3.152 \(\int \cot (c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=46 \[ \frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.038808, antiderivative size = 46, normalized size of antiderivative = 1., 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 = {2721, 43} \[ \frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2721

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)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

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])

Rubi steps

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

Mathematica [A]  time = 0.0234396, size = 46, normalized size = 1. \[ \frac{a^2 \log (\sin (c+d x))}{d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 45, normalized size = 1. \begin{align*}{\frac{{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{ab\sin \left ( dx+c \right ) }{d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}{b}^{2}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)*(a+b*sin(d*x+c))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.89331, size = 54, normalized size = 1.17 \begin{align*} \frac{b^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + 4 \, a b \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/2*(b^2*sin(d*x + c)^2 + 2*a^2*log(sin(d*x + c)) + 4*a*b*sin(d*x + c))/d

________________________________________________________________________________________

Fricas [A]  time = 1.57997, size = 108, normalized size = 2.35 \begin{align*} -\frac{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, a b \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (c + d x \right )}\right )^{2} \cot{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+b*sin(d*x+c))**2,x)

[Out]

Integral((a + b*sin(c + d*x))**2*cot(c + d*x), x)

________________________________________________________________________________________

Giac [A]  time = 2.32155, size = 55, normalized size = 1.2 \begin{align*} \frac{b^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 4 \, a b \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(b^2*sin(d*x + c)^2 + 2*a^2*log(abs(sin(d*x + c))) + 4*a*b*sin(d*x + c))/d