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

Optimal. Leaf size=24 \[ \frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2707, 43} \[ \frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.03, size = 26, normalized size = 1.08 \[ \frac {a (\sin (c+d x)+\log (\tan (c+d x))+\log (\cos (c+d x)))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 24, normalized size = 1.00 \[ \frac {a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + a \sin \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 23, normalized size = 0.96 \[ \frac {a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + a \sin \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(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 = 0.09, size = 25, normalized size = 1.04 \[ \frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {a \sin \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 22, normalized size = 0.92 \[ \frac {a \log \left (\sin \left (d x + c\right )\right ) + a \sin \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(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 = 6.58, size = 38, normalized size = 1.58 \[ \frac {a\,\left (\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )+\sin \left (c+d\,x\right )-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(Integral(sin(c + d*x)*cot(c + d*x), x) + Integral(cot(c + d*x), x))

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