3.564 \(\int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=203 \[ -\frac{5 a (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 f \sqrt{a \sin (e+f x)+a}}-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{5 \sqrt{a} (c+d)^3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{8 \sqrt{d} f} \]

[Out]

(-5*Sqrt[a]*(c + d)^3*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]
)])/(8*Sqrt[d]*f) - (5*a*(c + d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (5*
a*(c + d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*(c + d*Si
n[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])

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Rubi [A]  time = 0.414528, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2770, 2775, 205} \[ -\frac{5 a (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 f \sqrt{a \sin (e+f x)+a}}-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a \sin (e+f x)+a}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a \sin (e+f x)+a}}-\frac{5 \sqrt{a} (c+d)^3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{8 \sqrt{d} f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2),x]

[Out]

(-5*Sqrt[a]*(c + d)^3*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]
)])/(8*Sqrt[d]*f) - (5*a*(c + d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (5*
a*(c + d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*(c + d*Si
n[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])

Rule 2770

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

Rule 2775

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
(-2*b)/f, Subst[Int[1/(b + d*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}+\frac{1}{6} (5 (c+d)) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx\\ &=-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}+\frac{1}{8} \left (5 (c+d)^2\right ) \int \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx\\ &=-\frac{5 a (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 f \sqrt{a+a \sin (e+f x)}}-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}+\frac{1}{16} \left (5 (c+d)^3\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx\\ &=-\frac{5 a (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 f \sqrt{a+a \sin (e+f x)}}-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}-\frac{\left (5 a (c+d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+d x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{8 f}\\ &=-\frac{5 \sqrt{a} (c+d)^3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{8 \sqrt{d} f}-\frac{5 a (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{8 f \sqrt{a+a \sin (e+f x)}}-\frac{5 a (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 3.77205, size = 391, normalized size = 1.93 \[ -\frac{\sqrt{a (\sin (e+f x)+1)} \left (2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x)) \left (33 c^2+2 d (13 c+5 d) \sin (e+f x)+40 c d-4 d^2 \cos (2 (e+f x))+19 d^2\right )+\frac{15 (c+d)^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+i \cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (\log \left (\frac{e^{-i e} \left (2 \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}+2 \sqrt [4]{-1} c-2 (-1)^{3/4} d e^{i (e+f x)}\right )}{\sqrt{d}}\right )-\log \left (\frac{2 f e^{\frac{1}{2} i (e-2 f x)} \left (i \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}+\sqrt [4]{-1} c e^{i (e+f x)}+(-1)^{3/4} d\right )}{\sqrt{d}}\right )\right ) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt{d}}\right )}{48 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2),x]

[Out]

-(Sqrt[a*(1 + Sin[e + f*x])]*((15*(c + d)^3*(Log[(2*(-1)^(1/4)*c - 2*(-1)^(3/4)*d*E^(I*(e + f*x)) + 2*Sqrt[d]*
Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])/(Sqrt[d]*E^(I*e))] - Log[(2*E^((I/2)*(e - 2*f*x))*
((-1)^(3/4)*d + (-1)^(1/4)*c*E^(I*(e + f*x)) + I*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*
x)))])*f)/Sqrt[d]])*(I*Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e
+ f*x])])/Sqrt[d] + 2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])*(33*c^2 + 40*c*d + 19*d^2 - 4
*d^2*Cos[2*(e + f*x)] + 2*d*(13*c + 5*d)*Sin[e + f*x])))/(48*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c +
d*Sin[e + f*x]])

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+a\sin \left ( fx+e \right ) } \left ( c+d\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x)

[Out]

int((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(5/2), x)

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Fricas [B]  time = 5.82899, size = 3033, normalized size = 14.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

[1/192*(15*(c^3 + 3*c^2*d + 3*c*d^2 + d^3 + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*cos(f*x + e) + (c^3 + 3*c^2*d + 3*
c*d^2 + d^3)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*
d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3
- 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2
*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2
- (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^
3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e)
)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c
*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d
^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*
d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(8*
d^2*cos(f*x + e)^3 - 2*(13*c*d + d^2)*cos(f*x + e)^2 - 33*c^2 - 14*c*d - 13*d^2 - (33*c^2 + 40*c*d + 23*d^2)*c
os(f*x + e) - (8*d^2*cos(f*x + e)^2 - 33*c^2 - 14*c*d - 13*d^2 + 2*(13*c*d + 5*d^2)*cos(f*x + e))*sin(f*x + e)
)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(f*cos(f*x + e) + f*sin(f*x + e) + f), 1/96*(15*(c^3 + 3*
c^2*d + 3*c*d^2 + d^3 + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*cos(f*x + e) + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x
+ e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*s
in(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*s
in(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(8*d^2*cos(f*x + e)^3 - 2*(13*c*d + d^2)*cos(f*x +
e)^2 - 33*c^2 - 14*c*d - 13*d^2 - (33*c^2 + 40*c*d + 23*d^2)*cos(f*x + e) - (8*d^2*cos(f*x + e)^2 - 33*c^2 - 1
4*c*d - 13*d^2 + 2*(13*c*d + 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) +
c))/(f*cos(f*x + e) + f*sin(f*x + e) + f)]

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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))**(1/2)*(c+d*sin(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError