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

Optimal. Leaf size=145 \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}} \]

[Out]

(Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(14*a*c*f*(c - c*Sin[e + f*x])^(15/2)) + (Cos[e + f*x]*(a + a*Sin[e
+ f*x])^(9/2))/(84*a*c^2*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(840*a*c^3
*f*(c - c*Sin[e + f*x])^(11/2))

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Rubi [A]  time = 0.538983, antiderivative size = 145, normalized size of antiderivative = 1., 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 = {2841, 2743, 2742} \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(14*a*c*f*(c - c*Sin[e + f*x])^(15/2)) + (Cos[e + f*x]*(a + a*Sin[e
+ f*x])^(9/2))/(84*a*c^2*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(840*a*c^3
*f*(c - c*Sin[e + f*x])^(11/2))

Rule 2841

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[1/(a^(p/2)*c^(p/2)), Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(
n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p
/2]

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{\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{17/2}} \, dx &=\frac{\int \frac{(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{15/2}} \, dx}{a c}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac{\int \frac{(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{7 a c^2}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\int \frac{(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{84 a c^3}\\ &=\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{14 a c f (c-c \sin (e+f x))^{15/2}}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{84 a c^2 f (c-c \sin (e+f x))^{13/2}}+\frac{\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{840 a c^3 f (c-c \sin (e+f x))^{11/2}}\\ \end{align*}

Mathematica [B]  time = 6.83592, size = 419, normalized size = 2.89 \[ \frac{(a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}}{3 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{2 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}{f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{24 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}{5 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{16 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}{3 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{16 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}{7 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(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)) - (16*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(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])^(17/2)) + (24*(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)) - (2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(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)) + ((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(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])^(17/2))

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Maple [A]  time = 0.251, size = 243, normalized size = 1.7 \begin{align*}{\frac{ \left ( 9\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+63\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-216\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-406\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +790\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+448\,\sin \left ( fx+e \right ) -688 \right ) \sin \left ( fx+e \right ) \left ( \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) - \left ( \cos \left ( fx+e \right ) \right ) ^{2}-2\,\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +2 \right ) }{105\,f \left ( \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}+ \left ( \cos \left ( fx+e \right ) \right ) ^{4}-4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +3\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-4\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) -8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8\,\sin \left ( fx+e \right ) -4\,\cos \left ( fx+e \right ) +8 \right ) } \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{-{\frac{17}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105/f*(9*cos(f*x+e)^6+63*sin(f*x+e)*cos(f*x+e)^4-216*cos(f*x+e)^4-406*cos(f*x+e)^2*sin(f*x+e)+790*cos(f*x+e)
^2+448*sin(f*x+e)-688)*(a*(1+sin(f*x+e)))^(7/2)*sin(f*x+e)*(sin(f*x+e)*cos(f*x+e)-cos(f*x+e)^2-2*sin(f*x+e)-co
s(f*x+e)+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(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(17/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.99511, size = 517, normalized size = 3.57 \begin{align*} -\frac{{\left (35 \, a^{3} \cos \left (f x + e\right )^{4} - 154 \, a^{3} \cos \left (f x + e\right )^{2} + 128 \, a^{3} - 14 \,{\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 8 \, 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}}{105 \,{\left (7 \, c^{9} f \cos \left (f x + e\right )^{7} - 56 \, c^{9} f \cos \left (f x + e\right )^{5} + 112 \, c^{9} f \cos \left (f x + e\right )^{3} - 64 \, c^{9} f \cos \left (f x + e\right ) -{\left (c^{9} f \cos \left (f x + e\right )^{7} - 24 \, c^{9} f \cos \left (f x + e\right )^{5} + 80 \, c^{9} f \cos \left (f x + e\right )^{3} - 64 \, 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(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(17/2),x, algorithm="fricas")

[Out]

-1/105*(35*a^3*cos(f*x + e)^4 - 154*a^3*cos(f*x + e)^2 + 128*a^3 - 14*(5*a^3*cos(f*x + e)^2 - 8*a^3)*sin(f*x +
e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(7*c^9*f*cos(f*x + e)^7 - 56*c^9*f*cos(f*x + e)^5 + 11
2*c^9*f*cos(f*x + e)^3 - 64*c^9*f*cos(f*x + e) - (c^9*f*cos(f*x + e)^7 - 24*c^9*f*cos(f*x + e)^5 + 80*c^9*f*co
s(f*x + e)^3 - 64*c^9*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(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)/(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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{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(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(17/2),x, algorithm="giac")

[Out]

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