[c + d*x]]/(a*d) + Sin[c + d*x]/(a*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 75

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 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^3 (a-x)^2 (a+x)}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a^3}{x^3}-\frac {a^2}{x^2}-\frac {a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\sin (c+d x)}{a d}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 45, normalized size = 0.75 \[ -\frac {-2 \sin (c+d x)+\csc ^2(c+d x)-2 \csc (c+d x)+2 \log (\sin (c+d x))+3}{2 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

-1/2*(3 - 2*Csc[c + d*x] + Csc[c + d*x]^2 + 2*Log[Sin[c + d*x]] - 2*Sin[c + d*x])/(a*d)

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fricas [A]  time = 0.66, size = 61, normalized size = 1.02 \[ -\frac {2 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 2 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 1}{2 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 63, normalized size = 1.05 \[ -\frac {\frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{a} - \frac {3 \, \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 1}{a \sin \left (d x + c\right )^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/2*(2*log(abs(sin(d*x + c)))/a - 2*sin(d*x + c)/a - (3*sin(d*x + c)^2 + 2*sin(d*x + c) - 1)/(a*sin(d*x + c)^
2))/d

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maple [A]  time = 0.46, size = 61, normalized size = 1.02 \[ \frac {\sin \left (d x +c \right )}{a d}+\frac {1}{d a \sin \left (d x +c \right )}-\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {1}{2 a d \sin \left (d x +c \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^3/(a+a*sin(d*x+c)),x)

[Out]

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

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maxima [A]  time = 0.91, size = 52, normalized size = 0.87 \[ -\frac {\frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{a} - \frac {2 \, \sin \left (d x + c\right ) - 1}{a \sin \left (d x + c\right )^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*(2*log(sin(d*x + c))/a - 2*sin(d*x + c)/a - (2*sin(d*x + c) - 1)/(a*sin(d*x + c)^2))/d

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mupad [B]  time = 8.93, size = 150, normalized size = 2.50 \[ \frac {10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{2}}{d\,\left (4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d}+\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5/(sin(c + d*x)^3*(a + a*sin(c + d*x))),x)

[Out]

(2*tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2/2 + 10*tan(c/2 + (d*x)/2)^3 - 1/2)/(d*(4*a*tan(c/2 + (d*x)/2)^2 +
4*a*tan(c/2 + (d*x)/2)^4)) - log(tan(c/2 + (d*x)/2))/(a*d) - tan(c/2 + (d*x)/2)^2/(8*a*d) + tan(c/2 + (d*x)/2
)/(2*a*d) + log(tan(c/2 + (d*x)/2)^2 + 1)/(a*d)

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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)**5*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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