3.71 \(\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx\)
Optimal. Leaf size=201 \[ -\frac{7 c^4 (2 A-7 B) \cos (e+f x)}{2 a^3 f}-\frac{a^4 c^4 (A-B) \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}+\frac{2 a^2 c^4 (2 A-7 B) \cos ^7(e+f x)}{15 f (a \sin (e+f x)+a)^5}-\frac{7 c^4 (2 A-7 B) \cos ^3(e+f x)}{6 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac{7 c^4 x (2 A-7 B)}{2 a^3}-\frac{14 c^4 (2 A-7 B) \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^3} \]
[Out]
(-7*(2*A - 7*B)*c^4*x)/(2*a^3) - (7*(2*A - 7*B)*c^4*Cos[e + f*x])/(2*a^3*f) - (a^4*(A - B)*c^4*Cos[e + f*x]^9)
/(5*f*(a + a*Sin[e + f*x])^7) + (2*a^2*(2*A - 7*B)*c^4*Cos[e + f*x]^7)/(15*f*(a + a*Sin[e + f*x])^5) - (14*(2*
A - 7*B)*c^4*Cos[e + f*x]^5)/(15*f*(a + a*Sin[e + f*x])^3) - (7*(2*A - 7*B)*c^4*Cos[e + f*x]^3)/(6*f*(a^3 + a^
3*Sin[e + f*x]))
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Rubi [A] time = 0.391648, antiderivative size = 201, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used =
{2967, 2859, 2680, 2679, 2682, 8} \[ -\frac{7 c^4 (2 A-7 B) \cos (e+f x)}{2 a^3 f}-\frac{a^4 c^4 (A-B) \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}+\frac{2 a^2 c^4 (2 A-7 B) \cos ^7(e+f x)}{15 f (a \sin (e+f x)+a)^5}-\frac{7 c^4 (2 A-7 B) \cos ^3(e+f x)}{6 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac{7 c^4 x (2 A-7 B)}{2 a^3}-\frac{14 c^4 (2 A-7 B) \cos ^5(e+f x)}{15 f (a \sin (e+f x)+a)^3} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x])^3,x]
[Out]
(-7*(2*A - 7*B)*c^4*x)/(2*a^3) - (7*(2*A - 7*B)*c^4*Cos[e + f*x])/(2*a^3*f) - (a^4*(A - B)*c^4*Cos[e + f*x]^9)
/(5*f*(a + a*Sin[e + f*x])^7) + (2*a^2*(2*A - 7*B)*c^4*Cos[e + f*x]^7)/(15*f*(a + a*Sin[e + f*x])^5) - (14*(2*
A - 7*B)*c^4*Cos[e + f*x]^5)/(15*f*(a + a*Sin[e + f*x])^3) - (7*(2*A - 7*B)*c^4*Cos[e + f*x]^3)/(6*f*(a^3 + a^
3*Sin[e + f*x]))
Rule 2967
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Rule 2859
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m +
p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]
Rule 2680
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]
Rule 2679
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + p)), x] + Dist[(g^2*(p - 1))/(a*(m + p)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
&& LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p,
0] && IntegersQ[2*m, 2*p]
Rule 2682
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^3} \, dx &=\left (a^4 c^4\right ) \int \frac{\cos ^8(e+f x) (A+B \sin (e+f x))}{(a+a \sin (e+f x))^7} \, dx\\ &=-\frac{a^4 (A-B) c^4 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}-\frac{1}{5} \left (a^3 (2 A-7 B) c^4\right ) \int \frac{\cos ^8(e+f x)}{(a+a \sin (e+f x))^6} \, dx\\ &=-\frac{a^4 (A-B) c^4 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac{2 a^2 (2 A-7 B) c^4 \cos ^7(e+f x)}{15 f (a+a \sin (e+f x))^5}+\frac{1}{15} \left (7 a (2 A-7 B) c^4\right ) \int \frac{\cos ^6(e+f x)}{(a+a \sin (e+f x))^4} \, dx\\ &=-\frac{a^4 (A-B) c^4 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac{2 a^2 (2 A-7 B) c^4 \cos ^7(e+f x)}{15 f (a+a \sin (e+f x))^5}-\frac{14 (2 A-7 B) c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^3}-\frac{\left (7 (2 A-7 B) c^4\right ) \int \frac{\cos ^4(e+f x)}{(a+a \sin (e+f x))^2} \, dx}{3 a}\\ &=-\frac{a^4 (A-B) c^4 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac{2 a^2 (2 A-7 B) c^4 \cos ^7(e+f x)}{15 f (a+a \sin (e+f x))^5}-\frac{14 (2 A-7 B) c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^3}-\frac{7 (2 A-7 B) c^4 \cos ^3(e+f x)}{6 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{\left (7 (2 A-7 B) c^4\right ) \int \frac{\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{2 a^2}\\ &=-\frac{7 (2 A-7 B) c^4 \cos (e+f x)}{2 a^3 f}-\frac{a^4 (A-B) c^4 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac{2 a^2 (2 A-7 B) c^4 \cos ^7(e+f x)}{15 f (a+a \sin (e+f x))^5}-\frac{14 (2 A-7 B) c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^3}-\frac{7 (2 A-7 B) c^4 \cos ^3(e+f x)}{6 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac{\left (7 (2 A-7 B) c^4\right ) \int 1 \, dx}{2 a^3}\\ &=-\frac{7 (2 A-7 B) c^4 x}{2 a^3}-\frac{7 (2 A-7 B) c^4 \cos (e+f x)}{2 a^3 f}-\frac{a^4 (A-B) c^4 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac{2 a^2 (2 A-7 B) c^4 \cos ^7(e+f x)}{15 f (a+a \sin (e+f x))^5}-\frac{14 (2 A-7 B) c^4 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^3}-\frac{7 (2 A-7 B) c^4 \cos ^3(e+f x)}{6 f \left (a^3+a^3 \sin (e+f x)\right )}\\ \end{align*}
