3.6.64 \(\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx\) [564]

Optimal. Leaf size=305 \[ -\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{6435 d e}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{385 d}-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e} \]

[Out]

-34/6435*a*b*(53*a^2+38*b^2)*(e*cos(d*x+c))^(9/2)/d/e+2/385*(55*a^4+60*a^2*b^2+4*b^4)*e*(e*cos(d*x+c))^(5/2)*s
in(d*x+c)/d-2/715*b*(93*a^2+26*b^2)*(e*cos(d*x+c))^(9/2)*(a+b*sin(d*x+c))/d/e-14/65*a*b*(e*cos(d*x+c))^(9/2)*(
a+b*sin(d*x+c))^2/d/e-2/15*b*(e*cos(d*x+c))^(9/2)*(a+b*sin(d*x+c))^3/d/e+2/231*(55*a^4+60*a^2*b^2+4*b^4)*e^4*(
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*(55*a^4+60*a^2*b^2+4*b^4)*e^3*sin(d*x+c)*(e*cos(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.36, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{6435 d e}-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}+\frac {2 e^4 \left (55 a^4+60 a^2 b^2+4 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^3 \left (55 a^4+60 a^2 b^2+4 b^4\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}+\frac {2 e \left (55 a^4+60 a^2 b^2+4 b^4\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{385 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^4,x]

[Out]

(-34*a*b*(53*a^2 + 38*b^2)*(e*Cos[c + d*x])^(9/2))/(6435*d*e) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^4*Sqrt[Cos[
c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^3*Sqrt[
e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x]
)/(385*d) - (2*b*(93*a^2 + 26*b^2)*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(715*d*e) - (14*a*b*(e*Cos[c +
d*x])^(9/2)*(a + b*Sin[c + d*x])^2)/(65*d*e) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^3)/(15*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))^{7/2} (a+b \sin (c+d x))^4 \, dx &=-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}+\frac {2}{15} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \left (\frac {15 a^2}{2}+3 b^2+\frac {21}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}+\frac {4}{195} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (\frac {3}{4} a \left (65 a^2+54 b^2\right )+\frac {3}{4} b \left (93 a^2+26 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}+\frac {8 \int (e \cos (c+d x))^{7/2} \left (\frac {39}{8} \left (55 a^4+60 a^2 b^2+4 b^4\right )+\frac {51}{8} a b \left (53 a^2+38 b^2\right ) \sin (c+d x)\right ) \, dx}{2145}\\ &=-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{6435 d e}-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}+\frac {1}{55} \left (55 a^4+60 a^2 b^2+4 b^4\right ) \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{6435 d e}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{385 d}-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}+\frac {1}{77} \left (\left (55 a^4+60 a^2 b^2+4 b^4\right ) e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{6435 d e}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{385 d}-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}+\frac {1}{231} \left (\left (55 a^4+60 a^2 b^2+4 b^4\right ) e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{6435 d e}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{385 d}-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}+\frac {\left (\left (55 a^4+60 a^2 b^2+4 b^4\right ) e^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{231 \sqrt {e \cos (c+d x)}}\\ &=-\frac {34 a b \left (53 a^2+38 b^2\right ) (e \cos (c+d x))^{9/2}}{6435 d e}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e^4 \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 (55 a^4+60 a^2 b^2+4 b^4\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (55 a^4+60 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{385 d}-\frac {2 b \left (93 a^2+26 b^2\right ) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}-\frac {14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}\\ \end {align*}

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Mathematica [A]
time = 4.56, size = 251, normalized size = 0.82 \begin {gather*} \frac {(e \cos (c+d x))^{7/2} \left (-154 a b \left (26 a^2+11 b^2\right ) \sqrt {\cos (c+d x)}+104 \left (55 a^4+60 a^2 b^2+4 b^4\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {1}{120} \sqrt {\cos (c+d x)} \left (156 \left (5720 a^4+2460 a^2 b^2+87 b^4\right ) \sin (c+d x)+462 b^3 \cos (6 (c+d x)) (60 a+13 b \sin (c+d x))-28 b \cos (4 (c+d x)) \left (220 a \left (26 a^2-b^2\right )+39 b \left (180 a^2+b^2\right ) \sin (c+d x)\right )+\cos (2 (c+d x)) \left (-3080 \left (208 a^3 b+73 a b^3\right )+78 \left (2640 a^4-7200 a^2 b^2-557 b^4\right ) \sin (c+d x)\right )\right )\right )}{12012 d \cos ^{\frac {7}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^4,x]

[Out]

