3.306 \(\int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=308 \[ \frac{a^3 \left (3 A d (c+3 d)-B \left (15 c^2+25 c d+4 d^2\right )\right ) \cos (e+f x)}{4 d^3 f (c+d)^2 \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \left (A d (c+7 d)-B \left (5 c^2+7 c d-4 d^2\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 d^2 f (c+d)^2 (c+d \sin (e+f x))}-\frac{a^{5/2} \left (A d \left (3 c^2+10 c d+19 d^2\right )-B \left (30 c^2 d+15 c^3+7 c d^2-20 d^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{4 d^{7/2} f (c+d)^{5/2}}+\frac{a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 d f (c+d) (c+d \sin (e+f x))^2} \]

[Out]

-(a^(5/2)*(A*d*(3*c^2 + 10*c*d + 19*d^2) - B*(15*c^3 + 30*c^2*d + 7*c*d^2 - 20*d^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*
Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(7/2)*(c + d)^(5/2)*f) + (a^3*(3*A*d*(c + 3*d) - B
*(15*c^2 + 25*c*d + 4*d^2))*Cos[e + f*x])/(4*d^3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]) + (a*(B*c - A*d)*Cos[e
+ f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a^2*(A*d*(c + 7*d) - B*(5*c^2 + 7
*c*d - 4*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))

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Rubi [A]  time = 0.97193, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {2975, 2981, 2773, 208} \[ \frac{a^3 \left (3 A d (c+3 d)-B \left (15 c^2+25 c d+4 d^2\right )\right ) \cos (e+f x)}{4 d^3 f (c+d)^2 \sqrt{a \sin (e+f x)+a}}-\frac{a^2 \left (A d (c+7 d)-B \left (5 c^2+7 c d-4 d^2\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{4 d^2 f (c+d)^2 (c+d \sin (e+f x))}-\frac{a^{5/2} \left (A d \left (3 c^2+10 c d+19 d^2\right )-B \left (30 c^2 d+15 c^3+7 c d^2-20 d^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{4 d^{7/2} f (c+d)^{5/2}}+\frac{a (B c-A d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 d f (c+d) (c+d \sin (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]

[Out]

-(a^(5/2)*(A*d*(3*c^2 + 10*c*d + 19*d^2) - B*(15*c^3 + 30*c^2*d + 7*c*d^2 - 20*d^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*
Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(7/2)*(c + d)^(5/2)*f) + (a^3*(3*A*d*(c + 3*d) - B
*(15*c^2 + 25*c*d + 4*d^2))*Cos[e + f*x])/(4*d^3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]) + (a*(B*c - A*d)*Cos[e
+ f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a^2*(A*d*(c + 7*d) - B*(5*c^2 + 7
*c*d - 4*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))

Rule 2975

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*S
in[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])
^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n +
1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d
, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n]
|| 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 2773

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(-2*
b)/f, Subst[Int[1/(b*c + a*d - d*x^2), x], x, (b*Cos[e + f*x])/Sqrt[a + b*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 208

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

Rubi steps

\begin{align*} \int \frac{(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx &=\frac{a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac{\int \frac{(a+a \sin (e+f x))^{3/2} \left (-\frac{1}{2} a (3 B c-7 A d-4 B d)+\frac{1}{2} a (5 B c-A d+4 B d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)}\\ &=\frac{a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac{a^2 \left (A d (c+7 d)-B \left (5 c^2+7 c d-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac{\int \frac{\sqrt{a+a \sin (e+f x)} \left (\frac{1}{4} a^2 \left (A d (c+19 d)-B \left (5 c^2+3 c d-20 d^2\right )\right )-\frac{1}{4} a^2 \left (3 A d (c+3 d)-B \left (15 c^2+25 c d+4 d^2\right )\right ) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{2 d^2 (c+d)^2}\\ &=\frac{a^3 \left (3 A d (c+3 d)-B \left (15 c^2+25 c d+4 d^2\right )\right ) \cos (e+f x)}{4 d^3 (c+d)^2 f \sqrt{a+a \sin (e+f x)}}+\frac{a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac{a^2 \left (A d (c+7 d)-B \left (5 c^2+7 c d-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 d^2 (c+d)^2 f (c+d \sin (e+f x))}+\frac{\left (a^2 \left (A d \left (3 c^2+10 c d+19 d^2\right )-B \left (15 c^3+30 c^2 d+7 c d^2-20 d^3\right )\right )\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{8 d^3 (c+d)^2}\\ &=\frac{a^3 \left (3 A d (c+3 d)-B \left (15 c^2+25 c d+4 d^2\right )\right ) \cos (e+f x)}{4 d^3 (c+d)^2 f \sqrt{a+a \sin (e+f x)}}+\frac{a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac{a^2 \left (A d (c+7 d)-B \left (5 c^2+7 c d-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 d^2 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\left (a^3 \left (A d \left (3 c^2+10 c d+19 d^2\right )-B \left (15 c^3+30 c^2 d+7 c d^2-20 d^3\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a c+a d-d x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{4 d^3 (c+d)^2 f}\\ &=-\frac{a^{5/2} \left (A d \left (3 c^2+10 c d+19 d^2\right )-B \left (15 c^3+30 c^2 d+7 c d^2-20 d^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a+a \sin (e+f x)}}\right )}{4 d^{7/2} (c+d)^{5/2} f}+\frac{a^3 \left (3 A d (c+3 d)-B \left (15 c^2+25 c d+4 d^2\right )\right ) \cos (e+f x)}{4 d^3 (c+d)^2 f \sqrt{a+a \sin (e+f x)}}+\frac{a (B c-A d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac{a^2 \left (A d (c+7 d)-B \left (5 c^2+7 c d-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{4 d^2 (c+d)^2 f (c+d \sin (e+f x))}\\ \end{align*}

