3.73 \(\int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=244 \[ \frac{192 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9) \left (4 m^2+24 m+35\right )}+\frac{768 c^3 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right ) \sqrt{c-c \sin (e+f x)}}+\frac{24 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f \left (4 m^2+32 m+63\right )}+\frac{2 \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9)} \]

[Out]

(768*c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)*Sqrt[c - c*Si
n[e + f*x]]) + (192*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(9 + 2*m)*(35
+ 24*m + 4*m^2)) + (24*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(a*f*(63 + 32*
m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(5/2))/(a*f*(9 + 2*m))

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Rubi [A]  time = 0.615432, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2841, 2740, 2738} \[ \frac{192 c^2 \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9) \left (4 m^2+24 m+35\right )}+\frac{768 c^3 \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right ) \sqrt{c-c \sin (e+f x)}}+\frac{24 c \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^{m+1}}{a f \left (4 m^2+32 m+63\right )}+\frac{2 \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^{m+1}}{a f (2 m+9)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

(768*c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)*Sqrt[c - c*Si
n[e + f*x]]) + (192*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(9 + 2*m)*(35
+ 24*m + 4*m^2)) + (24*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(a*f*(63 + 32*
m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(5/2))/(a*f*(9 + 2*m))

Rule 2841

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[1/(a^(p/2)*c^(p/2)), Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(
n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p
/2]

Rule 2740

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Sim
p[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*(m + n)), x] + Dist[(a*(2*m - 1))/(m
+ n), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E
qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] &&  !LtQ[n, -1] &&  !(IGtQ[n - 1/2, 0] && LtQ[n, m])
&&  !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])

Rule 2738

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[
(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]

Rubi steps

\begin{align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx &=\frac{\int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{7/2} \, dx}{a c}\\ &=\frac{2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)}+\frac{12 \int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2} \, dx}{a (9+2 m)}\\ &=\frac{24 c \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2}}{a f \left (63+32 m+4 m^2\right )}+\frac{2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)}+\frac{(96 c) \int (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2} \, dx}{a \left (63+32 m+4 m^2\right )}\\ &=\frac{192 c^2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt{c-c \sin (e+f x)}}{a f (5+2 m) \left (63+32 m+4 m^2\right )}+\frac{24 c \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2}}{a f \left (63+32 m+4 m^2\right )}+\frac{2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)}+\frac{\left (384 c^2\right ) \int (a+a \sin (e+f x))^{1+m} \sqrt{c-c \sin (e+f x)} \, dx}{a (5+2 m) \left (63+32 m+4 m^2\right )}\\ &=\frac{768 c^3 \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (3+2 m) (5+2 m) \left (63+32 m+4 m^2\right ) \sqrt{c-c \sin (e+f x)}}+\frac{192 c^2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt{c-c \sin (e+f x)}}{a f (5+2 m) \left (63+32 m+4 m^2\right )}+\frac{24 c \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{3/2}}{a f \left (63+32 m+4 m^2\right )}+\frac{2 \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c-c \sin (e+f x))^{5/2}}{a f (9+2 m)}\\ \end{align*}