Mathematica [A] time = 1.53674, size = 348, normalized size = 1.73 \[ \frac{(c-c \sin (e+f x))^4 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (384 (A-B) \sin \left (\frac{1}{2} (e+f x)\right )-210 (2 A-7 B) (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5-60 (A-7 B) \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5+64 (29 A-79 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4+64 (8 A-13 B) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-128 (8 A-13 B) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2-192 (A-B) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )-15 B \sin (2 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5\right )}{60 a^3 f (\sin (e+f x)+1)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x])^3,x]
[Out]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^4*(384*(A - B)*Sin[(e + f*x)/2] - 192*(A - B)*(Cos
[(e + f*x)/2] + Sin[(e + f*x)/2]) - 128*(8*A - 13*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2
+ 64*(8*A - 13*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 64*(29*A - 79*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2
] + Sin[(e + f*x)/2])^4 - 210*(2*A - 7*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 60*(A - 7*B)*Cos
[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 15*B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sin[2*(e + f*
x)]))/(60*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(1 + Sin[e + f*x])^3)
________________________________________________________________________________________
Maple [B] time = 0.155, size = 474, normalized size = 2.4 \begin{align*}{\frac{B{c}^{4}}{f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-2\,{\frac{{c}^{4} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}A}{f{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}+14\,{\frac{{c}^{4} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}B}{f{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{B{c}^{4}}{f{a}^{3}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-2\,{\frac{A{c}^{4}}{f{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}+14\,{\frac{B{c}^{4}}{f{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}+49\,{\frac{{c}^{4}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) B}{f{a}^{3}}}-14\,{\frac{{c}^{4}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) A}{f{a}^{3}}}+64\,{\frac{A{c}^{4}}{f{a}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{4}}}-64\,{\frac{B{c}^{4}}{f{a}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{4}}}-16\,{\frac{A{c}^{4}}{f{a}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }}+48\,{\frac{B{c}^{4}}{f{a}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }}-{\frac{128\,A{c}^{4}}{5\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-5}}+{\frac{128\,B{c}^{4}}{5\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-5}}-{\frac{128\,A{c}^{4}}{3\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-3}}+{\frac{64\,B{c}^{4}}{3\,f{a}^{3}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-3}}+32\,{\frac{B{c}^{4}}{f{a}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x)
[Out]
1/f*c^4/a^3/(1+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f*x+1/2*e)^3*B-2/f*c^4/a^3/(1+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f
*x+1/2*e)^2*A+14/f*c^4/a^3/(1+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f*x+1/2*e)^2*B-1/f*c^4/a^3/(1+tan(1/2*f*x+1/2*e)
^2)^2*B*tan(1/2*f*x+1/2*e)-2/f*c^4/a^3/(1+tan(1/2*f*x+1/2*e)^2)^2*A+14/f*c^4/a^3/(1+tan(1/2*f*x+1/2*e)^2)^2*B+
49/f*c^4/a^3*arctan(tan(1/2*f*x+1/2*e))*B-14/f*c^4/a^3*arctan(tan(1/2*f*x+1/2*e))*A+64/f*c^4/a^3/(tan(1/2*f*x+
1/2*e)+1)^4*A-64/f*c^4/a^3/(tan(1/2*f*x+1/2*e)+1)^4*B-16/f*c^4/a^3/(tan(1/2*f*x+1/2*e)+1)*A+48/f*c^4/a^3/(tan(
1/2*f*x+1/2*e)+1)*B-128/5/f*c^4/a^3/(tan(1/2*f*x+1/2*e)+1)^5*A+128/5/f*c^4/a^3/(tan(1/2*f*x+1/2*e)+1)^5*B-128/
3/f*c^4/a^3/(tan(1/2*f*x+1/2*e)+1)^3*A+64/3/f*c^4/a^3/(tan(1/2*f*x+1/2*e)+1)^3*B+32/f*c^4/a^3*B/(tan(1/2*f*x+1