Mathematica [A]  time = 8.11959, size = 504, normalized size = 1.64 \[ \frac{(a (\sin (e+f x)+1))^{5/2} \left (-\frac{4 \sqrt{d} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (d \left (A d \left (-5 c^2-6 c d+11 d^2\right )+B \left (34 c^2 d+25 c^3+c d^2+4 d^3\right )\right ) \sin (e+f x)-8 A c^2 d^2-3 A c^3 d+9 A c d^3+2 A d^4-B c^2 d^2+20 B c^3 d+15 B c^4-4 B d^2 (c+d)^2 \cos (2 (e+f x))+10 B c d^3+4 B d^4\right )}{(c+d)^2 (c+d \sin (e+f x))^2}+\frac{\left (A d \left (3 c^2+10 c d+19 d^2\right )-B \left (30 c^2 d+15 c^3+7 c d^2-20 d^3\right )\right ) \left (2 \log \left (\sqrt{d} \sqrt{c+d} \left (\tan ^2\left (\frac{1}{4} (e+f x)\right )+2 \tan \left (\frac{1}{4} (e+f x)\right )-1\right )+(c+d) \sec ^2\left (\frac{1}{4} (e+f x)\right )\right )-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+e+f x\right )}{(c+d)^{5/2}}+\frac{\left (B \left (30 c^2 d+15 c^3+7 c d^2-20 d^3\right )-A d \left (3 c^2+10 c d+19 d^2\right )\right ) \left (2 \log \left (-\sec ^2\left (\frac{1}{4} (e+f x)\right ) \left (-\sqrt{d} \sqrt{c+d} \sin \left (\frac{1}{2} (e+f x)\right )+\sqrt{d} \sqrt{c+d} \cos \left (\frac{1}{2} (e+f x)\right )+c+d\right )\right )-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+e+f x\right )}{(c+d)^{5/2}}\right )}{16 d^{7/2} f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3,x]

[Out]

((a*(1 + Sin[e + f*x]))^(5/2)*(((-(A*d*(3*c^2 + 10*c*d + 19*d^2)) + B*(15*c^3 + 30*c^2*d + 7*c*d^2 - 20*d^3))*
(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[-(Sec[(e + f*x)/4]^2*(c + d + Sqrt[d]*Sqrt[c + d]*Cos[(e + f*x)/2
] - Sqrt[d]*Sqrt[c + d]*Sin[(e + f*x)/2]))]))/(c + d)^(5/2) + ((A*d*(3*c^2 + 10*c*d + 19*d^2) - B*(15*c^3 + 30
*c^2*d + 7*c*d^2 - 20*d^3))*(e + f*x - 2*Log[Sec[(e + f*x)/4]^2] + 2*Log[(c + d)*Sec[(e + f*x)/4]^2 + Sqrt[d]*
Sqrt[c + d]*(-1 + 2*Tan[(e + f*x)/4] + Tan[(e + f*x)/4]^2)]))/(c + d)^(5/2) - (4*Sqrt[d]*(Cos[(e + f*x)/2] - S
in[(e + f*x)/2])*(15*B*c^4 - 3*A*c^3*d + 20*B*c^3*d - 8*A*c^2*d^2 - B*c^2*d^2 + 9*A*c*d^3 + 10*B*c*d^3 + 2*A*d
^4 + 4*B*d^4 - 4*B*d^2*(c + d)^2*Cos[2*(e + f*x)] + d*(A*d*(-5*c^2 - 6*c*d + 11*d^2) + B*(25*c^3 + 34*c^2*d +
c*d^2 + 4*d^3))*Sin[e + f*x]))/((c + d)^2*(c + d*Sin[e + f*x])^2)))/(16*d^(7/2)*f*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])^5)

