3.204 \(\int \sin ^2(a+b x) \sin ^3(c+d x) \, dx\)

Optimal. Leaf size=144 \[ \frac {\cos (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}-\frac {3 \cos (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac {3 \cos (2 a+x (2 b+d)+c)}{16 (2 b+d)}-\frac {\cos (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}-\frac {3 \cos (c+d x)}{8 d}+\frac {\cos (3 c+3 d x)}{24 d} \]

[Out]

1/16*cos(2*a-3*c+(2*b-3*d)*x)/(2*b-3*d)-3/16*cos(2*a-c+(2*b-d)*x)/(2*b-d)-3/8*cos(d*x+c)/d+1/24*cos(3*d*x+3*c)
/d+3/16*cos(2*a+c+(2*b+d)*x)/(2*b+d)-1/16*cos(2*a+3*c+(2*b+3*d)*x)/(2*b+3*d)

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Rubi [A]  time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4569, 2638} \[ \frac {\cos (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}-\frac {3 \cos (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac {3 \cos (2 a+x (2 b+d)+c)}{16 (2 b+d)}-\frac {\cos (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}-\frac {3 \cos (c+d x)}{8 d}+\frac {\cos (3 c+3 d x)}{24 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^2*Sin[c + d*x]^3,x]

[Out]

Cos[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) - (3*Cos[2*a - c + (2*b - d)*x])/(16*(2*b - d)) - (3*Cos[c + d
*x])/(8*d) + Cos[3*c + 3*d*x]/(24*d) + (3*Cos[2*a + c + (2*b + d)*x])/(16*(2*b + d)) - Cos[2*a + 3*c + (2*b +
3*d)*x]/(16*(2*b + 3*d))

Rule 2638

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

Rule 4569

Int[Sin[v_]^(p_.)*Sin[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Sin[w]^q, x], x] /; ((PolynomialQ[
v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])) && IGtQ[p, 0] && IGtQ[q
, 0]

Rubi steps

\begin {align*} \int \sin ^2(a+b x) \sin ^3(c+d x) \, dx &=\int \left (-\frac {1}{16} \sin (2 a-3 c+(2 b-3 d) x)+\frac {3}{16} \sin (2 a-c+(2 b-d) x)+\frac {3}{8} \sin (c+d x)-\frac {1}{8} \sin (3 c+3 d x)-\frac {3}{16} \sin (2 a+c+(2 b+d) x)+\frac {1}{16} \sin (2 a+3 c+(2 b+3 d) x)\right ) \, dx\\ &=-\left (\frac {1}{16} \int \sin (2 a-3 c+(2 b-3 d) x) \, dx\right )+\frac {1}{16} \int \sin (2 a+3 c+(2 b+3 d) x) \, dx-\frac {1}{8} \int \sin (3 c+3 d x) \, dx+\frac {3}{16} \int \sin (2 a-c+(2 b-d) x) \, dx-\frac {3}{16} \int \sin (2 a+c+(2 b+d) x) \, dx+\frac {3}{8} \int \sin (c+d x) \, dx\\ &=\frac {\cos (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}-\frac {3 \cos (2 a-c+(2 b-d) x)}{16 (2 b-d)}-\frac {3 \cos (c+d x)}{8 d}+\frac {\cos (3 c+3 d x)}{24 d}+\frac {3 \cos (2 a+c+(2 b+d) x)}{16 (2 b+d)}-\frac {\cos (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)}\\ \end {align*}

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Mathematica [A]  time = 1.57, size = 158, normalized size = 1.10 \[ \frac {1}{48} \left (\frac {3 \cos (2 a+2 b x-3 c-3 d x)}{2 b-3 d}-\frac {9 \cos (2 a+2 b x-c-d x)}{2 b-d}+\frac {9 \cos (2 a+2 b x+c+d x)}{2 b+d}-\frac {3 \cos (2 a+2 b x+3 c+3 d x)}{2 b+3 d}+\frac {18 \sin (c) \sin (d x)}{d}-\frac {2 \sin (3 c) \sin (3 d x)}{d}-\frac {18 \cos (c) \cos (d x)}{d}+\frac {2 \cos (3 c) \cos (3 d x)}{d}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^2*Sin[c + d*x]^3,x]