Mathematica [C]  time = 6.57638, size = 695, normalized size = 2.85 \[ \frac{(c-c \sin (e+f x))^{5/2} (a (\sin (e+f x)+1))^m \left (\frac{\left (8 m^3+108 m^2+590 m+2205\right ) \left (\left (\frac{3}{8}-\frac{3 i}{8}\right ) \sin \left (\frac{1}{2} (e+f x)\right )+\left (\frac{3}{8}+\frac{3 i}{8}\right ) \cos \left (\frac{1}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left (8 m^3+108 m^2+590 m+2205\right ) \left (\left (\frac{3}{8}+\frac{3 i}{8}\right ) \sin \left (\frac{1}{2} (e+f x)\right )+\left (\frac{3}{8}-\frac{3 i}{8}\right ) \cos \left (\frac{1}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left (4 m^3+48 m^2+191 m\right ) \left ((1-i) \cos \left (\frac{3}{2} (e+f x)\right )-(1+i) \sin \left (\frac{3}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{\left (4 m^3+48 m^2+191 m\right ) \left ((1+i) \cos \left (\frac{3}{2} (e+f x)\right )-(1-i) \sin \left (\frac{3}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7) (2 m+9)}+\frac{(2 m+21) \left (\left (\frac{3}{2}-\frac{3 i}{2}\right ) \sin \left (\frac{5}{2} (e+f x)\right )+\left (\frac{3}{2}+\frac{3 i}{2}\right ) \cos \left (\frac{5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7) (2 m+9)}+\frac{(2 m+21) \left (\left (\frac{3}{2}+\frac{3 i}{2}\right ) \sin \left (\frac{5}{2} (e+f x)\right )+\left (\frac{3}{2}-\frac{3 i}{2}\right ) \cos \left (\frac{5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7) (2 m+9)}+\frac{(2 m+15) \left (\left (\frac{3}{16}-\frac{3 i}{16}\right ) \cos \left (\frac{7}{2} (e+f x)\right )-\left (\frac{3}{16}+\frac{3 i}{16}\right ) \sin \left (\frac{7}{2} (e+f x)\right )\right )}{(2 m+7) (2 m+9)}+\frac{(2 m+15) \left (\left (\frac{3}{16}+\frac{3 i}{16}\right ) \cos \left (\frac{7}{2} (e+f x)\right )-\left (\frac{3}{16}-\frac{3 i}{16}\right ) \sin \left (\frac{7}{2} (e+f x)\right )\right )}{(2 m+7) (2 m+9)}+\frac{\left (-\frac{1}{16}+\frac{i}{16}\right ) \cos \left (\frac{9}{2} (e+f x)\right )-\left (\frac{1}{16}+\frac{i}{16}\right ) \sin \left (\frac{9}{2} (e+f x)\right )}{2 m+9}+\frac{\left (-\frac{1}{16}-\frac{i}{16}\right ) \cos \left (\frac{9}{2} (e+f x)\right )-\left (\frac{1}{16}-\frac{i}{16}\right ) \sin \left (\frac{9}{2} (e+f x)\right )}{2 m+9}\right )}{f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((2205 + 590*m + 108*m^2 + 8*m^3)*((3/8 + (3*I)/8)*Cos[(
e + f*x)/2] + (3/8 - (3*I)/8)*Sin[(e + f*x)/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((2205 + 590*m +
108*m^2 + 8*m^3)*((3/8 - (3*I)/8)*Cos[(e + f*x)/2] + (3/8 + (3*I)/8)*Sin[(e + f*x)/2]))/((3 + 2*m)*(5 + 2*m)*(
7 + 2*m)*(9 + 2*m)) + ((191*m + 48*m^2 + 4*m^3)*((1 - I)*Cos[(3*(e + f*x))/2] - (1 + I)*Sin[(3*(e + f*x))/2]))
/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((191*m + 48*m^2 + 4*m^3)*((1 + I)*Cos[(3*(e + f*x))/2] - (1 - I)
*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((21 + 2*m)*((3/2 + (3*I)/2)*Cos[(5*(e + f
*x))/2] + (3/2 - (3*I)/2)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((21 + 2*m)*((3/2 - (3*I)/2
)*Cos[(5*(e + f*x))/2] + (3/2 + (3*I)/2)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7 + 2*m)*(9 + 2*m)) + ((15 + 2*m)*
((3/16 - (3*I)/16)*Cos[(7*(e + f*x))/2] - (3/16 + (3*I)/16)*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*m)) + ((1
5 + 2*m)*((3/16 + (3*I)/16)*Cos[(7*(e + f*x))/2] - (3/16 - (3*I)/16)*Sin[(7*(e + f*x))/2]))/((7 + 2*m)*(9 + 2*
m)) + ((-1/16 + I/16)*Cos[(9*(e + f*x))/2] - (1/16 + I/16)*Sin[(9*(e + f*x))/2])/(9 + 2*m) + ((-1/16 - I/16)*C
os[(9*(e + f*x))/2] - (1/16 - I/16)*Sin[(9*(e + f*x))/2])/(9 + 2*m)))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])
^5)

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Maple [F]  time = 0.362, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x)

[Out]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x)

________________________________________________________________________________________

Maxima [B]  time = 2.32389, size = 753, normalized size = 3.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