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Maple [B]  time = 2.445, size = 1587, normalized size = 5.2 \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))^(5/2)*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3,x)

[Out]

-1/4*a*(8*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a*d^4+8*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*
(c+d)*d)^(1/2)*sin(f*x+e)^2*a*c^2*d^2+16*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)^2*a*c*d^3+1
6*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a*c^3*d+32*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d
)^(1/2)*sin(f*x+e)*a*c^2*d^2+16*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*sin(f*x+e)*a*c*d^3-15*a^2*arcta
nh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*B*c^5-11*A*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*d^4
-4*B*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*d^4-30*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1
/2))*sin(f*x+e)*a^2*c^4*d-60*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*a^2*c^3*d^2+
40*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*a^2*c*d^4-3*A*(-a*(-1+sin(f*x+e)))^(1/
2)*(a*(c+d)*d)^(1/2)*a*c^3*d-13*A*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a*c^2*d^2+3*A*(-a*(-1+sin(f*x+e
)))^(1/2)*(a*(c+d)*d)^(1/2)*a*c*d^3+29*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a*c^3*d-3*B*(-a*(-1+sin(
f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a*c^2*d^2-13*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a*c*d^3+38*A*arct
anh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*a^2*c*d^4+3*A*arctanh((-a*(-1+sin(f*x+e)))^(1/2
)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*a^2*c^2*d^3+10*A*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*s
in(f*x+e)^2*a^2*c*d^4-15*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*a^2*c^3*d^2-30
*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*a^2*c^2*d^3-7*B*arctanh((-a*(-1+sin(f*
x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*a^2*c*d^4+6*A*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^
(1/2))*sin(f*x+e)*a^2*c^3*d^2+20*A*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*a^2*c^2*
d^3-14*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)*a^2*c^2*d^3+3*A*arctanh((-a*(-1+si
n(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^2*c^4*d+10*A*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a
^2*c^3*d^2+19*A*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^2*c^2*d^3-9*B*(-a*(-1+sin(f*x+e)))^(
3/2)*(a*(c+d)*d)^(1/2)*c^3*d-2*B*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*c^2*d^2+19*A*arctanh((-a*(-1+sin
(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*sin(f*x+e)^2*a^2*d^5+20*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d
)^(1/2))*sin(f*x+e)^2*a^2*d^5+5*A*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*c^2*d^2+6*A*(-a*(-1+sin(f*x+e))