/2*e)+1)^2
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Maxima [B] time = 1.71925, size = 3232, normalized size = 16.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="maxima")
[Out]
1/15*(B*c^4*((1325*sin(f*x + e)/(cos(f*x + e) + 1) + 2673*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3805*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 4329*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3575*sin(f*x + e)^5/(cos(f*x + e) + 1)
^5 + 2275*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 975*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 195*sin(f*x + e)^8/(
cos(f*x + e) + 1)^8 + 304)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^3*sin(f*x + e)^2/(cos(f*x + e)
+ 1)^2 + 20*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 26*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26*a^3*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + 20*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 12*a^3*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 5*a^3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 195*arcta
n(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) - 6*A*c^4*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 189*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 + 200*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 160*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 75*s
in(f*x + e)^5/(cos(f*x + e) + 1)^5 + 15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 24)/(a^3 + 5*a^3*sin(f*x + e)/(c
os(f*x + e) + 1) + 11*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1
5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 11*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*a^3*sin(f*x + e)^6/
(cos(f*x + e) + 1)^6 + a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a
^3) + 24*B*c^4*((105*sin(f*x + e)/(cos(f*x + e) + 1) + 189*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 200*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 160*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 75*sin(f*x + e)^5/(cos(f*x + e) + 1)^5
+ 15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 24)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 11*a^3*sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 + 15*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*a^3*sin(f*x + e)^4/(cos(f*x + e)
+ 1)^4 + 11*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^3*sin(f*x
+ e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) - 8*A*c^4*((95*sin(f*x + e)/(co
s(f*x + e) + 1) + 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*
x + e)^4/(cos(f*x + e) + 1)^4 + 22)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3
*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + 12*B*c^4*((95*sin(f*
x + e)/(cos(f*x + e) + 1) + 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 +
15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 22)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x +
e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) +
1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^3) - 2*A*c^4*(2
0*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 30*sin(f*x + e)^3/(cos(f*x + e) +
1)^3 + 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 7)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(
f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x +
e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 24*A*c^4*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 10*sin(f
*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3
*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 16*B*c^4*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 10*sin(f*x + e)^2/(cos(f
*x + e) + 1)^2 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2
+ 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/
(cos(f*x + e) + 1)^5) + 24*A*c^4*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 +
5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^
2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^
4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) - 6*B*c^4*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 + 5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1)