[Out]

((-18*Cos[c]*Cos[d*x])/d + (2*Cos[3*c]*Cos[3*d*x])/d + (3*Cos[2*a - 3*c + 2*b*x - 3*d*x])/(2*b - 3*d) - (9*Cos
[2*a - c + 2*b*x - d*x])/(2*b - d) + (9*Cos[2*a + c + 2*b*x + d*x])/(2*b + d) - (3*Cos[2*a + 3*c + 2*b*x + 3*d
*x])/(2*b + 3*d) + (18*Sin[c]*Sin[d*x])/d - (2*Sin[3*c]*Sin[3*d*x])/d)/48

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fricas [A]  time = 0.61, size = 192, normalized size = 1.33 \[ \frac {{\left (8 \, b^{4} - 38 \, b^{2} d^{2} + 9 \, d^{4} + 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left ({\left (4 \, b^{3} d - b d^{3}\right )} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{2} - {\left (4 \, b^{3} d - 7 \, b d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 3 \, {\left (8 \, b^{4} - 26 \, b^{2} d^{2} + 9 \, d^{4} + 3 \, {\left (4 \, b^{2} d^{2} - 3 \, d^{4}\right )} \cos \left (b x + a\right )^{2}\right )} \cos \left (d x + c\right )}{3 \, {\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^2*sin(d*x+c)^3,x, algorithm="fricas")

[Out]

1/3*((8*b^4 - 38*b^2*d^2 + 9*d^4 + 9*(4*b^2*d^2 - d^4)*cos(b*x + a)^2)*cos(d*x + c)^3 + 6*((4*b^3*d - b*d^3)*c
os(b*x + a)*cos(d*x + c)^2 - (4*b^3*d - 7*b*d^3)*cos(b*x + a))*sin(b*x + a)*sin(d*x + c) - 3*(8*b^4 - 26*b^2*d
^2 + 9*d^4 + 3*(4*b^2*d^2 - 3*d^4)*cos(b*x + a)^2)*cos(d*x + c))/(16*b^4*d - 40*b^2*d^3 + 9*d^5)

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giac [A]  time = 2.53, size = 129, normalized size = 0.90 \[ -\frac {\cos \left (2 \, b x + 3 \, d x + 2 \, a + 3 \, c\right )}{16 \, {\left (2 \, b + 3 \, d\right )}} + \frac {3 \, \cos \left (2 \, b x + d x + 2 \, a + c\right )}{16 \, {\left (2 \, b + d\right )}} - \frac {3 \, \cos \left (2 \, b x - d x + 2 \, a - c\right )}{16 \, {\left (2 \, b - d\right )}} + \frac {\cos \left (2 \, b x - 3 \, d x + 2 \, a - 3 \, c\right )}{16 \, {\left (2 \, b - 3 \, d\right )}} + \frac {\cos \left (3 \, d x + 3 \, c\right )}{24 \, d} - \frac {3 \, \cos \left (d x + c\right )}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^2*sin(d*x+c)^3,x, algorithm="giac")

[Out]

-1/16*cos(2*b*x + 3*d*x + 2*a + 3*c)/(2*b + 3*d) + 3/16*cos(2*b*x + d*x + 2*a + c)/(2*b + d) - 3/16*cos(2*b*x
- d*x + 2*a - c)/(2*b - d) + 1/16*cos(2*b*x - 3*d*x + 2*a - 3*c)/(2*b - 3*d) + 1/24*cos(3*d*x + 3*c)/d - 3/8*c
os(d*x + c)/d

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maple [A]  time = 0.28, size = 133, normalized size = 0.92 \[ \frac {\cos \left (2 a -3 c +\left (2 b -3 d \right ) x \right )}{32 b -48 d}-\frac {3 \cos \left (2 a -c +\left (2 b -d \right ) x \right )}{16 \left (2 b -d \right )}-\frac {3 \cos \left (d x +c \right )}{8 d}+\frac {\cos \left (3 d x +3 c \right )}{24 d}+\frac {3 \cos \left (2 a +c +\left (2 b +d \right ) x \right )}{16 \left (2 b +d \right )}-\frac {\cos \left (2 a +3 c +\left (2 b +3 d \right ) x \right )}{16 \left (2 b +3 d \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^2*sin(d*x+c)^3,x)