)^(3/2)*(a*(c+d)*d)^(1/2)*c*d^3+15*B*(-a*(-1+sin(f*x+e)))^(3/2)*(a*(c+d)*d)^(1/2)*c*d^3+13*A*(-a*(-1+sin(f*x+e
)))^(1/2)*(a*(c+d)*d)^(1/2)*a*d^4+15*B*(-a*(-1+sin(f*x+e)))^(1/2)*(a*(c+d)*d)^(1/2)*a*c^4+4*B*(-a*(-1+sin(f*x+
e)))^(1/2)*(a*(c+d)*d)^(1/2)*a*d^4-30*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^2*c^4*d-7*B*
arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/(a*(c+d)*d)^(1/2))*a^2*c^3*d^2+20*B*arctanh((-a*(-1+sin(f*x+e)))^(1/2)*d/
(a*(c+d)*d)^(1/2))*a^2*c^2*d^3)*(-a*(-1+sin(f*x+e)))^(1/2)*(1+sin(f*x+e))/(a*(c+d)*d)^(1/2)/(c+d*sin(f*x+e))^2
/(c+d)^2/d^3/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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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((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 21.8934, size = 6692, normalized size = 21.73 \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)*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/16*((15*B*a^2*c^5 - 3*(A - 20*B)*a^2*c^4*d - 2*(8*A - 41*B)*a^2*c^3*d^2 - 6*(7*A - 4*B)*a^2*c^2*d^3 - 3*(16
*A + 11*B)*a^2*c*d^4 - (19*A + 20*B)*a^2*d^5 - (15*B*a^2*c^3*d^2 - 3*(A - 10*B)*a^2*c^2*d^3 - (10*A - 7*B)*a^2
*c*d^4 - (19*A + 20*B)*a^2*d^5)*cos(f*x + e)^3 - (30*B*a^2*c^4*d - 3*(2*A - 25*B)*a^2*c^3*d^2 - (23*A - 44*B)*
a^2*c^2*d^3 - 3*(16*A + 11*B)*a^2*c*d^4 - (19*A + 20*B)*a^2*d^5)*cos(f*x + e)^2 + (15*B*a^2*c^5 - 3*(A - 10*B)
*a^2*c^4*d - 2*(5*A - 11*B)*a^2*c^3*d^2 - 2*(11*A - 5*B)*a^2*c^2*d^3 - (10*A - 7*B)*a^2*c*d^4 - (19*A + 20*B)*
a^2*d^5)*cos(f*x + e) + (15*B*a^2*c^5 - 3*(A - 20*B)*a^2*c^4*d - 2*(8*A - 41*B)*a^2*c^3*d^2 - 6*(7*A - 4*B)*a^
2*c^2*d^3 - 3*(16*A + 11*B)*a^2*c*d^4 - (19*A + 20*B)*a^2*d^5 - (15*B*a^2*c^3*d^2 - 3*(A - 10*B)*a^2*c^2*d^3 -
(10*A - 7*B)*a^2*c*d^4 - (19*A + 20*B)*a^2*d^5)*cos(f*x + e)^2 + 2*(15*B*a^2*c^4*d - 3*(A - 10*B)*a^2*c^3*d^2
- (10*A - 7*B)*a^2*c^2*d^3 - (19*A + 20*B)*a^2*c*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a/(c*d + d^2))*log((a*
d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3
- (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 +
d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*c
os(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x +
e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*c
os(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(15*B*a^2*c^4 - (3*A + 5*B)*a^2*c^3
*d - (3*A + 31*B)*a^2*c^2*d^2 + (15*A + 17*B)*a^2*c*d^3 - (9*A - 4*B)*a^2*d^4 - 8*(B*a^2*c^2*d^2 + 2*B*a^2*c*d
^3 + B*a^2*d^4)*cos(f*x + e)^3 + (25*B*a^2*c^3*d - (5*A - 26*B)*a^2*c^2*d^2 - 3*(2*A + 5*B)*a^2*c*d^3 + (11*A
- 4*B)*a^2*d^4)*cos(f*x + e)^2 + (15*B*a^2*c^4 - (3*A - 20*B)*a^2*c^3*d - (8*A - 3*B)*a^2*c^2*d^2 + 9*(A + 2*B
)*a^2*c*d^3 + 2*(A + 4*B)*a^2*d^4)*cos(f*x + e) - (15*B*a^2*c^4 - (3*A + 5*B)*a^2*c^3*d - (3*A + 31*B)*a^2*c^2
*d^2 + (15*A + 17*B)*a^2*c*d^3 - (9*A - 4*B)*a^2*d^4 - 8*(B*a^2*c^2*d^2 + 2*B*a^2*c*d^3 + B*a^2*d^4)*cos(f*x +
e)^2 - (25*B*a^2*c^3*d - (5*A - 34*B)*a^2*c^2*d^2 - (6*A - B)*a^2*c*d^3 + (11*A + 4*B)*a^2*d^4)*cos(f*x + e))
