3.193 \(\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=25 \[ \frac {a \log (\sin (c+d x))}{d}-\frac {a \csc (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2833, 12, 43} \[ \frac {a \log (\sin (c+d x))}{d}-\frac {a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/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)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a+x)}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {a+x}{x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {a}{x^2}+\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 33, normalized size = 1.32 \[ \frac {a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}-\frac {a \csc (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Csc[c + d*x])/d) + (a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d

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fricas [A]  time = 0.47, size = 33, normalized size = 1.32 \[ \frac {a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - a}{d \sin \left (d x + c\right )} \]

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)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.13, size = 26, normalized size = 1.04 \[ \frac {a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {a}{\sin \left (d x + c\right )}}{d} \]

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)),x, algorithm="giac")

[Out]

(a*log(abs(sin(d*x + c))) - a/sin(d*x + c))/d

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maple [A]  time = 0.08, size = 28, normalized size = 1.12 \[ -\frac {a}{d \sin \left (d x +c \right )}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d} \]

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)),x)

[Out]

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

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maxima [A]  time = 0.30, size = 25, normalized size = 1.00 \[ \frac {a \log \left (\sin \left (d x + c\right )\right ) - \frac {a}{\sin \left (d x + c\right )}}{d} \]

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)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 8.58, size = 55, normalized size = 2.20 \[ -\frac {a\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )+2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {1}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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