3.159 \(\int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx\)

Optimal. Leaf size=196 \[ \frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{960 c^3 f (c-c \sin (e+f x))^{7/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{160 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{40 c f (c-c \sin (e+f x))^{11/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}} \]

[Out]

1/12*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/f/(c-c*sin(f*x+e))^(13/2)+1/40*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e)
)^(5/2)/c/f/(c-c*sin(f*x+e))^(11/2)+1/160*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c^2/f/(c-c*sin(f*x+e))^(9/
2)+1/960*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c^3/f/(c-c*sin(f*x+e))^(7/2)

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Rubi [A]  time = 0.48, antiderivative size = 196, normalized size of antiderivative = 1.00, 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} \[ \frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{960 c^3 f (c-c \sin (e+f x))^{7/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{160 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{40 c f (c-c \sin (e+f x))^{11/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(13/2),x]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(12*f*(c - c*Sin[e + f*x])^(13/2)) + ((A - 3*B)*Cos[e + f*x]
*(a + a*Sin[e + f*x])^(5/2))/(40*c*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x
])^(5/2))/(160*c^2*f*(c - c*Sin[e + f*x])^(9/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(960*c^
3*f*(c - c*Sin[e + f*x])^(7/2))

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)]

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 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]

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{4 c}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{40 c f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{20 c^2}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{40 c f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{160 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac {(A-3 B) \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx}{160 c^3}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{40 c f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{160 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{960 c^3 f (c-c \sin (e+f x))^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 5.62, size = 144, normalized size = 0.73 \[ \frac {a^2 \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (6 (6 A+7 B) \sin (e+f x)-15 (A+B) \cos (2 (e+f x))+29 A-10 B \sin (3 (e+f x))+13 B)}{120 c^6 f (\sin (e+f x)-1)^6 \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(13/2),x]

[Out]

(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(29*A + 13*B - 15*(A + B)*Cos[2*(e + f*x
)] + 6*(6*A + 7*B)*Sin[e + f*x] - 10*B*Sin[3*(e + f*x)]))/(120*c^6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1
+ Sin[e + f*x])^6*Sqrt[c - c*Sin[e + f*x]])

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fricas [A]  time = 0.47, size = 196, normalized size = 1.00 \[ \frac {{\left (15 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (11 \, A + 7 \, B\right )} a^{2} + 2 \, {\left (10 \, B a^{2} \cos \left (f x + e\right )^{2} - {\left (9 \, A + 13 \, B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{60 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} - 32 \, c^{7} f \cos \left (f x + e\right ) + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{5} - 16 \, c^{7} f \cos \left (f x + e\right )^{3} + 16 \, c^{7} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2),x, algorithm="fricas")

[Out]

1/60*(15*(A + B)*a^2*cos(f*x + e)^2 - 2*(11*A + 7*B)*a^2 + 2*(10*B*a^2*cos(f*x + e)^2 - (9*A + 13*B)*a^2)*sin(
f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^7*f*cos(f*x + e)^7 - 18*c^7*f*cos(f*x + e)^5 +
48*c^7*f*cos(f*x + e)^3 - 32*c^7*f*cos(f*x + e) + 2*(3*c^7*f*cos(f*x + e)^5 - 16*c^7*f*cos(f*x + e)^3 + 16*c^
7*f*cos(f*x + e))*sin(f*x + e))

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giac [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))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.72, size = 423, normalized size = 2.16 \[ \frac {\sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \left (444 A \sin \left (f x +e \right )-202 A \sin \left (f x +e \right ) \cos \left (f x +e \right )+29 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-444 A +52 B -B \left (\cos ^{6}\left (f x +e \right )\right )+46 B \sin \left (f x +e \right ) \cos \left (f x +e \right )+49 A \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-7 A \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )+B \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-75 B \left (\cos ^{2}\left (f x +e \right )\right )-17 B \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-7 B \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+119 A \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-52 B \sin \left (f x +e \right )-343 A \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+242 A \cos \left (f x +e \right )+24 B \left (\cos ^{4}\left (f x +e \right )\right )-6 B \cos \left (f x +e \right )-168 A \left (\cos ^{4}\left (f x +e \right )\right )+42 A \left (\cos ^{5}\left (f x +e \right )\right )-6 B \left (\cos ^{5}\left (f x +e \right )\right )-224 A \left (\cos ^{3}\left (f x +e \right )\right )+12 B \left (\cos ^{3}\left (f x +e \right )\right )+545 A \left (\cos ^{2}\left (f x +e \right )\right )+7 A \left (\cos ^{6}\left (f x +e \right )\right )\right )}{60 f \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {13}{2}} \left (\cos ^{3}\left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-3 \left (\cos ^{2}\left (f x +e \right )\right )-2 \sin \left (f x +e \right ) \cos \left (f x +e \right )-2 \cos \left (f x +e \right )+4 \sin \left (f x +e \right )+4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2),x)