-2*((8*m^3 + 108*m^2 + 526*m + 957)*a^m*c^(5/2) - 3*(8*m^3 + 76*m^2 + 142*m - 315)*a^m*c^(5/2)*sin(f*x + e)/(c
os(f*x + e) + 1) - 24*(4*m^2 + 16*m - 81)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 16*(4*m^3 + 36*m^2
+ 95*m + 315)*a^m*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6*(8*m^3 + 60*m^2 + 206*m - 567)*a^m*c^(5/2)*
sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 6*(8*m^3 + 60*m^2 + 206*m - 567)*a^m*c^(5/2)*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 + 16*(4*m^3 + 36*m^2 + 95*m + 315)*a^m*c^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 24*(4*m^2 + 16*m
- 81)*a^m*c^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3*(8*m^3 + 76*m^2 + 142*m - 315)*a^m*c^(5/2)*sin(f*x
+ e)^8/(cos(f*x + e) + 1)^8 + (8*m^3 + 108*m^2 + 526*m + 957)*a^m*c^(5/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)
*e^(2*m*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1) - m*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/((16*m^4 +
192*m^3 + 824*m^2 + 1488*m + 2*(16*m^4 + 192*m^3 + 824*m^2 + 1488*m + 945)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2
+ (16*m^4 + 192*m^3 + 824*m^2 + 1488*m + 945)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 945)*f*(sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 + 1)^(5/2))

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Fricas [A]  time = 2.02952, size = 979, normalized size = 4.01 \begin{align*} -\frac{2 \,{\left ({\left (8 \, c^{2} m^{3} + 60 \, c^{2} m^{2} + 142 \, c^{2} m + 105 \, c^{2}\right )} \cos \left (f x + e\right )^{5} -{\left (8 \, c^{2} m^{3} + 108 \, c^{2} m^{2} + 334 \, c^{2} m + 285 \, c^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (8 \, c^{2} m^{3} + 84 \, c^{2} m^{2} + 334 \, c^{2} m + 339 \, c^{2}\right )} \cos \left (f x + e\right )^{3} - 384 \, c^{2} \cos \left (f x + e\right ) - 96 \,{\left (2 \, c^{2} m - c^{2}\right )} \cos \left (f x + e\right )^{2} - 768 \, c^{2} +{\left ({\left (8 \, c^{2} m^{3} + 60 \, c^{2} m^{2} + 142 \, c^{2} m + 105 \, c^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (8 \, c^{2} m^{3} + 84 \, c^{2} m^{2} + 238 \, c^{2} m + 195 \, c^{2}\right )} \cos \left (f x + e\right )^{3} - 384 \, c^{2} \cos \left (f x + e\right ) - 96 \,{\left (2 \, c^{2} m + 3 \, c^{2}\right )} \cos \left (f x + e\right )^{2} - 768 \, c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m +{\left (16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + 945 \, f\right )} \cos \left (f x + e\right ) -{\left (16 \, f m^{4} + 192 \, f m^{3} + 824 \, f m^{2} + 1488 \, f m + 945 \, f\right )} \sin \left (f x + e\right ) + 945 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-2*((8*c^2*m^3 + 60*c^2*m^2 + 142*c^2*m + 105*c^2)*cos(f*x + e)^5 - (8*c^2*m^3 + 108*c^2*m^2 + 334*c^2*m + 285
*c^2)*cos(f*x + e)^4 - 2*(8*c^2*m^3 + 84*c^2*m^2 + 334*c^2*m + 339*c^2)*cos(f*x + e)^3 - 384*c^2*cos(f*x + e)
- 96*(2*c^2*m - c^2)*cos(f*x + e)^2 - 768*c^2 + ((8*c^2*m^3 + 60*c^2*m^2 + 142*c^2*m + 105*c^2)*cos(f*x + e)^4
+ 2*(8*c^2*m^3 + 84*c^2*m^2 + 238*c^2*m + 195*c^2)*cos(f*x + e)^3 - 384*c^2*cos(f*x + e) - 96*(2*c^2*m + 3*c^
2)*cos(f*x + e)^2 - 768*c^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(16*f*m^4 + 192*f*
m^3 + 824*f*m^2 + 1488*f*m + (16*f*m^4 + 192*f*m^3 + 824*f*m^2 + 1488*f*m + 945*f)*cos(f*x + e) - (16*f*m^4 +
192*f*m^3 + 824*f*m^2 + 1488*f*m + 945*f)*sin(f*x + e) + 945*f)

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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(f*x+e)**2*(a+a*sin(f*x+e))**m*(c-c*sin(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [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(f*x+e)^2*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Timed out