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

Optimal. Leaf size=275 \[ -\frac {64 c^3 (B (5-2 m)-A (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}-\frac {16 c^2 (B (5-2 m)-A (2 m+7)) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) \left (4 m^2+16 m+15\right )}-\frac {2 c (B (5-2 m)-A (2 m+7)) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)} \]

[Out]

-2*c*(B*(5-2*m)-A*(7+2*m))*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(3/2)/f/(4*m^2+24*m+35)-2*B*cos(f*x+
e)*(a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2)/f/(7+2*m)-64*c^3*(B*(5-2*m)-A*(7+2*m))*cos(f*x+e)*(a+a*sin(f*x+e)
)^m/f/(5+2*m)/(7+2*m)/(4*m^2+8*m+3)/(c-c*sin(f*x+e))^(1/2)-16*c^2*(B*(5-2*m)-A*(7+2*m))*cos(f*x+e)*(a+a*sin(f*
x+e))^m*(c-c*sin(f*x+e))^(1/2)/f/(7+2*m)/(4*m^2+16*m+15)

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Rubi [A]  time = 0.50, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2973, 2740, 2738} \[ -\frac {16 c^2 (B (5-2 m)-A (2 m+7)) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) \left (4 m^2+16 m+15\right )}-\frac {64 c^3 (B (5-2 m)-A (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}-\frac {2 c (B (5-2 m)-A (2 m+7)) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-64*c^3*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*
m^2)*Sqrt[c - c*Sin[e + f*x]]) - (16*c^2*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[
c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(15 + 16*m + 4*m^2)) - (2*c*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*
Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m
*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m))

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)]

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 2973

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

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx &=-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}+\frac {\left (B c \left (-\frac {5}{2}+m\right )+A c \left (\frac {7}{2}+m\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx}{c \left (\frac {7}{2}+m\right )}\\ &=-\frac {2 c (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}-\frac {(8 c (B (5-2 m)-A (7+2 m))) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx}{(5+2 m) (7+2 m)}\\ &=-\frac {16 c^2 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac {2 c (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}-\frac {\left (32 c^2 (B (5-2 m)-A (7+2 m))\right ) \int (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx}{(3+2 m) (5+2 m) (7+2 m)}\\ &=-\frac {64 c^3 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \sqrt {c-c \sin (e+f x)}}-\frac {16 c^2 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac {2 c (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}\\ \end {align*}

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Mathematica [C]  time = 6.82, size = 667, normalized size = 2.43 \[ \frac {(c-c \sin (e+f x))^{5/2} (a (\sin (e+f x)+1))^m \left (\frac {\left (24 A m^2+184 A m+350 A-12 B m^2-104 B m-385 B\right ) \left (\left (\frac {1}{8}-\frac {i}{8}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7)}+\frac {\left (24 A m^2+184 A m+350 A-12 B m^2-104 B m-385 B\right ) \left (\left (\frac {1}{8}+\frac {i}{8}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7)}+\frac {\left (32 A m^3+304 A m^2+1272 A m+2100 A-8 B m^3-68 B m^2-110 B m-1575 B\right ) \left (\left (\frac {1}{8}-\frac {i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{8}+\frac {i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{(2 m+1) (2 m+3) (2 m+5) (2 m+7)}+\frac {\left (32 A m^3+304 A m^2+1272 A m+2100 A-8 B m^3-68 B m^2-110 B m-1575 B\right ) \left (\left (\frac {1}{8}+\frac {i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{(2 m+1) (2 m+3) (2 m+5) (2 m+7)}+\frac {(4 A m+14 A-6 B m-35 B) \left (\left (-\frac {1}{8}+\frac {i}{8}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7)}+\frac {(4 A m+14 A-6 B m-35 B) \left (\left (-\frac {1}{8}-\frac {i}{8}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7)}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) B \cos \left (\frac {7}{2} (e+f x)\right )-\left (\frac {1}{8}+\frac {i}{8}\right ) B \sin \left (\frac {7}{2} (e+f x)\right )}{2 m+7}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) B \cos \left (\frac {7}{2} (e+f x)\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) B \sin \left (\frac {7}{2} (e+f x)\right )}{2 m+7}\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[(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(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)*(((2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 - 68*
B*m^2 + 32*A*m^3 - 8*B*m^3)*((1/8 + I/8)*Cos[(e + f*x)/2] + (1/8 - I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m
)*(5 + 2*m)*(7 + 2*m)) + ((2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 - 68*B*m^2 + 32*A*m^3 - 8*B*m^3)*(
(1/8 - I/8)*Cos[(e + f*x)/2] + (1/8 + I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((35
0*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2)*((1/8 - I/8)*Cos[(3*(e + f*x))/2] - (1/8 + I/8)*Sin[(3*
(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((350*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2)*(
(1/8 + I/8)*Cos[(3*(e + f*x))/2] - (1/8 - I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((14*A
- 35*B + 4*A*m - 6*B*m)*((-1/8 + I/8)*Cos[(5*(e + f*x))/2] - (1/8 + I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7
+ 2*m)) + ((14*A - 35*B + 4*A*m - 6*B*m)*((-1/8 - I/8)*Cos[(5*(e + f*x))/2] - (1/8 - I/8)*Sin[(5*(e + f*x))/2
]))/((5 + 2*m)*(7 + 2*m)) + ((1/8 - I/8)*B*Cos[(7*(e + f*x))/2] - (1/8 + I/8)*B*Sin[(7*(e + f*x))/2])/(7 + 2*m
) + ((1/8 + I/8)*B*Cos[(7*(e + f*x))/2] - (1/8 - I/8)*B*Sin[(7*(e + f*x))/2])/(7 + 2*m)))/(f*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^5)

