3.1065 \(\int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=96 \[ \frac{\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}+\frac{a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+b^2 (-x) \]

[Out]

-(b^2*x) + (a*b*ArcTanh[Cos[c + d*x]])/d + ((a^2 - 2*b^2)*Cot[c + d*x])/(3*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]
)/(3*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.389709, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2889, 3048, 3031, 3021, 2735, 3770} \[ \frac{\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}+\frac{a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+b^2 (-x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*x) + (a*b*ArcTanh[Cos[c + d*x]])/d + ((a^2 - 2*b^2)*Cot[c + d*x])/(3*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]
)/(3*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(3*d)

Rule 2889

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

Rule 3048

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m
- 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2*(
m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] &
& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3031

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 3770

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

Rubi steps

\begin{align*} \int \cot ^2(c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \csc ^4(c+d x) (a+b \sin (c+d x))^2 \left (1-\sin ^2(c+d x)\right ) \, dx\\ &=-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}+\frac{1}{3} \int \csc ^3(c+d x) (a+b \sin (c+d x)) \left (2 b-a \sin (c+d x)-3 b \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-\frac{1}{6} \int \csc ^2(c+d x) \left (2 \left (a^2-2 b^2\right )+6 a b \sin (c+d x)+6 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac{\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac{a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-\frac{1}{6} \int \csc (c+d x) \left (6 a b+6 b^2 \sin (c+d x)\right ) \, dx\\ &=-b^2 x+\frac{\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac{a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}-(a b) \int \csc (c+d x) \, dx\\ &=-b^2 x+\frac{a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac{\left (a^2-2 b^2\right ) \cot (c+d x)}{3 d}-\frac{a b \cot (c+d x) \csc (c+d x)}{3 d}-\frac{\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^2}{3 d}\\ \end{align*}

Mathematica [B]  time = 6.16953, size = 538, normalized size = 5.6 \[ \frac{\sin ^2(c+d x) \csc \left (\frac{1}{2} (c+d x)\right ) \left (a^2 \cos \left (\frac{1}{2} (c+d x)\right )-3 b^2 \cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{6 d (a+b \sin (c+d x))^2}+\frac{\sin ^2(c+d x) \sec \left (\frac{1}{2} (c+d x)\right ) \left (3 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-a^2 \sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{6 d (a+b \sin (c+d x))^2}-\frac{a^2 \sin ^2(c+d x) \cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{24 d (a+b \sin (c+d x))^2}+\frac{a^2 \sin ^2(c+d x) \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{24 d (a+b \sin (c+d x))^2}-\frac{b^2 (c+d x) \sin ^2(c+d x) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2}-\frac{a b \sin ^2(c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{4 d (a+b \sin (c+d x))^2}-\frac{a b \sin ^2(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2}+\frac{a b \sin ^2(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{4 d (a+b \sin (c+d x))^2}+\frac{a b \sin ^2(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*(c + d*x)*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(d*(a + b*Sin[c + d*x])^2)) + ((a^2*Cos[(c + d*x)/2] -
3*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2]*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(6*d*(a + b*Sin[c + d*x])^2)
- (a*b*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(4*d*(a + b*Sin[c + d*x])^2) - (a^2*Cot[(c +
d*x)/2]*Csc[(c + d*x)/2]^2*(b + a*Csc[c + d*x])^2*Sin[c + d*x]^2)/(24*d*(a + b*Sin[c + d*x])^2) + (a*b*(b + a*
Csc[c + d*x])^2*Log[Cos[(c + d*x)/2]]*Sin[c + d*x]^2)/(d*(a + b*Sin[c + d*x])^2) - (a*b*(b + a*Csc[c + d*x])^2
*Log[Sin[(c + d*x)/2]]*Sin[c + d*x]^2)/(d*(a + b*Sin[c + d*x])^2) + (a*b*(b + a*Csc[c + d*x])^2*Sec[(c + d*x)/
2]^2*Sin[c + d*x]^2)/(4*d*(a + b*Sin[c + d*x])^2) + ((b + a*Csc[c + d*x])^2*Sec[(c + d*x)/2]*(-(a^2*Sin[(c + d
*x)/2]) + 3*b^2*Sin[(c + d*x)/2])*Sin[c + d*x]^2)/(6*d*(a + b*Sin[c + d*x])^2) + (a^2*(b + a*Csc[c + d*x])^2*S
ec[(c + d*x)/2]^2*Sin[c + d*x]^2*Tan[(c + d*x)/2])/(24*d*(a + b*Sin[c + d*x])^2)

________________________________________________________________________________________

Maple [A]  time = 0.077, size = 114, normalized size = 1.2 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{ab\cos \left ( dx+c \right ) }{d}}-{\frac{ab\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-{b}^{2}x-{\frac{{b}^{2}\cot \left ( dx+c \right ) }{d}}-{\frac{{b}^{2}c}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x)

[Out]

-1/3/d*a^2/sin(d*x+c)^3*cos(d*x+c)^3-1/d*a*b/sin(d*x+c)^2*cos(d*x+c)^3-a*b*cos(d*x+c)/d-1/d*a*b*ln(csc(d*x+c)-
cot(d*x+c))-b^2*x-b^2*cot(d*x+c)/d-1/d*b^2*c

________________________________________________________________________________________

Maxima [A]  time = 1.56373, size = 111, normalized size = 1.16 \begin{align*} -\frac{6 \,{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} b^{2} - 3 \, a b{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{2 \, a^{2}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/6*(6*(d*x + c + 1/tan(d*x + c))*b^2 - 3*a*b*(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) + log(cos(d*x + c) + 1) -
log(cos(d*x + c) - 1)) + 2*a^2/tan(d*x + c)^3)/d

________________________________________________________________________________________

Fricas [A]  time = 1.51407, size = 423, normalized size = 4.41 \begin{align*} \frac{2 \,{\left (a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, b^{2} \cos \left (d x + c\right ) + 3 \,{\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 3 \,{\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 6 \,{\left (b^{2} d x \cos \left (d x + c\right )^{2} - b^{2} d x - a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/6*(2*(a^2 - 3*b^2)*cos(d*x + c)^3 + 6*b^2*cos(d*x + c) + 3*(a*b*cos(d*x + c)^2 - a*b)*log(1/2*cos(d*x + c) +
1/2)*sin(d*x + c) - 3*(a*b*cos(d*x + c)^2 - a*b)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 6*(b^2*d*x*cos(d
*x + c)^2 - b^2*d*x - a*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))

________________________________________________________________________________________

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)**2*csc(d*x+c)**4*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.26827, size = 225, normalized size = 2.34 \begin{align*} \frac{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \,{\left (d x + c\right )} b^{2} - 24 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{44 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 6 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/24*(a^2*tan(1/2*d*x + 1/2*c)^3 + 6*a*b*tan(1/2*d*x + 1/2*c)^2 - 24*(d*x + c)*b^2 - 24*a*b*log(abs(tan(1/2*d*
x + 1/2*c))) - 3*a^2*tan(1/2*d*x + 1/2*c) + 12*b^2*tan(1/2*d*x + 1/2*c) + (44*a*b*tan(1/2*d*x + 1/2*c)^3 + 3*a
^2*tan(1/2*d*x + 1/2*c)^2 - 12*b^2*tan(1/2*d*x + 1/2*c)^2 - 6*a*b*tan(1/2*d*x + 1/2*c) - a^2)/tan(1/2*d*x + 1/
2*c)^3)/d