3.173 \(\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx\)

Optimal. Leaf size=246 \[ \frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8960 c^4 f (c-c \sin (e+f x))^{9/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}} \]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(16*f*(c - c*Sin[e + f*x])^(17/2)) + ((A - 3*B)*Cos[e + f*x]
*(a + a*Sin[e + f*x])^(7/2))/(56*c*f*(c - c*Sin[e + f*x])^(15/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x
])^(7/2))/(224*c^2*f*(c - c*Sin[e + f*x])^(13/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(1120*
c^3*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8960*c^4*f*(c - c*Si
n[e + f*x])^(9/2))

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Rubi [A]  time = 0.590887, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2972, 2743, 2742} \[ \frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8960 c^4 f (c-c \sin (e+f x))^{9/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(17/2),x]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(16*f*(c - c*Sin[e + f*x])^(17/2)) + ((A - 3*B)*Cos[e + f*x]
*(a + a*Sin[e + f*x])^(7/2))/(56*c*f*(c - c*Sin[e + f*x])^(15/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x
])^(7/2))/(224*c^2*f*(c - c*Sin[e + f*x])^(13/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(1120*
c^3*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8960*c^4*f*(c - c*Si
n[e + f*x])^(9/2))

Rule 2972

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

Rule 2743

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(a*f*(2*m + 1)), x] + Dist[(m + n + 1)/(a*(2*m
+ 1)), 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, m, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m
, 1] ||  !SumSimplerQ[n, 1])

Rule 2742

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(a*f*(2*m + 1)), x] /; FreeQ[{a, b, c, d, e, f
, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[m, -2^(-1)]

Rubi steps

\begin{align*} \int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac{(A-3 B) \int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx}{4 c}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{(3 (A-3 B)) \int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{56 c^2}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{(A-3 B) \int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{112 c^3}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{(A-3 B) \int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{1120 c^4}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{8960 c^4 f (c-c \sin (e+f x))^{9/2}}\\ \end{align*}

Mathematica [A]  time = 7.1124, size = 436, normalized size = 1.77 \[ \frac{(-A-7 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}{5 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{(A+3 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{4 (3 A+5 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}{7 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{(A+B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{B (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}{4 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(17/2),x]

[Out]

((A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f
*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - (4*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e
+ f*x]))^(7/2))/(7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + ((A + 3*B)*(Cos[(e
+ f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c -
c*Sin[e + f*x])^(17/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(
5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f*x
)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)
)

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Maple [B]  time = 0.405, size = 560, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2),x)

[Out]

-1/140/f*(3076*A-268*B+96*A*cos(f*x+e)^7-8*B*cos(f*x+e)^7-3076*A*sin(f*x+e)-5348*A*cos(f*x+e)^2-1332*A*cos(f*x
+e)^4*sin(f*x+e)+3880*A*cos(f*x+e)^2*sin(f*x+e)+372*A*cos(f*x+e)^5*sin(f*x+e)+111*B*sin(f*x+e)*cos(f*x+e)^4+10
8*A*cos(f*x+e)^6*sin(f*x+e)-1548*A*cos(f*x+e)^3*sin(f*x+e)-300*B*cos(f*x+e)^2*sin(f*x+e)-1608*A*cos(f*x+e)+164
*B*cos(f*x+e)^3*sin(f*x+e)-B*cos(f*x+e)^8-31*B*cos(f*x+e)^5*sin(f*x+e)+2332*A*cos(f*x+e)^3-136*B*cos(f*x+e)^3+
64*B*cos(f*x+e)-480*A*cos(f*x+e)^6+40*B*cos(f*x+e)^6-204*B*sin(f*x+e)*cos(f*x+e)+1468*A*sin(f*x+e)*cos(f*x+e)-
9*B*cos(f*x+e)^6*sin(f*x+e)-960*A*cos(f*x+e)^5+80*B*cos(f*x+e)^5+B*cos(f*x+e)^7*sin(f*x+e)-12*A*cos(f*x+e)^7*s
in(f*x+e)+2880*A*cos(f*x+e)^4-275*B*cos(f*x+e)^4+504*B*cos(f*x+e)^2+268*B*sin(f*x+e)+12*A*cos(f*x+e)^8)*sin(f*
x+e)*(a*(1+sin(f*x+e)))^(7/2)/(sin(f*x+e)*cos(f*x+e)^3+cos(f*x+e)^4-4*cos(f*x+e)^2*sin(f*x+e)+3*cos(f*x+e)^3-4
*sin(f*x+e)*cos(f*x+e)-8*cos(f*x+e)^2+8*sin(f*x+e)-4*cos(f*x+e)+8)/(-c*(-1+sin(f*x+e)))^(17/2)

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Maxima [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((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 2.19861, size = 613, normalized size = 2.49 \begin{align*} \frac{{\left (35 \, B a^{3} \cos \left (f x + e\right )^{4} - 56 \,{\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \,{\left (17 \, A + 19 \, B\right )} a^{3} - 4 \,{\left (7 \,{\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 2 \,{\left (9 \, A + 8 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{140 \,{\left (c^{9} f \cos \left (f x + e\right )^{9} - 32 \, c^{9} f \cos \left (f x + e\right )^{7} + 160 \, c^{9} f \cos \left (f x + e\right )^{5} - 256 \, c^{9} f \cos \left (f x + e\right )^{3} + 128 \, c^{9} f \cos \left (f x + e\right ) + 8 \,{\left (c^{9} f \cos \left (f x + e\right )^{7} - 10 \, c^{9} f \cos \left (f x + e\right )^{5} + 24 \, c^{9} f \cos \left (f x + e\right )^{3} - 16 \, c^{9} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2),x, algorithm="fricas")

[Out]

1/140*(35*B*a^3*cos(f*x + e)^4 - 56*(A + 2*B)*a^3*cos(f*x + e)^2 + 4*(17*A + 19*B)*a^3 - 4*(7*(A + 2*B)*a^3*co
s(f*x + e)^2 - 2*(9*A + 8*B)*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^9*f*cos(
f*x + e)^9 - 32*c^9*f*cos(f*x + e)^7 + 160*c^9*f*cos(f*x + e)^5 - 256*c^9*f*cos(f*x + e)^3 + 128*c^9*f*cos(f*x
+ e) + 8*(c^9*f*cos(f*x + e)^7 - 10*c^9*f*cos(f*x + e)^5 + 24*c^9*f*cos(f*x + e)^3 - 16*c^9*f*cos(f*x + e))*s
in(f*x + e))

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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((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(17/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{17}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2),x, algorithm="giac")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(17/2), x)