3.166 \(\int \frac{(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=271 \[ \frac{2 a^3 (3 A+5 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a^2 (3 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^4 (3 A+5 B) \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a (3 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}} \]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2*f*(c - c*Sin[e + f*x])^(3/2)) + (4*a^4*(3*A + 5*B)*Cos[e
+ f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^3*(3*A + 5*B)*Cos
[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) + (a^2*(3*A + 5*B)*Cos[e + f*x]*(a + a*Sin[
e + f*x])^(3/2))/(2*c*f*Sqrt[c - c*Sin[e + f*x]]) + (a*(3*A + 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6
*c*f*Sqrt[c - c*Sin[e + f*x]])

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Rubi [A]  time = 0.593029, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2972, 2740, 2737, 2667, 31} \[ \frac{2 a^3 (3 A+5 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a^2 (3 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{4 a^4 (3 A+5 B) \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{a (3 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}+\frac{(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}} \]

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])^(3/2),x]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2*f*(c - c*Sin[e + f*x])^(3/2)) + (4*a^4*(3*A + 5*B)*Cos[e
+ f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^3*(3*A + 5*B)*Cos
[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) + (a^2*(3*A + 5*B)*Cos[e + f*x]*(a + a*Sin[
e + f*x])^(3/2))/(2*c*f*Sqrt[c - c*Sin[e + f*x]]) + (a*(3*A + 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6
*c*f*Sqrt[c - c*Sin[e + f*x]])

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 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 2737

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

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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))^{3/2}} \, dx &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}}-\frac{(3 A+5 B) \int \frac{(a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx}{2 c}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}}+\frac{a (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}-\frac{(a (3 A+5 B)) \int \frac{(a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}}+\frac{a^2 (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{a (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}-\frac{\left (2 a^2 (3 A+5 B)\right ) \int \frac{(a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (3 A+5 B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a^2 (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{a (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}-\frac{\left (4 a^3 (3 A+5 B)\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (3 A+5 B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a^2 (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{a (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}-\frac{\left (4 a^4 (3 A+5 B) \cos (e+f x)\right ) \int \frac{\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (3 A+5 B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a^2 (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{a (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}+\frac{\left (4 a^4 (3 A+5 B) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 f (c-c \sin (e+f x))^{3/2}}+\frac{4 a^4 (3 A+5 B) \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 a^3 (3 A+5 B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{c f \sqrt{c-c \sin (e+f x)}}+\frac{a^2 (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c f \sqrt{c-c \sin (e+f x)}}+\frac{a (3 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 c f \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 3.56677, size = 292, normalized size = 1.08 \[ \frac{a^3 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (-2 (27 A+59 B) \cos (2 (e+f x))-117 A \sin (e+f x)-3 A \sin (3 (e+f x))-576 A \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+576 A \sin (e+f x) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-132 A-279 B \sin (e+f x)-13 B \sin (3 (e+f x))+B \cos (4 (e+f x))-960 B \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+960 B \sin (e+f x) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-45 B\right )}{24 c f (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2),x]

[Out]

(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-132*A - 45*B - 2*(27*A + 59*B)*Cos[2*(
e + f*x)] + B*Cos[4*(e + f*x)] - 576*A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 960*B*Log[Cos[(e + f*x)/2] -
Sin[(e + f*x)/2]] - 117*A*Sin[e + f*x] - 279*B*Sin[e + f*x] + 576*A*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*
Sin[e + f*x] + 960*B*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 3*A*Sin[3*(e + f*x)] - 13*B*Sin[3
*(e + f*x)]))/(24*c*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])

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Maple [B]  time = 0.281, size = 927, normalized size = 3.4 \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))^(3/2),x)

[Out]

-1/6/f*(-102*A-166*B+102*A*sin(f*x+e)+99*A*cos(f*x+e)^2-24*A*cos(f*x+e)^2*sin(f*x+e)+72*A*cos(f*x+e)*ln(2/(cos
(f*x+e)+1))+120*B*cos(f*x+e)^2*ln(2/(cos(f*x+e)+1))+2*B*sin(f*x+e)*cos(f*x+e)^4-3*A*cos(f*x+e)^3*sin(f*x+e)-48
*B*cos(f*x+e)^2*sin(f*x+e)+27*A*cos(f*x+e)+144*A*cos(f*x+e)*sin(f*x+e)*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+
e))-72*A*cos(f*x+e)*sin(f*x+e)*ln(2/(cos(f*x+e)+1))-13*B*cos(f*x+e)^3*sin(f*x+e)-144*A*cos(f*x+e)*ln(-(-1+cos(
f*x+e)+sin(f*x+e))/sin(f*x+e))-240*B*cos(f*x+e)^2*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-27*A*cos(f*x+e)^3
-61*B*cos(f*x+e)^3+59*B*cos(f*x+e)-120*B*ln(2/(cos(f*x+e)+1))*sin(f*x+e)*cos(f*x+e)+240*B*ln(-(-1+cos(f*x+e)+s
in(f*x+e))/sin(f*x+e))*sin(f*x+e)*cos(f*x+e)+144*A*sin(f*x+e)*ln(2/(cos(f*x+e)+1))-288*A*sin(f*x+e)*ln(-(-1+co
s(f*x+e)+sin(f*x+e))/sin(f*x+e))-107*B*sin(f*x+e)*cos(f*x+e)+120*B*cos(f*x+e)*ln(2/(cos(f*x+e)+1))-240*B*cos(f
*x+e)*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))+240*B*sin(f*x+e)*ln(2/(cos(f*x+e)+1))-480*B*sin(f*x+e)*ln(-(-
1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-75*A*sin(f*x+e)*cos(f*x+e)-144*A*cos(f*x+e)^2*ln(-(-1+cos(f*x+e)+sin(f*x+
e))/sin(f*x+e))+72*A*cos(f*x+e)^2*ln(2/(cos(f*x+e)+1))+2*B*cos(f*x+e)^5+3*A*cos(f*x+e)^4+11*B*cos(f*x+e)^4+155
*B*cos(f*x+e)^2-144*A*ln(2/(cos(f*x+e)+1))+288*A*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))-240*B*ln(2/(cos(f*
x+e)+1))+480*B*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))+166*B*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)))^(3/2)

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Maxima [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{3}{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))^(3/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (B a^{3} \cos \left (f x + e\right )^{4} -{\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \,{\left (A + B\right )} a^{3} -{\left ({\left (A + 3 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 4 \,{\left (A + 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}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c^{2} \sin \left (f x + e\right ) - 2 \, c^{2}}, x\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))^(3/2),x, algorithm="fricas")

[Out]

integral(-(B*a^3*cos(f*x + e)^4 - (3*A + 5*B)*a^3*cos(f*x + e)^2 + 4*(A + B)*a^3 - ((A + 3*B)*a^3*cos(f*x + e)
^2 - 4*(A + B)*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^2*cos(f*x + e)^2 + 2*c
^2*sin(f*x + e) - 2*c^2), x)

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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))**(3/2),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{3}{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))^(3/2),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)^(3/2), x)