3.23 \(\int (a-a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^2(c+d x) \, dx\)

Optimal. Leaf size=42 \[ \frac{2 a A \cos (c+d x)}{d}-\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a A x}{2} \]

[Out]

(3*a*A*x)/2 + (2*a*A*Cos[c + d*x])/d - (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)

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Rubi [A]  time = 0.0687744, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {21, 3788, 2638, 4045, 8} \[ \frac{2 a A \cos (c+d x)}{d}-\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a A x}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*A*x)/2 + (2*a*A*Cos[c + d*x])/d - (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3788

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Dist[(2*a*b)/
d, Int[(d*Csc[e + f*x])^(n + 1), x], x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b
, d, e, f, n}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4045

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0553226, size = 35, normalized size = 0.83 \[ \frac{a A (6 (c+d x)-\sin (2 (c+d x))+8 \cos (c+d x))}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*(6*(c + d*x) + 8*Cos[c + d*x] - Sin[2*(c + d*x)]))/(4*d)

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Maple [A]  time = 0.041, size = 49, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( Aa \left ( -{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,Aa\cos \left ( dx+c \right ) +Aa \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 0.994249, size = 63, normalized size = 1.5 \begin{align*} \frac{{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 4 \,{\left (d x + c\right )} A a + 8 \, A a \cos \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)^2,x, algorithm="maxima")

[Out]

1/4*((2*d*x + 2*c - sin(2*d*x + 2*c))*A*a + 4*(d*x + c)*A*a + 8*A*a*cos(d*x + c))/d

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Fricas [A]  time = 0.472881, size = 97, normalized size = 2.31 \begin{align*} \frac{3 \, A a d x - A a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, A a \cos \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)^2,x, algorithm="fricas")

[Out]

1/2*(3*A*a*d*x - A*a*cos(d*x + c)*sin(d*x + c) + 4*A*a*cos(d*x + c))/d

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Sympy [A]  time = 10.545, size = 105, normalized size = 2.5 \begin{align*} \begin{cases} \frac{A a x \cot ^{2}{\left (c + d x \right )}}{2 \csc ^{2}{\left (c + d x \right )}} + A a x + \frac{A a x}{2 \csc ^{2}{\left (c + d x \right )}} + \frac{2 A a \cot{\left (c + d x \right )}}{d \csc{\left (c + d x \right )}} - \frac{A a \cot{\left (c + d x \right )}}{2 d \csc ^{2}{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x \left (- A \csc{\left (c \right )} + A\right ) \left (- a \csc{\left (c \right )} + a\right )}{\csc ^{2}{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)**2,x)

[Out]

Piecewise((A*a*x*cot(c + d*x)**2/(2*csc(c + d*x)**2) + A*a*x + A*a*x/(2*csc(c + d*x)**2) + 2*A*a*cot(c + d*x)/
(d*csc(c + d*x)) - A*a*cot(c + d*x)/(2*d*csc(c + d*x)**2), Ne(d, 0)), (x*(-A*csc(c) + A)*(-a*csc(c) + a)/csc(c
)**2, True))

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Giac [B]  time = 1.43383, size = 107, normalized size = 2.55 \begin{align*} \frac{3 \,{\left (d x + c\right )} A a + \frac{2 \,{\left (A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4 \, A a\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*csc(d*x+c))*(A-A*csc(d*x+c))/csc(d*x+c)^2,x, algorithm="giac")

[Out]

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