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

Optimal. Leaf size=258 \[ -\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e} \]

[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

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Rubi [A]
time = 0.33, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2771, 2941, 2748, 2715, 2721, 2720} \begin {gather*} -\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)} F\left (\left .\frac {1}{2} (c+d x)\right |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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx &=-\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)} F\left (\left .\frac {1}{2} (c+d x)\right |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*}

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Mathematica [A]
time = 2.78, size = 189, normalized size = 0.73 \begin {gather*} \frac {(e \cos (c+d x))^{3/2} \left (240 \left (77 a^4+132 a^2 b^2+12 b^4\right ) F\left (\left .\frac {1}{2} (c+d x)\right |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)} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(638\) vs. \(2(258)=516\).
time = 9.99, size = 639, normalized size = 2.48

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

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.14, size = 216, normalized size = 0.84 \begin {gather*} \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} \cos \left (d x + c\right )^{4} e^{\frac {3}{2}} - 2772 \, {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 15 \, {\left (21 \, b^{4} \cos \left (d x + c\right )^{4} e^{\frac {3}{2}} - 3 \, {\left (66 \, a^{2} b^{2} + 13 \, b^{4}\right )} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + {\left (77 \, a^{4} + 132 \, a^{2} b^{2} + 12 \, b^{4}\right )} e^{\frac {3}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{3465 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*cos(d*x + c)^4*e^(3/2) - 2772*(a^3*b + a*b^3)*cos(d*x + c)^2*e^(3/2) + 15*(21*b^4
*cos(d*x + c)^4*e^(3/2) - 3*(66*a^2*b^2 + 13*b^4)*cos(d*x + c)^2*e^(3/2) + (77*a^4 + 132*a^2*b^2 + 12*b^4)*e^(
3/2))*sin(d*x + c))*sqrt(cos(d*x + c)))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7319 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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