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

Optimal. Leaf size=377 \[ -\frac{a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{128 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{5/2} (c+d)^3 \left (3 c^2-34 c d+283 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{128 d^{5/2} f}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f} \]

[Out]

-(a^(5/2)*(c + d)^3*(3*c^2 - 34*c*d + 283*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]
*Sqrt[c + d*Sin[e + f*x]])])/(128*d^(5/2)*f) - (a^3*(c + d)^2*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*Sqrt[c +
d*Sin[e + f*x]])/(128*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(c + d)*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*
(c + d*Sin[e + f*x])^(3/2))/(192*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x
]*(c + d*Sin[e + f*x])^(5/2))/(240*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*a^3*(c - 7*d)*Cos[e + f*x]*(c + d*Sin[
e + f*x])^(7/2))/(40*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e
+ f*x])^(7/2))/(5*d*f)

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Rubi [A]  time = 0.854936, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2763, 2981, 2770, 2775, 205} \[ -\frac{a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{128 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^{5/2} (c+d)^3 \left (3 c^2-34 c d+283 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{128 d^{5/2} f}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}{5 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^(5/2)*(c + d)^3*(3*c^2 - 34*c*d + 283*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]
*Sqrt[c + d*Sin[e + f*x]])])/(128*d^(5/2)*f) - (a^3*(c + d)^2*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*Sqrt[c +
d*Sin[e + f*x]])/(128*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(c + d)*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*
(c + d*Sin[e + f*x])^(3/2))/(192*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x
]*(c + d*Sin[e + f*x])^(5/2))/(240*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*a^3*(c - 7*d)*Cos[e + f*x]*(c + d*Sin[
e + f*x])^(7/2))/(40*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e
+ f*x])^(7/2))/(5*d*f)

Rule 2763

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*
(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(
m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m, 2*
n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

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 (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac{\int \sqrt{a+a \sin (e+f x)} \left (\frac{1}{2} a^2 (c+17 d)-\frac{3}{2} a^2 (c-7 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2} \, dx}{5 d}\\ &=\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac{\left (a^2 \left (3 c^2-34 c d+283 d^2\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx}{80 d^2}\\ &=-\frac{a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac{\left (a^2 (c+d) \left (3 c^2-34 c d+283 d^2\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx}{96 d^2}\\ &=-\frac{a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac{\left (a^2 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right )\right ) \int \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx}{128 d^2}\\ &=-\frac{a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{128 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac{\left (a^2 (c+d)^3 \left (3 c^2-34 c d+283 d^2\right )\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx}{256 d^2}\\ &=-\frac{a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{128 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}-\frac{\left (a^3 (c+d)^3 \left (3 c^2-34 c d+283 d^2\right )\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 )}{128 d^2 f}\\ &=-\frac{a^{5/2} (c+d)^3 \left (3 c^2-34 c d+283 d^2\right ) \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 )}{128 d^{5/2} f}-\frac{a^3 (c+d)^2 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{128 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^3 (c+d) \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{192 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^3 \left (3 c^2-34 c d+283 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{240 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{3 a^3 (c-7 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{40 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\\ \end{align*}

Mathematica [A]  time = 2.96373, size = 395, normalized size = 1.05 \[ \frac{(a (\sin (e+f x)+1))^{5/2} \left (\frac{2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{c+d \sin (e+f x)} \left (-3322 c^2 d^2 \sin (e+f x)+4 d^2 \left (93 c^2+488 c d+331 d^2\right ) \cos (2 (e+f x))-8396 c^2 d^2-30 c^3 d \sin (e+f x)-390 c^3 d+45 c^4-7774 c d^3 \sin (e+f x)+252 c d^3 \sin (3 (e+f x))-12762 c d^3-3874 d^4 \sin (e+f x)+348 d^4 \sin (3 (e+f x))-48 d^4 \cos (4 (e+f x))-5521 d^4\right )}{15 d^2}+\frac{\left (3 c^2-34 c d+283 d^2\right ) (c+d)^3 \left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sin \left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{\sqrt{c+d \sin (e+f x)}}\right )-\log \left (\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )+\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \cos \left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{\sqrt{c+d \sin (e+f x)}}\right )\right )}{d^{5/2}}\right )}{256 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*(1 + Sin[e + f*x]))^(5/2)*(((c + d)^3*(3*c^2 - 34*c*d + 283*d^2)*(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi
+ 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e
+ f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(5/2) + (2*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(45*c^4 - 390*c^3*d - 8396*c^2*d^2 - 12762*c*d^3 - 5521*d
^4 + 4*d^2*(93*c^2 + 488*c*d + 331*d^2)*Cos[2*(e + f*x)] - 48*d^4*Cos[4*(e + f*x)] - 30*c^3*d*Sin[e + f*x] - 3
322*c^2*d^2*Sin[e + f*x] - 7774*c*d^3*Sin[e + f*x] - 3874*d^4*Sin[e + f*x] + 252*c*d^3*Sin[3*(e + f*x)] + 348*
d^4*Sin[3*(e + f*x)]))/(15*d^2)))/(256*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)