[Out]

1/16*cos(2*a-3*c+(2*b-3*d)*x)/(2*b-3*d)-3/16*cos(2*a-c+(2*b-d)*x)/(2*b-d)-3/8*cos(d*x+c)/d+1/24*cos(3*d*x+3*c)
/d+3/16*cos(2*a+c+(2*b+d)*x)/(2*b+d)-1/16*cos(2*a+3*c+(2*b+3*d)*x)/(2*b+3*d)

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maxima [B]  time = 0.42, size = 1362, normalized size = 9.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^2*sin(d*x+c)^3,x, algorithm="maxima")

[Out]

-1/96*(3*(8*b^3*d*cos(3*c) - 12*b^2*d^2*cos(3*c) - 2*b*d^3*cos(3*c) + 3*d^4*cos(3*c))*cos((2*b + 3*d)*x + 2*a
+ 6*c) + 3*(8*b^3*d*cos(3*c) - 12*b^2*d^2*cos(3*c) - 2*b*d^3*cos(3*c) + 3*d^4*cos(3*c))*cos((2*b + 3*d)*x + 2*
a) - 9*(8*b^3*d*cos(3*c) - 4*b^2*d^2*cos(3*c) - 18*b*d^3*cos(3*c) + 9*d^4*cos(3*c))*cos((2*b + d)*x + 2*a + 4*
c) - 9*(8*b^3*d*cos(3*c) - 4*b^2*d^2*cos(3*c) - 18*b*d^3*cos(3*c) + 9*d^4*cos(3*c))*cos((2*b + d)*x + 2*a - 2*
c) + 9*(8*b^3*d*cos(3*c) + 4*b^2*d^2*cos(3*c) - 18*b*d^3*cos(3*c) - 9*d^4*cos(3*c))*cos(-(2*b - d)*x - 2*a + 4
*c) + 9*(8*b^3*d*cos(3*c) + 4*b^2*d^2*cos(3*c) - 18*b*d^3*cos(3*c) - 9*d^4*cos(3*c))*cos(-(2*b - d)*x - 2*a -
2*c) - 3*(8*b^3*d*cos(3*c) + 12*b^2*d^2*cos(3*c) - 2*b*d^3*cos(3*c) - 3*d^4*cos(3*c))*cos(-(2*b - 3*d)*x - 2*a
+ 6*c) - 3*(8*b^3*d*cos(3*c) + 12*b^2*d^2*cos(3*c) - 2*b*d^3*cos(3*c) - 3*d^4*cos(3*c))*cos(-(2*b - 3*d)*x -
2*a) - 2*(16*b^4*cos(3*c) - 40*b^2*d^2*cos(3*c) + 9*d^4*cos(3*c))*cos(3*d*x) - 2*(16*b^4*cos(3*c) - 40*b^2*d^2
*cos(3*c) + 9*d^4*cos(3*c))*cos(3*d*x + 6*c) + 18*(16*b^4*cos(3*c) - 40*b^2*d^2*cos(3*c) + 9*d^4*cos(3*c))*cos
(d*x + 4*c) + 18*(16*b^4*cos(3*c) - 40*b^2*d^2*cos(3*c) + 9*d^4*cos(3*c))*cos(d*x - 2*c) + 3*(8*b^3*d*sin(3*c)
- 12*b^2*d^2*sin(3*c) - 2*b*d^3*sin(3*c) + 3*d^4*sin(3*c))*sin((2*b + 3*d)*x + 2*a + 6*c) - 3*(8*b^3*d*sin(3*
c) - 12*b^2*d^2*sin(3*c) - 2*b*d^3*sin(3*c) + 3*d^4*sin(3*c))*sin((2*b + 3*d)*x + 2*a) - 9*(8*b^3*d*sin(3*c) -
4*b^2*d^2*sin(3*c) - 18*b*d^3*sin(3*c) + 9*d^4*sin(3*c))*sin((2*b + d)*x + 2*a + 4*c) + 9*(8*b^3*d*sin(3*c) -
4*b^2*d^2*sin(3*c) - 18*b*d^3*sin(3*c) + 9*d^4*sin(3*c))*sin((2*b + d)*x + 2*a - 2*c) + 9*(8*b^3*d*sin(3*c) +
4*b^2*d^2*sin(3*c) - 18*b*d^3*sin(3*c) - 9*d^4*sin(3*c))*sin(-(2*b - d)*x - 2*a + 4*c) - 9*(8*b^3*d*sin(3*c)
+ 4*b^2*d^2*sin(3*c) - 18*b*d^3*sin(3*c) - 9*d^4*sin(3*c))*sin(-(2*b - d)*x - 2*a - 2*c) - 3*(8*b^3*d*sin(3*c)
+ 12*b^2*d^2*sin(3*c) - 2*b*d^3*sin(3*c) - 3*d^4*sin(3*c))*sin(-(2*b - 3*d)*x - 2*a + 6*c) + 3*(8*b^3*d*sin(3
*c) + 12*b^2*d^2*sin(3*c) - 2*b*d^3*sin(3*c) - 3*d^4*sin(3*c))*sin(-(2*b - 3*d)*x - 2*a) + 2*(16*b^4*sin(3*c)
- 40*b^2*d^2*sin(3*c) + 9*d^4*sin(3*c))*sin(3*d*x) - 2*(16*b^4*sin(3*c) - 40*b^2*d^2*sin(3*c) + 9*d^4*sin(3*c)
)*sin(3*d*x + 6*c) + 18*(16*b^4*sin(3*c) - 40*b^2*d^2*sin(3*c) + 9*d^4*sin(3*c))*sin(d*x + 4*c) - 18*(16*b^4*s
in(3*c) - 40*b^2*d^2*sin(3*c) + 9*d^4*sin(3*c))*sin(d*x - 2*c))/(9*(cos(3*c)^2 + sin(3*c)^2)*d^5 - 40*(b^2*cos
(3*c)^2 + b^2*sin(3*c)^2)*d^3 + 16*(b^4*cos(3*c)^2 + b^4*sin(3*c)^2)*d)

