3.172 \(\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6720 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}} \]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x])^(15/2)) + ((3*A - 11*B)*Cos[e + f
*x]*(a + a*Sin[e + f*x])^(7/2))/(168*c*f*(c - c*Sin[e + f*x])^(13/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[
e + f*x])^(7/2))/(840*c^2*f*(c - c*Sin[e + f*x])^(11/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/
2))/(6720*c^3*f*(c - c*Sin[e + f*x])^(9/2))

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Rubi [A]  time = 0.49066, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2972, 2743, 2742} \[ \frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6720 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}} \]

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])^(15/2),x]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x])^(15/2)) + ((3*A - 11*B)*Cos[e + f
*x]*(a + a*Sin[e + f*x])^(7/2))/(168*c*f*(c - c*Sin[e + f*x])^(13/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[
e + f*x])^(7/2))/(840*c^2*f*(c - c*Sin[e + f*x])^(11/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/
2))/(6720*c^3*f*(c - c*Sin[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))^{15/2}} \, dx &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac{(3 A-11 B) \int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{14 c}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac{(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac{(3 A-11 B) \int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{84 c^2}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac{(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac{(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{(3 A-11 B) \int \frac{(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{840 c^3}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac{(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac{(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac{(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6720 c^3 f (c-c \sin (e+f x))^{9/2}}\\ \end{align*}

Mathematica [B]  time = 7.14892, size = 442, normalized size = 2.19 \[ \frac{(-A-7 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}{4 f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{6 (A+3 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{5 f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{2 (3 A+5 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}{3 f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{8 (A+B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{7 f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{B (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}{3 f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]

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])^(15/2),x]

[Out]

(8*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(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])^(15/2)) - (2*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Si
n[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + (6*(A + 3*B)*(
Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(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])^(15/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(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])^(15/2)) + (B*(Cos[(e + f*x)/2] - Sin[
(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x]
)^(15/2))

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Maple [B]  time = 0.366, size = 505, normalized size = 2.5 \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))^(15/2),x)

[Out]

-1/420/f*(5016*A-472*B+39*A*cos(f*x+e)^7-3*B*cos(f*x+e)^7-5016*A*sin(f*x+e)-7404*A*cos(f*x+e)^2-1209*A*cos(f*x
+e)^4*sin(f*x+e)+5136*A*cos(f*x+e)^2*sin(f*x+e)+273*A*cos(f*x+e)^5*sin(f*x+e)+93*B*sin(f*x+e)*cos(f*x+e)^4+39*
A*cos(f*x+e)^6*sin(f*x+e)-1911*A*cos(f*x+e)^3*sin(f*x+e)-352*B*cos(f*x+e)^2*sin(f*x+e)-2748*A*cos(f*x+e)+287*B
*cos(f*x+e)^3*sin(f*x+e)-21*B*cos(f*x+e)^5*sin(f*x+e)+3225*A*cos(f*x+e)^3-65*B*cos(f*x+e)^3-4*B*cos(f*x+e)-312
*A*cos(f*x+e)^6+24*B*cos(f*x+e)^6-476*B*sin(f*x+e)*cos(f*x+e)+2268*A*sin(f*x+e)*cos(f*x+e)-3*B*cos(f*x+e)^6*si
n(f*x+e)-936*A*cos(f*x+e)^5+72*B*cos(f*x+e)^5+3120*A*cos(f*x+e)^4-380*B*cos(f*x+e)^4+828*B*cos(f*x+e)^2+472*B*
sin(f*x+e))*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)))^(15/
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))^(15/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 2.13455, size = 585, normalized size = 2.9 \begin{align*} -\frac{{\left (140 \, B a^{3} \cos \left (f x + e\right )^{4} - 7 \,{\left (27 \, A + 61 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \,{\left (57 \, A + 71 \, B\right )} a^{3} - 7 \,{\left (5 \,{\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 4 \,{\left (9 \, A + 7 \, 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}}{420 \,{\left (7 \, c^{8} f \cos \left (f x + e\right )^{7} - 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right ) -{\left (c^{8} f \cos \left (f x + e\right )^{7} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} 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))^(15/2),x, algorithm="fricas")

[Out]

-1/420*(140*B*a^3*cos(f*x + e)^4 - 7*(27*A + 61*B)*a^3*cos(f*x + e)^2 + 4*(57*A + 71*B)*a^3 - 7*(5*(3*A + 5*B)
*a^3*cos(f*x + e)^2 - 4*(9*A + 7*B)*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(7*c
^8*f*cos(f*x + e)^7 - 56*c^8*f*cos(f*x + e)^5 + 112*c^8*f*cos(f*x + e)^3 - 64*c^8*f*cos(f*x + e) - (c^8*f*cos(
f*x + e)^7 - 24*c^8*f*cos(f*x + e)^5 + 80*c^8*f*cos(f*x + e)^3 - 64*c^8*f*cos(f*x + e))*sin(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))**(15/2),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{15}{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))^(15/2),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)^(15/2), x)