[Out]

1/60/f*sin(f*x+e)*(a*(1+sin(f*x+e)))^(5/2)*(-17*B*cos(f*x+e)^3*sin(f*x+e)+444*A*sin(f*x+e)-202*A*sin(f*x+e)*co
s(f*x+e)+119*A*cos(f*x+e)^3*sin(f*x+e)+49*A*cos(f*x+e)^4*sin(f*x+e)-7*B*sin(f*x+e)*cos(f*x+e)^4-343*A*cos(f*x+
e)^2*sin(f*x+e)+29*B*cos(f*x+e)^2*sin(f*x+e)+24*B*cos(f*x+e)^4-444*A+52*B-224*A*cos(f*x+e)^3+46*B*sin(f*x+e)*c
os(f*x+e)+42*A*cos(f*x+e)^5-6*B*cos(f*x+e)^5-B*cos(f*x+e)^6+B*cos(f*x+e)^5*sin(f*x+e)-7*A*cos(f*x+e)^5*sin(f*x
+e)-75*B*cos(f*x+e)^2-168*A*cos(f*x+e)^4+12*B*cos(f*x+e)^3-52*B*sin(f*x+e)+242*A*cos(f*x+e)-6*B*cos(f*x+e)+7*A
*cos(f*x+e)^6+545*A*cos(f*x+e)^2)/(-c*(sin(f*x+e)-1))^(13/2)/(cos(f*x+e)^3-cos(f*x+e)^2*sin(f*x+e)-3*cos(f*x+e
)^2-2*sin(f*x+e)*cos(f*x+e)-2*cos(f*x+e)+4*sin(f*x+e)+4)

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maxima [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))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 20.71, size = 357, normalized size = 1.82 \[ \frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (A\,29{}\mathrm {i}+B\,13{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,16{}\mathrm {i}}{15\,c^7\,f}-\frac {a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\left (A\,1{}\mathrm {i}+B\,1{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,16{}\mathrm {i}}{c^7\,f}-\frac {32\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\left (6\,A+7\,B\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^7\,f}+\frac {32\,B\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,c^7\,f}\right )}{-858\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+858\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )-130\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )+2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (7\,e+7\,f\,x\right )+1144\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )-416\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )+24\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(13/2),x)

[Out]

((c - c*sin(e + f*x))^(1/2)*((a^2*exp(e*7i + f*x*7i)*(A*29i + B*13i)*(a + a*sin(e + f*x))^(1/2)*16i)/(15*c^7*f
) - (a^2*exp(e*7i + f*x*7i)*cos(2*e + 2*f*x)*(A*1i + B*1i)*(a + a*sin(e + f*x))^(1/2)*16i)/(c^7*f) - (32*a^2*e
xp(e*7i + f*x*7i)*sin(e + f*x)*(6*A + 7*B)*(a + a*sin(e + f*x))^(1/2))/(5*c^7*f) + (32*B*a^2*exp(e*7i + f*x*7i
)*sin(3*e + 3*f*x)*(a + a*sin(e + f*x))^(1/2))/(3*c^7*f)))/(858*exp(e*7i + f*x*7i)*cos(3*e + 3*f*x) - 858*cos(
e + f*x)*exp(e*7i + f*x*7i) - 130*exp(e*7i + f*x*7i)*cos(5*e + 5*f*x) + 2*exp(e*7i + f*x*7i)*cos(7*e + 7*f*x)
+ 1144*exp(e*7i + f*x*7i)*sin(2*e + 2*f*x) - 416*exp(e*7i + f*x*7i)*sin(4*e + 4*f*x) + 24*exp(e*7i + f*x*7i)*s
in(6*e + 6*f*x))

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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))**(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(13/2),x)

[Out]

Timed out

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