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mupad [B]  time = 1.87, size = 469, normalized size = 3.26 \[ {\mathrm {e}}^{a\,2{}\mathrm {i}-c\,3{}\mathrm {i}+b\,x\,2{}\mathrm {i}-d\,x\,3{}\mathrm {i}}\,\left (\frac {3\,d\,\left (2\,b+3\,d\right )}{384\,b^2\,d-864\,d^3}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (8\,b^2-18\,d^2\right )}{384\,b^2\,d-864\,d^3}-\frac {3\,d\,{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (2\,b-3\,d\right )}{384\,b^2\,d-864\,d^3}\right )+{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,3{}\mathrm {i}+b\,x\,2{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (-\frac {3\,d\,\left (2\,b-3\,d\right )}{384\,b^2\,d-864\,d^3}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (8\,b^2-18\,d^2\right )}{384\,b^2\,d-864\,d^3}+\frac {3\,d\,{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (2\,b+3\,d\right )}{384\,b^2\,d-864\,d^3}\right )-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,\left (2\,b+d\right )}{32\,\left (4\,b^2-d^2\right )}-\frac {3\,{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (2\,b-d\right )}{32\,\left (4\,b^2-d^2\right )}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (24\,b^2-6\,d^2\right )}{32\,d\,\left (4\,b^2-d^2\right )}\right )-{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (-\frac {3\,\left (2\,b-d\right )}{32\,\left (4\,b^2-d^2\right )}+\frac {3\,{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (2\,b+d\right )}{32\,\left (4\,b^2-d^2\right )}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (24\,b^2-6\,d^2\right )}{32\,d\,\left (4\,b^2-d^2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^2*sin(c + d*x)^3,x)

