3.200 \(\int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0546732, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{a^2 \sin (c+d x)}{d}-\frac{a^2 \csc (c+d x)}{d}+\frac{2 a^2 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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])

Rubi steps

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

Mathematica [A]  time = 0.0228808, size = 38, normalized size = 0.88 \[ a^2 \left (\frac{\sin (c+d x)}{d}-\frac{\csc (c+d x)}{d}+\frac{2 \log (\sin (c+d x))}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.036, size = 46, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}}{d\sin \left ( dx+c \right ) }}+2\,{\frac{{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

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))^2,x)

[Out]

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

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Maxima [A]  time = 1.18064, size = 54, normalized size = 1.26 \begin{align*} \frac{2 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + a^{2} \sin \left (d x + c\right ) - \frac{a^{2}}{\sin \left (d x + c\right )}}{d} \end{align*}

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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6657, size = 112, normalized size = 2.6 \begin{align*} -\frac{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{d \sin \left (d x + c\right )} \end{align*}

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))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.29898, size = 55, normalized size = 1.28 \begin{align*} \frac{2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + a^{2} \sin \left (d x + c\right ) - \frac{a^{2}}{\sin \left (d x + c\right )}}{d} \end{align*}

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))^2,x, algorithm="giac")

[Out]

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