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fricas [B]  time = 0.49, size = 562, normalized size = 2.04 \[ \frac {2 \, {\left ({\left (8 \, B c^{2} m^{3} + 36 \, B c^{2} m^{2} + 46 \, B c^{2} m + 15 \, B c^{2}\right )} \cos \left (f x + e\right )^{4} + 64 \, {\left (A + B\right )} c^{2} m - {\left (8 \, {\left (A - 2 \, B\right )} c^{2} m^{3} + 4 \, {\left (11 \, A - 28 \, B\right )} c^{2} m^{2} + 2 \, {\left (31 \, A - 86 \, B\right )} c^{2} m + 3 \, {\left (7 \, A - 20 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{3} + 32 \, {\left (7 \, A - 5 \, B\right )} c^{2} + {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 4 \, {\left (19 \, A - 11 \, B\right )} c^{2} m^{2} + 190 \, {\left (A - B\right )} c^{2} m + {\left (77 \, A - 85 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 60 \, {\left (A - B\right )} c^{2} m^{2} + 2 \, {\left (79 \, A - 63 \, B\right )} c^{2} m + {\left (161 \, A - 145 \, B\right )} c^{2}\right )} \cos \left (f x + e\right ) + {\left (64 \, {\left (A + B\right )} c^{2} m - {\left (8 \, B c^{2} m^{3} + 36 \, B c^{2} m^{2} + 46 \, B c^{2} m + 15 \, B c^{2}\right )} \cos \left (f x + e\right )^{3} + 32 \, {\left (7 \, A - 5 \, B\right )} c^{2} - {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 4 \, {\left (11 \, A - 19 \, B\right )} c^{2} m^{2} + 2 \, {\left (31 \, A - 63 \, B\right )} c^{2} m + 3 \, {\left (7 \, A - 15 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 60 \, {\left (A - B\right )} c^{2} m^{2} + 2 \, {\left (63 \, A - 79 \, B\right )} c^{2} m + {\left (49 \, A - 65 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )\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} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \cos \left (f x + e\right ) - {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \sin \left (f x + e\right ) + 105 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((8*B*c^2*m^3 + 36*B*c^2*m^2 + 46*B*c^2*m + 15*B*c^2)*cos(f*x + e)^4 + 64*(A + B)*c^2*m - (8*(A - 2*B)*c^2*m
^3 + 4*(11*A - 28*B)*c^2*m^2 + 2*(31*A - 86*B)*c^2*m + 3*(7*A - 20*B)*c^2)*cos(f*x + e)^3 + 32*(7*A - 5*B)*c^2
+ (8*(A - B)*c^2*m^3 + 4*(19*A - 11*B)*c^2*m^2 + 190*(A - B)*c^2*m + (77*A - 85*B)*c^2)*cos(f*x + e)^2 + 2*(8
*(A - B)*c^2*m^3 + 60*(A - B)*c^2*m^2 + 2*(79*A - 63*B)*c^2*m + (161*A - 145*B)*c^2)*cos(f*x + e) + (64*(A + B
)*c^2*m - (8*B*c^2*m^3 + 36*B*c^2*m^2 + 46*B*c^2*m + 15*B*c^2)*cos(f*x + e)^3 + 32*(7*A - 5*B)*c^2 - (8*(A - B
)*c^2*m^3 + 4*(11*A - 19*B)*c^2*m^2 + 2*(31*A - 63*B)*c^2*m + 3*(7*A - 15*B)*c^2)*cos(f*x + e)^2 - 2*(8*(A - B
)*c^2*m^3 + 60*(A - B)*c^2*m^2 + 2*(63*A - 79*B)*c^2*m + (49*A - 65*B)*c^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-
c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(16*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f*m + (16*f*m^4 + 128*f*m^3
+ 344*f*m^2 + 352*f*m + 105*f)*cos(f*x + e) - (16*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f*m + 105*f)*sin(f*x +
e) + 105*f)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.34, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.55, size = 725, normalized size = 2.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(((4*m^2 + 24*m + 43)*a^m*c^(5/2) - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 2*(4
*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2*(4*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (4*m^2 + 24
*m + 43)*a^m*c^(5/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)*A*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))/((8*m^3 + 36*m^2 + 46*m + 15)*(sin(f*x + e)^2/(cos(f*x + e) +
1)^2 + 1)^(5/2)) - 2*((4*m^2 + 40*m + 115)*a^m*c^(5/2) - 2*(4*m^3 + 40*m^2 + 115*m)*a^m*c^(5/2)*sin(f*x + e)/(
cos(f*x + e) + 1) + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - (16*m^3
+ 76*m^2 + 260*m - 175)*a^m*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - (16*m^3 + 76*m^2 + 260*m - 175)*a^m
*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^m*c^(5/2)*sin(f*x + e)^5/(co
s(f*x + e) + 1)^5 - 2*(4*m^3 + 40*m^2 + 115*m)*a^m*c^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + (4*m^2 + 40*m
+ 115)*a^m*c^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*B*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 + 128*m^3 + 344*m^2 + 352*m + (16*m^4 + 128*m^3 + 344*
m^2 + 352*m + 105)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 105)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2))
)/f