[Out]

exp(a*2i - c*3i + b*x*2i - d*x*3i)*((3*d*(2*b + 3*d))/(384*b^2*d - 864*d^3) + (exp(- a*2i - b*x*2i)*(8*b^2 - 1
8*d^2))/(384*b^2*d - 864*d^3) - (3*d*exp(- a*4i - b*x*4i)*(2*b - 3*d))/(384*b^2*d - 864*d^3)) + exp(a*2i + c*3
i + b*x*2i + d*x*3i)*((exp(- a*2i - b*x*2i)*(8*b^2 - 18*d^2))/(384*b^2*d - 864*d^3) - (3*d*(2*b - 3*d))/(384*b
^2*d - 864*d^3) + (3*d*exp(- a*4i - b*x*4i)*(2*b + 3*d))/(384*b^2*d - 864*d^3)) - exp(a*2i - c*1i + b*x*2i - d
*x*1i)*((3*(2*b + d))/(32*(4*b^2 - d^2)) - (3*exp(- a*4i - b*x*4i)*(2*b - d))/(32*(4*b^2 - d^2)) + (exp(- a*2i
- b*x*2i)*(24*b^2 - 6*d^2))/(32*d*(4*b^2 - d^2))) - exp(a*2i + c*1i + b*x*2i + d*x*1i)*((3*exp(- a*4i - b*x*4
i)*(2*b + d))/(32*(4*b^2 - d^2)) - (3*(2*b - d))/(32*(4*b^2 - d^2)) + (exp(- a*2i - b*x*2i)*(24*b^2 - 6*d^2))/
(32*d*(4*b^2 - d^2)))

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sympy [A]  time = 113.79, size = 1999, normalized size = 13.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)**2*sin(d*x+c)**3,x)

[Out]