+ 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)
^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5))/f
________________________________________________________________________________________
Fricas [B] time = 1.83411, size = 961, normalized size = 4.78 \begin{align*} \frac{15 \, B c^{4} \cos \left (f x + e\right )^{5} - 30 \,{\left (A - 6 \, B\right )} c^{4} \cos \left (f x + e\right )^{4} + 420 \,{\left (2 \, A - 7 \, B\right )} c^{4} f x + 96 \,{\left (A - B\right )} c^{4} -{\left (105 \,{\left (2 \, A - 7 \, B\right )} c^{4} f x +{\left (554 \, A - 1819 \, B\right )} c^{4}\right )} \cos \left (f x + e\right )^{3} -{\left (315 \,{\left (2 \, A - 7 \, B\right )} c^{4} f x - 2 \,{\left (134 \, A - 619 \, B\right )} c^{4}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (35 \,{\left (2 \, A - 7 \, B\right )} c^{4} f x + 2 \,{\left (74 \, A - 249 \, B\right )} c^{4}\right )} \cos \left (f x + e\right ) -{\left (15 \, B c^{4} \cos \left (f x + e\right )^{4} + 15 \,{\left (2 \, A - 11 \, B\right )} c^{4} \cos \left (f x + e\right )^{3} - 420 \,{\left (2 \, A - 7 \, B\right )} c^{4} f x + 96 \,{\left (A - B\right )} c^{4} +{\left (105 \,{\left (2 \, A - 7 \, B\right )} c^{4} f x - 2 \,{\left (262 \, A - 827 \, B\right )} c^{4}\right )} \cos \left (f x + e\right )^{2} - 6 \,{\left (35 \,{\left (2 \, A - 7 \, B\right )} c^{4} f x + 2 \,{\left (66 \, A - 241 \, B\right )} c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{30 \,{\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f +{\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
[Out]
1/30*(15*B*c^4*cos(f*x + e)^5 - 30*(A - 6*B)*c^4*cos(f*x + e)^4 + 420*(2*A - 7*B)*c^4*f*x + 96*(A - B)*c^4 - (
105*(2*A - 7*B)*c^4*f*x + (554*A - 1819*B)*c^4)*cos(f*x + e)^3 - (315*(2*A - 7*B)*c^4*f*x - 2*(134*A - 619*B)*
c^4)*cos(f*x + e)^2 + 6*(35*(2*A - 7*B)*c^4*f*x + 2*(74*A - 249*B)*c^4)*cos(f*x + e) - (15*B*c^4*cos(f*x + e)^
4 + 15*(2*A - 11*B)*c^4*cos(f*x + e)^3 - 420*(2*A - 7*B)*c^4*f*x + 96*(A - B)*c^4 + (105*(2*A - 7*B)*c^4*f*x -
2*(262*A - 827*B)*c^4)*cos(f*x + e)^2 - 6*(35*(2*A - 7*B)*c^4*f*x + 2*(66*A - 241*B)*c^4)*cos(f*x + e))*sin(f
*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)
^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*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+B*sin(f*x+e))*(c-c*sin(f*x+e))**4/(a+a*sin(f*x+e))**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [A] time = 1.23538, size = 412, normalized size = 2.05 \begin{align*} -\frac{\frac{105 \,{\left (2 \, A c^{4} - 7 \, B c^{4}\right )}{\left (f x + e\right )}}{a^{3}} - \frac{30 \,{\left (B c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 2 \, A c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 14 \, B c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - B c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 2 \, A c^{4} + 14 \, B c^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}^{2} a^{3}} + \frac{32 \,{\left (15 \, A c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 45 \, B c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 60 \, A c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 210 \, B c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 130 \, A c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 380 \, B c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 80 \, A c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 250 \, B c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 19 \, A c^{4} - 59 \, B c^{4}\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{5}}}{30 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm="giac")
[Out]
-1/30*(105*(2*A*c^4 - 7*B*c^4)*(f*x + e)/a^3 - 30*(B*c^4*tan(1/2*f*x + 1/2*e)^3 - 2*A*c^4*tan(1/2*f*x + 1/2*e)
^2 + 14*B*c^4*tan(1/2*f*x + 1/2*e)^2 - B*c^4*tan(1/2*f*x + 1/2*e) - 2*A*c^4 + 14*B*c^4)/((tan(1/2*f*x + 1/2*e)
^2 + 1)^2*a^3) + 32*(15*A*c^4*tan(1/2*f*x + 1/2*e)^4 - 45*B*c^4*tan(1/2*f*x + 1/2*e)^4 + 60*A*c^4*tan(1/2*f*x
+ 1/2*e)^3 - 210*B*c^4*tan(1/2*f*x + 1/2*e)^3 + 130*A*c^4*tan(1/2*f*x + 1/2*e)^2 - 380*B*c^4*tan(1/2*f*x + 1/2
*e)^2 + 80*A*c^4*tan(1/2*f*x + 1/2*e) - 250*B*c^4*tan(1/2*f*x + 1/2*e) + 19*A*c^4 - 59*B*c^4)/(a^3*(tan(1/2*f*
x + 1/2*e) + 1)^5))/f