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mupad [B]  time = 21.06, size = 749, normalized size = 2.72 \[ -\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {B\,c^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c^2\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (2100\,A-1575\,B+1272\,A\,m-110\,B\,m+304\,A\,m^2+32\,A\,m^3-68\,B\,m^2-8\,B\,m^3\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,2100{}\mathrm {i}-B\,1575{}\mathrm {i}+A\,m\,1272{}\mathrm {i}-B\,m\,110{}\mathrm {i}+A\,m^2\,304{}\mathrm {i}+A\,m^3\,32{}\mathrm {i}-B\,m^2\,68{}\mathrm {i}-B\,m^3\,8{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c^2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (350\,A-385\,B+184\,A\,m-104\,B\,m+24\,A\,m^2-12\,B\,m^2\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c^2\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,350{}\mathrm {i}-B\,385{}\mathrm {i}+A\,m\,184{}\mathrm {i}-B\,m\,104{}\mathrm {i}+A\,m^2\,24{}\mathrm {i}-B\,m^2\,12{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {B\,c^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (8\,m^3+36\,m^2+46\,m+15\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )\,\left (14\,A-35\,B+4\,A\,m-6\,B\,m\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )\,\left (A\,14{}\mathrm {i}-B\,35{}\mathrm {i}+A\,m\,4{}\mathrm {i}-B\,m\,6{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}\right )}{{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (m^4\,16{}\mathrm {i}+m^3\,128{}\mathrm {i}+m^2\,344{}\mathrm {i}+m\,352{}\mathrm {i}+105{}\mathrm {i}\right )}{16\,m^4+128\,m^3+344\,m^2+352\,m+105}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c - c*sin(e + f*x))^(1/2)*((B*c^2*(a + a*sin(e + f*x))^m*(m*46i + m^2*36i + m^3*8i + 15i))/(4*f*(352*m + 34
4*m^2 + 128*m^3 + 16*m^4 + 105)) - (c^2*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^m*(2100*A - 1575*B + 1272*A*m
- 110*B*m + 304*A*m^2 + 32*A*m^3 - 68*B*m^2 - 8*B*m^3))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) + (c^
2*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x))^m*(A*2100i - B*1575i + A*m*1272i - B*m*110i + A*m^2*304i + A*m^3*32i
- B*m^2*68i - B*m^3*8i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - (c^2*exp(e*5i + f*x*5i)*(2*m + 1)
*(a + a*sin(e + f*x))^m*(350*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2))/(4*f*(352*m + 344*m^2 + 128
*m^3 + 16*m^4 + 105)) + (c^2*exp(e*2i + f*x*2i)*(2*m + 1)*(a + a*sin(e + f*x))^m*(A*350i - B*385i + A*m*184i -
B*m*104i + A*m^2*24i - B*m^2*12i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) - (B*c^2*exp(e*7i + f*x*7
i)*(a + a*sin(e + f*x))^m*(46*m + 36*m^2 + 8*m^3 + 15))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105)) + (c^
2*exp(e*1i + f*x*1i)*(a + a*sin(e + f*x))^m*(8*m + 4*m^2 + 3)*(14*A - 35*B + 4*A*m - 6*B*m))/(4*f*(352*m + 344
*m^2 + 128*m^3 + 16*m^4 + 105)) - (c^2*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^m*(8*m + 4*m^2 + 3)*(A*14i - B*
35i + A*m*4i - B*m*6i))/(4*f*(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105))))/(exp(e*4i + f*x*4i) - (exp(e*3i + f
*x*3i)*(m*352i + m^2*344i + m^3*128i + m^4*16i + 105i))/(352*m + 344*m^2 + 128*m^3 + 16*m^4 + 105))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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