Piecewise((x*sin(a)**2*sin(c)**3, Eq(b, 0) & Eq(d, 0)), (x*sin(a - 3*d*x/2)**2*sin(c + d*x)**3/16 - 3*x*sin(a
- 3*d*x/2)**2*sin(c + d*x)*cos(c + d*x)**2/16 - 3*x*sin(a - 3*d*x/2)*sin(c + d*x)**2*cos(a - 3*d*x/2)*cos(c +
d*x)/8 + x*sin(a - 3*d*x/2)*cos(a - 3*d*x/2)*cos(c + d*x)**3/8 - x*sin(c + d*x)**3*cos(a - 3*d*x/2)**2/16 + 3*
x*sin(c + d*x)*cos(a - 3*d*x/2)**2*cos(c + d*x)**2/16 - sin(a - 3*d*x/2)**2*sin(c + d*x)**2*cos(c + d*x)/d - 5
*sin(a - 3*d*x/2)**2*cos(c + d*x)**3/(48*d) - sin(a - 3*d*x/2)*sin(c + d*x)**3*cos(a - 3*d*x/2)/(24*d) + 5*sin
(a - 3*d*x/2)*sin(c + d*x)*cos(a - 3*d*x/2)*cos(c + d*x)**2/(4*d) - 9*cos(a - 3*d*x/2)**2*cos(c + d*x)**3/(16*
d), Eq(b, -3*d/2)), (3*x*sin(a - d*x/2)**2*sin(c + d*x)**3/16 + 3*x*sin(a - d*x/2)**2*sin(c + d*x)*cos(c + d*x
)**2/16 - 3*x*sin(a - d*x/2)*sin(c + d*x)**2*cos(a - d*x/2)*cos(c + d*x)/8 - 3*x*sin(a - d*x/2)*cos(a - d*x/2)
*cos(c + d*x)**3/8 - 3*x*sin(c + d*x)**3*cos(a - d*x/2)**2/16 - 3*x*sin(c + d*x)*cos(a - d*x/2)**2*cos(c + d*x
)**2/16 - sin(a - d*x/2)**2*sin(c + d*x)**2*cos(c + d*x)/d - 31*sin(a - d*x/2)**2*cos(c + d*x)**3/(48*d) - 3*s
in(a - d*x/2)*sin(c + d*x)**3*cos(a - d*x/2)/(8*d) - sin(a - d*x/2)*sin(c + d*x)*cos(a - d*x/2)*cos(c + d*x)**
2/(4*d) - cos(a - d*x/2)**2*cos(c + d*x)**3/(48*d), Eq(b, -d/2)), (3*x*sin(a + d*x/2)**2*sin(c + d*x)**3/16 +
3*x*sin(a + d*x/2)**2*sin(c + d*x)*cos(c + d*x)**2/16 + 3*x*sin(a + d*x/2)*sin(c + d*x)**2*cos(a + d*x/2)*cos(
c + d*x)/8 + 3*x*sin(a + d*x/2)*cos(a + d*x/2)*cos(c + d*x)**3/8 - 3*x*sin(c + d*x)**3*cos(a + d*x/2)**2/16 -
3*x*sin(c + d*x)*cos(a + d*x/2)**2*cos(c + d*x)**2/16 - sin(a + d*x/2)**2*sin(c + d*x)**2*cos(c + d*x)/d - 31*
sin(a + d*x/2)**2*cos(c + d*x)**3/(48*d) + 3*sin(a + d*x/2)*sin(c + d*x)**3*cos(a + d*x/2)/(8*d) + sin(a + d*x
/2)*sin(c + d*x)*cos(a + d*x/2)*cos(c + d*x)**2/(4*d) - cos(a + d*x/2)**2*cos(c + d*x)**3/(48*d), Eq(b, d/2)),
(x*sin(a + 3*d*x/2)**2*sin(c + d*x)**3/16 - 3*x*sin(a + 3*d*x/2)**2*sin(c + d*x)*cos(c + d*x)**2/16 + 3*x*sin
(a + 3*d*x/2)*sin(c + d*x)**2*cos(a + 3*d*x/2)*cos(c + d*x)/8 - x*sin(a + 3*d*x/2)*cos(a + 3*d*x/2)*cos(c + d*
x)**3/8 - x*sin(c + d*x)**3*cos(a + 3*d*x/2)**2/16 + 3*x*sin(c + d*x)*cos(a + 3*d*x/2)**2*cos(c + d*x)**2/16 -
sin(a + 3*d*x/2)**2*sin(c + d*x)**2*cos(c + d*x)/d - 5*sin(a + 3*d*x/2)**2*cos(c + d*x)**3/(48*d) + sin(a + 3
*d*x/2)*sin(c + d*x)**3*cos(a + 3*d*x/2)/(24*d) - 5*sin(a + 3*d*x/2)*sin(c + d*x)*cos(a + 3*d*x/2)*cos(c + d*x
)**2/(4*d) - 9*cos(a + 3*d*x/2)**2*cos(c + d*x)**3/(16*d), Eq(b, 3*d/2)), ((x*sin(a + b*x)**2/2 + x*cos(a + b*
x)**2/2 - sin(a + b*x)*cos(a + b*x)/(2*b))*sin(c)**3, Eq(d, 0)), (-24*b**4*sin(a + b*x)**2*sin(c + d*x)**2*cos
(c + d*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 16*b**4*sin(a + b*x)**2*cos(c + d*x)**3/(48*b**4*d - 120*b**
2*d**3 + 27*d**5) - 24*b**4*sin(c + d*x)**2*cos(a + b*x)**2*cos(c + d*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5)
- 16*b**4*cos(a + b*x)**2*cos(c + d*x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 24*b**3*d*sin(a + b*x)*sin(
c + d*x)**3*cos(a + b*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 78*b**2*d**2*sin(a + b*x)**2*sin(c + d*x)**2*
cos(c + d*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 40*b**2*d**2*sin(a + b*x)**2*cos(c + d*x)**3/(48*b**4*d -
120*b**2*d**3 + 27*d**5) + 42*b**2*d**2*sin(c + d*x)**2*cos(a + b*x)**2*cos(c + d*x)/(48*b**4*d - 120*b**2*d*
*3 + 27*d**5) + 40*b**2*d**2*cos(a + b*x)**2*cos(c + d*x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 42*b*d**3
*sin(a + b*x)*sin(c + d*x)**3*cos(a + b*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 36*b*d**3*sin(a + b*x)*sin(
c + d*x)*cos(a + b*x)*cos(c + d*x)**2/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 27*d**4*sin(a + b*x)**2*sin(c +
d*x)**2*cos(c + d*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 18*d**4*sin(a + b*x)**2*cos(c + d*x)**3/(48*b**4*
d - 120*b**2*d**3 + 27*d**5), True))

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