((e*Cos[c + d*x])^(7/2)*(-154*a*b*(26*a^2 + 11*b^2)*Sqrt[Cos[c + d*x]] + 104*(55*a^4 + 60*a^2*b^2 + 4*b^4)*Ell
ipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(156*(5720*a^4 + 2460*a^2*b^2 + 87*b^4)*Sin[c + d*x] + 462*b^3*Co
s[6*(c + d*x)]*(60*a + 13*b*Sin[c + d*x]) - 28*b*Cos[4*(c + d*x)]*(220*a*(26*a^2 - b^2) + 39*b*(180*a^2 + b^2)
*Sin[c + d*x]) + Cos[2*(c + d*x)]*(-3080*(208*a^3*b + 73*a*b^3) + 78*(2640*a^4 - 7200*a^2*b^2 - 557*b^4)*Sin[c
+ d*x])))/120))/(12012*d*Cos[c + d*x]^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(862\) vs. \(2(301)=602\).
time = 12.36, size = 863, normalized size = 2.83

method	result	size
default	\(\text {Expression too large to display}\)	\(863\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

-2/45045/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^4*(8673280*a*b^3*sin(1/2*d*x+1/2*c)^11-26906
88*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+768768*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^16+3739008*b^
4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-2620800*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+102960*a^4*cos
(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+946608*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-154440*a^4*cos(1/2*d*x
+1/2*c)*sin(1/2*d*x+1/2*c)^6+11700*(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+3931200*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10-3818880*a^2*b^2*cos(
1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+1797120*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-360360*a^2*b^2*cos
(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+11700*a^2*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+10725*(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+780*(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^4-1572480*a^2*b^2*cos(1
/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-144456*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+120120*a^4*cos(1/2*d*x+
1/2*c)*sin(1/2*d*x+1/2*c)^4-34320*a^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+780*b^4*cos(1/2*d*x+1/2*c)*sin(1
/2*d*x+1/2*c)^2-6209280*a*b^3*sin(1/2*d*x+1/2*c)^13+1774080*a*b^3*sin(1/2*d*x+1/2*c)^15-640640*a^3*b*sin(1/2*d
*x+1/2*c)^11+1601600*a^3*b*sin(1/2*d*x+1/2*c)^9-6160000*a*b^3*sin(1/2*d*x+1/2*c)^9-1601600*a^3*b*sin(1/2*d*x+1
/2*c)^7+2279200*a*b^3*sin(1/2*d*x+1/2*c)^7+800800*a^3*b*sin(1/2*d*x+1/2*c)^5-363440*a*b^3*sin(1/2*d*x+1/2*c)^5
-200200*a^3*b*sin(1/2*d*x+1/2*c)^3-6160*a*b^3*sin(1/2*d*x+1/2*c)^3+20020*a^3*b*sin(1/2*d*x+1/2*c)+6160*a*b^3*s
in(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))^(7/2)*(a+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

e^(7/2)*integrate((b*sin(d*x + c) + a)^4*cos(d*x + c)^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 248, normalized size = 0.81 \begin {gather*} \frac {-195 i \, \sqrt {2} {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (13860 \, a b^{3} \cos \left (d x + c\right )^{6} e^{\frac {7}{2}} - 20020 \, {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} + 39 \, {\left (77 \, b^{4} \cos \left (d x + c\right )^{6} e^{\frac {7}{2}} - 7 \, {\left (90 \, a^{2} b^{2} + 17 \, b^{4}\right )} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} + 3 \, {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} + 5 \, {\left (55 \, a^{4} + 60 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {7}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{45045 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/45045*(-195*I*sqrt(2)*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(
d*x + c)) + 195*I*sqrt(2)*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*si
n(d*x + c)) + 2*(13860*a*b^3*cos(d*x + c)^6*e^(7/2) - 20020*(a^3*b + a*b^3)*cos(d*x + c)^4*e^(7/2) + 39*(77*b^
4*cos(d*x + c)^6*e^(7/2) - 7*(90*a^2*b^2 + 17*b^4)*cos(d*x + c)^4*e^(7/2) + 3*(55*a^4 + 60*a^2*b^2 + 4*b^4)*co
s(d*x + c)^2*e^(7/2) + 5*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^(7/2))*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))**(7/2)*(a+b*sin(d*x+c))**4,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8856 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))^(7/2)*(a+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^4*cos(d*x + c)^(7/2)*e^(7/2), 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 )}^{7/2}\,{\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))^(7/2)*(a + b*sin(c + d*x))^4,x)

[Out]

int((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^4, x)

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