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

[Out]

int((a+a*sin(f*x+e))^(5/2)*(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{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}{\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))^(5/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 21.0004, size = 4905, normalized size = 13.01 \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))^(5/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

[1/15360*(15*(3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5 + (3*
a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*cos(f*x + e) + (3*a^
2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*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 - 2
5*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)*co
s(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*s
in(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) + (1
28*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*(384*a^2*d^4*cos(f*x + e)^5 - 45*a^2*c
^4 + 360*a^2*c^3*d + 5446*a^2*c^2*d^2 + 6688*a^2*c*d^3 + 2671*a^2*d^4 - 1008*(a^2*c*d^3 + a^2*d^4)*cos(f*x + e
)^4 - 8*(93*a^2*c^2*d^2 + 488*a^2*c*d^3 + 379*a^2*d^4)*cos(f*x + e)^3 + 2*(15*a^2*c^3*d + 1289*a^2*c^2*d^2 + 2
565*a^2*c*d^3 + 1291*a^2*d^4)*cos(f*x + e)^2 - (45*a^2*c^4 - 390*a^2*c^3*d - 8768*a^2*c^2*d^2 - 14714*a^2*c*d^
3 - 6893*a^2*d^4)*cos(f*x + e) - (384*a^2*d^4*cos(f*x + e)^4 - 45*a^2*c^4 + 360*a^2*c^3*d + 5446*a^2*c^2*d^2 +
6688*a^2*c*d^3 + 2671*a^2*d^4 + 48*(21*a^2*c*d^3 + 29*a^2*d^4)*cos(f*x + e)^3 - 8*(93*a^2*c^2*d^2 + 362*a^2*c
*d^3 + 205*a^2*d^4)*cos(f*x + e)^2 - 2*(15*a^2*c^3*d + 1661*a^2*c^2*d^2 + 4013*a^2*c*d^3 + 2111*a^2*d^4)*cos(f
*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x
+ e) + d^2*f), 1/7680*(15*(3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*
a^2*d^5 + (3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*cos(f*x
+ e) + (3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*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*sin
(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)*sin
(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*(384*a^2*d^4*cos(f*x + e)^5 - 45*a^2*c^4 + 360*a^2*c^
3*d + 5446*a^2*c^2*d^2 + 6688*a^2*c*d^3 + 2671*a^2*d^4 - 1008*(a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^4 - 8*(93*a^2
*c^2*d^2 + 488*a^2*c*d^3 + 379*a^2*d^4)*cos(f*x + e)^3 + 2*(15*a^2*c^3*d + 1289*a^2*c^2*d^2 + 2565*a^2*c*d^3 +
1291*a^2*d^4)*cos(f*x + e)^2 - (45*a^2*c^4 - 390*a^2*c^3*d - 8768*a^2*c^2*d^2 - 14714*a^2*c*d^3 - 6893*a^2*d^
4)*cos(f*x + e) - (384*a^2*d^4*cos(f*x + e)^4 - 45*a^2*c^4 + 360*a^2*c^3*d + 5446*a^2*c^2*d^2 + 6688*a^2*c*d^3
+ 2671*a^2*d^4 + 48*(21*a^2*c*d^3 + 29*a^2*d^4)*cos(f*x + e)^3 - 8*(93*a^2*c^2*d^2 + 362*a^2*c*d^3 + 205*a^2*
d^4)*cos(f*x + e)^2 - 2*(15*a^2*c^3*d + 1661*a^2*c^2*d^2 + 4013*a^2*c*d^3 + 2111*a^2*d^4)*cos(f*x + e))*sin(f*
x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*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))**(5/2)*(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: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError