\(\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx\) [566]

Optimal result

Integrand size = 25, antiderivative size = 258 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e} \]

[Out]

-26/3465*a*b*(79*a^2+74*b^2)*(e*cos(d*x+c))^(5/2)/d/e-2/693*b*(167*a^2+54*b^2)*(e*cos(d*x+c))^(5/2)*(a+b*sin(d
*x+c))/d/e-34/99*a*b*(e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^2/d/e-2/11*b*(e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^
3/d/e+2/231*(77*a^4+132*a^2*b^2+12*b^4)*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*
d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/d/(e*cos(d*x+c))^(1/2)+2/231*(77*a^4+132*a^2*b^2+12*b^4)*e*sin(d*x+c)*(e*
cos(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2771, 2941, 2748, 2715, 2721, 2720} \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}+\frac {2 e^2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {2 e \left (77 a^4+132 a^2 b^2+12 b^4\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e} \]

[In]

Int[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4,x]

[Out]

(-26*a*b*(79*a^2 + 74*b^2)*(e*Cos[c + d*x])^(5/2))/(3465*d*e) + (2*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e^2*Sqrt[Co
s[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (2*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e*Sqr
t[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) - (2*b*(167*a^2 + 54*b^2)*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))
/(693*d*e) - (34*a*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2)/(99*d*e) - (2*b*(e*Cos[c + d*x])^(5/2)*(a
+ b*Sin[c + d*x])^3)/(11*d*e)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2771

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-b)*(
g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Dist[1/(m + p), Int[(g*Cos[e + f*x]
)^p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x], x] /; FreeQ
[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ
[m])

Rule 2941

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x
] + Dist[1/(m + p + 1), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(m + p + 1) + b*d*m + (a*
d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] &&
GtQ[m, 0] &&  !LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])

Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {2}{11} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \left (\frac {11 a^2}{2}+3 b^2+\frac {17}{2} a b \sin (c+d x)\right ) \, dx \\ & = -\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {4}{99} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (\frac {1}{4} a \left (99 a^2+122 b^2\right )+\frac {1}{4} b \left (167 a^2+54 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {8}{693} \int (e \cos (c+d x))^{3/2} \left (\frac {9}{8} \left (77 a^4+132 a^2 b^2+12 b^4\right )+\frac {13}{8} a b \left (79 a^2+74 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {1}{77} \left (77 a^4+132 a^2 b^2+12 b^4\right ) \int (e \cos (c+d x))^{3/2} \, dx \\ & = -\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {1}{231} \left (\left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {\left (\left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{231 \sqrt {e \cos (c+d x)}} \\ & = -\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.80 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.73 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\frac {(e \cos (c+d x))^{3/2} \left (240 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \left (-1848 b \left (12 a^3+7 a b^2\right )-2464 \left (9 a^3 b+4 a b^3\right ) \cos (2 (c+d x))+3080 a b^3 \cos (4 (c+d x))+30 \left (616 a^4+660 a^2 b^2+39 b^4\right ) \sin (c+d x)-45 b \left (264 a^2 b+31 b^3\right ) \sin (3 (c+d x))+315 b^4 \sin (5 (c+d x))\right )\right )}{27720 d \cos ^{\frac {3}{2}}(c+d x)} \]

[In]

Integrate[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4,x]

[Out]

((e*Cos[c + d*x])^(3/2)*(240*(77*a^4 + 132*a^2*b^2 + 12*b^4)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(-
1848*b*(12*a^3 + 7*a*b^2) - 2464*(9*a^3*b + 4*a*b^3)*Cos[2*(c + d*x)] + 3080*a*b^3*Cos[4*(c + d*x)] + 30*(616*
a^4 + 660*a^2*b^2 + 39*b^4)*Sin[c + d*x] - 45*b*(264*a^2*b + 31*b^3)*Sin[3*(c + d*x)] + 315*b^4*Sin[5*(c + d*x
)])))/(27720*d*Cos[c + d*x]^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(638\) vs. \(2(258)=516\).

Time = 12.33 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.48

method	result	size
default	\(-\frac {2 e^{2} \left (20160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-50400 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+49280 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-47520 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+41040 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-123200 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+71280 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}-11160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}-22176 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b +101024 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+4620 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}-27720 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+33264 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b -28336 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}-2310 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}+1980 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{2}+180 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}+1155 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{4}+1980 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2} b^{2}+180 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{4}-16632 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b -1232 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{3}+2772 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b +1232 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{3}\right )}{3465 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(639\)
parts	\(\text {Expression too large to display}\)	\(726\)

[In]

int((e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

-2/3465/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^2*(20160*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^12*b^4-50400*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^4+49280*sin(1/2*d*x+1/2*c)^11*a*b^3-47520*cos(1/2*d*
x+1/2*c)*sin(1/2*d*x+1/2*c)^8*a^2*b^2+41040*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8*b^4-123200*sin(1/2*d*x+1/2
*c)^9*a*b^3+71280*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*a^2*b^2-11160*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^
6*b^4-22176*sin(1/2*d*x+1/2*c)^7*a^3*b+101024*sin(1/2*d*x+1/2*c)^7*a*b^3+4620*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1
/2*c)^4*a^4-27720*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a^2*b^2+33264*sin(1/2*d*x+1/2*c)^5*a^3*b-28336*sin(1
/2*d*x+1/2*c)^5*a*b^3-2310*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^4+1980*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^2*a^2*b^2+180*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b^4+1155*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x
+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^4+1980*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+
1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^2+180*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*
x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b^4-16632*sin(1/2*d*x+1/2*c)^3*a^3*b-1232*sin(1/2*d*
x+1/2*c)^3*a*b^3+2772*sin(1/2*d*x+1/2*c)*a^3*b+1232*sin(1/2*d*x+1/2*c)*a*b^3)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.83 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\frac {-15 i \, \sqrt {2} {\left (77 \, a^{4} + 132 \, a^{2} b^{2} + 12 \, b^{4}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} {\left (77 \, a^{4} + 132 \, a^{2} b^{2} + 12 \, b^{4}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (1540 \, a b^{3} e \cos \left (d x + c\right )^{4} - 2772 \, {\left (a^{3} b + a b^{3}\right )} e \cos \left (d x + c\right )^{2} + 15 \, {\left (21 \, b^{4} e \cos \left (d x + c\right )^{4} - 3 \, {\left (66 \, a^{2} b^{2} + 13 \, b^{4}\right )} e \cos \left (d x + c\right )^{2} + {\left (77 \, a^{4} + 132 \, a^{2} b^{2} + 12 \, b^{4}\right )} e\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{3465 \, d} \]

[In]

integrate((e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/3465*(-15*I*sqrt(2)*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(
d*x + c)) + 15*I*sqrt(2)*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*s
in(d*x + c)) + 2*(1540*a*b^3*e*cos(d*x + c)^4 - 2772*(a^3*b + a*b^3)*e*cos(d*x + c)^2 + 15*(21*b^4*e*cos(d*x +
c)^4 - 3*(66*a^2*b^2 + 13*b^4)*e*cos(d*x + c)^2 + (77*a^4 + 132*a^2*b^2 + 12*b^4)*e)*sin(d*x + c))*sqrt(e*cos
(d*x + c)))/d

Sympy [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\text {Timed out} \]

[In]

integrate((e*cos(d*x+c))**(3/2)*(a+b*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [F]

\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(3/2)*(b*sin(d*x + c) + a)^4, x)

Giac [F]

\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((e*cos(d*x + c))^(3/2)*(b*sin(d*x + c) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]

[In]

int((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^4,x)

[Out]

int((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^4, x)