*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c^2*d^5 + 2*c*d^6 + d^7)*f*cos(f*x + e)^3 + (2*c^3*d^4 + 5*c^2*d^5
+ 4*c*d^6 + d^7)*f*cos(f*x + e)^2 - (c^4*d^3 + 2*c^3*d^4 + 2*c^2*d^5 + 2*c*d^6 + d^7)*f*cos(f*x + e) - (c^4*d^
3 + 4*c^3*d^4 + 6*c^2*d^5 + 4*c*d^6 + d^7)*f + ((c^2*d^5 + 2*c*d^6 + d^7)*f*cos(f*x + e)^2 - 2*(c^3*d^4 + 2*c^
2*d^5 + c*d^6)*f*cos(f*x + e) - (c^4*d^3 + 4*c^3*d^4 + 6*c^2*d^5 + 4*c*d^6 + d^7)*f)*sin(f*x + e)), -1/8*((15*
B*a^2*c^5 - 3*(A - 20*B)*a^2*c^4*d - 2*(8*A - 41*B)*a^2*c^3*d^2 - 6*(7*A - 4*B)*a^2*c^2*d^3 - 3*(16*A + 11*B)*
a^2*c*d^4 - (19*A + 20*B)*a^2*d^5 - (15*B*a^2*c^3*d^2 - 3*(A - 10*B)*a^2*c^2*d^3 - (10*A - 7*B)*a^2*c*d^4 - (1
9*A + 20*B)*a^2*d^5)*cos(f*x + e)^3 - (30*B*a^2*c^4*d - 3*(2*A - 25*B)*a^2*c^3*d^2 - (23*A - 44*B)*a^2*c^2*d^3
- 3*(16*A + 11*B)*a^2*c*d^4 - (19*A + 20*B)*a^2*d^5)*cos(f*x + e)^2 + (15*B*a^2*c^5 - 3*(A - 10*B)*a^2*c^4*d
- 2*(5*A - 11*B)*a^2*c^3*d^2 - 2*(11*A - 5*B)*a^2*c^2*d^3 - (10*A - 7*B)*a^2*c*d^4 - (19*A + 20*B)*a^2*d^5)*co
s(f*x + e) + (15*B*a^2*c^5 - 3*(A - 20*B)*a^2*c^4*d - 2*(8*A - 41*B)*a^2*c^3*d^2 - 6*(7*A - 4*B)*a^2*c^2*d^3 -
3*(16*A + 11*B)*a^2*c*d^4 - (19*A + 20*B)*a^2*d^5 - (15*B*a^2*c^3*d^2 - 3*(A - 10*B)*a^2*c^2*d^3 - (10*A - 7*
B)*a^2*c*d^4 - (19*A + 20*B)*a^2*d^5)*cos(f*x + e)^2 + 2*(15*B*a^2*c^4*d - 3*(A - 10*B)*a^2*c^3*d^2 - (10*A -
7*B)*a^2*c^2*d^3 - (19*A + 20*B)*a^2*c*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a
*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*(15*B*a^2*c^4 - (3*A
+ 5*B)*a^2*c^3*d - (3*A + 31*B)*a^2*c^2*d^2 + (15*A + 17*B)*a^2*c*d^3 - (9*A - 4*B)*a^2*d^4 - 8*(B*a^2*c^2*d^2
+ 2*B*a^2*c*d^3 + B*a^2*d^4)*cos(f*x + e)^3 + (25*B*a^2*c^3*d - (5*A - 26*B)*a^2*c^2*d^2 - 3*(2*A + 5*B)*a^2*
c*d^3 + (11*A - 4*B)*a^2*d^4)*cos(f*x + e)^2 + (15*B*a^2*c^4 - (3*A - 20*B)*a^2*c^3*d - (8*A - 3*B)*a^2*c^2*d^
2 + 9*(A + 2*B)*a^2*c*d^3 + 2*(A + 4*B)*a^2*d^4)*cos(f*x + e) - (15*B*a^2*c^4 - (3*A + 5*B)*a^2*c^3*d - (3*A +
31*B)*a^2*c^2*d^2 + (15*A + 17*B)*a^2*c*d^3 - (9*A - 4*B)*a^2*d^4 - 8*(B*a^2*c^2*d^2 + 2*B*a^2*c*d^3 + B*a^2*
d^4)*cos(f*x + e)^2 - (25*B*a^2*c^3*d - (5*A - 34*B)*a^2*c^2*d^2 - (6*A - B)*a^2*c*d^3 + (11*A + 4*B)*a^2*d^4)
*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c^2*d^5 + 2*c*d^6 + d^7)*f*cos(f*x + e)^3 + (2*c^3*d^
4 + 5*c^2*d^5 + 4*c*d^6 + d^7)*f*cos(f*x + e)^2 - (c^4*d^3 + 2*c^3*d^4 + 2*c^2*d^5 + 2*c*d^6 + d^7)*f*cos(f*x
+ e) - (c^4*d^3 + 4*c^3*d^4 + 6*c^2*d^5 + 4*c*d^6 + d^7)*f + ((c^2*d^5 + 2*c*d^6 + d^7)*f*cos(f*x + e)^2 - 2*(
c^3*d^4 + 2*c^2*d^5 + c*d^6)*f*cos(f*x + e) - (c^4*d^3 + 4*c^3*d^4 + 6*c^2*d^5 + 4*c*d^6 + d^7)*f)*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((a+a*sin(f*x+e))**(5/2)*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))**3,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))^(5/2)*(A+B*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

Exception raised: TypeError