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]

1/14*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)/a/c/f/(c-c*sin(f*x+e))^(15/2)+1/84*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)/a/
c^2/f/(c-c*sin(f*x+e))^(13/2)+1/840*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)/a/c^3/f/(c-c*sin(f*x+e))^(11/2)

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Rubi [A]  time = 0.54, antiderivative size = 145, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [B]  time = 6.87, 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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fricas [A]  time = 0.51, size = 205, normalized size = 1.41 \[ -\frac {{\left (35 \, a^{3} \cos \left (f x + e\right )^{4} - 154 \, a^{3} \cos \left (f x + e\right )^{2} + 128 \, a^{3} - 14 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 8 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, {\left (7 \, c^{9} f \cos \left (f x + e\right )^{7} - 56 \, c^{9} f \cos \left (f x + e\right )^{5} + 112 \, c^{9} f \cos \left (f x + e\right )^{3} - 64 \, c^{9} f \cos \left (f x + e\right ) - {\left (c^{9} f \cos \left (f x + e\right )^{7} - 24 \, c^{9} f \cos \left (f x + e\right )^{5} + 80 \, c^{9} f \cos \left (f x + e\right )^{3} - 64 \, c^{9} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]

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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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(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]

Timed out

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maple [A]  time = 0.43, size = 240, normalized size = 1.66 \[ -\frac {\left (9 \left (\cos ^{6}\left (f x +e \right )\right )+63 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-216 \left (\cos ^{4}\left (f x +e \right )\right )-406 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+790 \left (\cos ^{2}\left (f x +e \right )\right )+448 \sin \left (f x +e \right )-688\right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )-\sin \left (f x +e \right ) \cos \left (f x +e \right )+\cos \left (f x +e \right )+2 \sin \left (f x +e \right )-2\right )}{105 f \left (\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )+\cos ^{4}\left (f x +e \right )-4 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \left (\cos ^{3}\left (f x +e \right )\right )-4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-8 \left (\cos ^{2}\left (f x +e \right )\right )+8 \sin \left (f x +e \right )-4 \cos \left (f x +e \right )+8\right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {17}{2}}} \]

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)*(cos(f*x+e)^2-sin(f*x+e)*cos(f*x+e)+cos(f*x+e)+2*s
in(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*(sin(f*x+e)-1))^(17/2)

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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(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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mupad [B]  time = 14.93, size = 764, normalized size = 5.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

-((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*((a^3*exp(e*4i + f*x*4i)*(a + a*((ex
p(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*16i)/(3*c^9*f) + (64*a^3*exp(e*5i + f*x*5i)*(a +
a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(3*c^9*f) - (a^3*exp(e*6i + f*x*6i)*(a + a
*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*1088i)/(15*c^9*f) - (576*a^3*exp(e*7i + f*x*
7i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(5*c^9*f) + (a^3*exp(e*8i + f*x*8
i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*5472i)/(35*c^9*f) + (576*a^3*exp(e*
9i + f*x*9i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(5*c^9*f) - (a^3*exp(e*1
0i + f*x*10i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*1088i)/(15*c^9*f) - (64*
a^3*exp(e*11i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(3*c^9*f) +
(a^3*exp(e*12i + f*x*12i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*16i)/(3*c^9*
f)))/(exp(e*1i + f*x*1i)*14i - 90*exp(e*2i + f*x*2i) - exp(e*3i + f*x*3i)*350i + 910*exp(e*4i + f*x*4i) + exp(
e*5i + f*x*5i)*1638i - 2002*exp(e*6i + f*x*6i) - exp(e*7i + f*x*7i)*1430i - exp(e*9i + f*x*9i)*1430i + 2002*ex
p(e*10i + f*x*10i) + exp(e*11i + f*x*11i)*1638i - 910*exp(e*12i + f*x*12i) - exp(e*13i + f*x*13i)*350i + 90*ex
p(e*14i + f*x*14i) + exp(e*15i + f*x*15i)*14i - exp(e*16i + f*x*16i) + 1)

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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(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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