3.6.55 \(\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx\) [555]

Optimal. Leaf size=237 \[ -\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e} \]

[Out]

-2/1287*b*(177*a^2+44*b^2)*(e*cos(d*x+c))^(9/2)/d/e+2/77*a*(11*a^2+6*b^2)*e*(e*cos(d*x+c))^(5/2)*sin(d*x+c)/d-
34/143*a*b*(e*cos(d*x+c))^(9/2)*(a+b*sin(d*x+c))/d/e-2/13*b*(e*cos(d*x+c))^(9/2)*(a+b*sin(d*x+c))^2/d/e+10/231
*a*(11*a^2+6*b^2)*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))*co
s(d*x+c)^(1/2)/d/(e*cos(d*x+c))^(1/2)+10/231*a*(11*a^2+6*b^2)*e^3*sin(d*x+c)*(e*cos(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.21, antiderivative size = 237, 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 {10 a e^4 \left (11 a^2+6 b^2\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 {10 a e^3 \left (11 a^2+6 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {2 a e \left (11 a^2+6 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3,x]

[Out]

(-2*b*(177*a^2 + 44*b^2)*(e*Cos[c + d*x])^(9/2))/(1287*d*e) + (10*a*(11*a^2 + 6*b^2)*e^4*Sqrt[Cos[c + d*x]]*El
lipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (10*a*(11*a^2 + 6*b^2)*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c
+ d*x])/(231*d) + (2*a*(11*a^2 + 6*b^2)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (34*a*b*(e*Cos[c + d*x
])^(9/2)*(a + b*Sin[c + d*x]))/(143*d*e) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^2)/(13*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))^3 \, dx &=-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {2}{13} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (\frac {13 a^2}{2}+2 b^2+\frac {17}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {4}{143} \int (e \cos (c+d x))^{7/2} \left (\frac {13}{4} a \left (11 a^2+6 b^2\right )+\frac {1}{4} b \left (177 a^2+44 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{11} \left (a \left (11 a^2+6 b^2\right )\right ) \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{77} \left (5 a \left (11 a^2+6 b^2\right ) e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{231} \left (5 a \left (11 a^2+6 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {\left (5 a \left (11 a^2+6 b^2\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 {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\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 {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\\ \end {align*}

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Mathematica [A]
time = 2.04, size = 205, normalized size = 0.86 \begin {gather*} \frac {(e \cos (c+d x))^{7/2} \left (-154 b \left (78 a^2+11 b^2\right ) \sqrt {\cos (c+d x)}+2080 \left (11 a^3+6 a b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {1}{3} \sqrt {\cos (c+d x)} \left (-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))+154 b \left (-78 a^2+b^2\right ) \cos (4 (c+d x))+693 b^3 \cos (6 (c+d x))+156 a \left (506 a^2+213 b^2\right ) \sin (c+d x)+234 a \left (44 a^2-39 b^2\right ) \sin (3 (c+d x))-4914 a b^2 \sin (5 (c+d x))\right )\right )}{48048 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])^3,x]

[Out]

((e*Cos[c + d*x])^(7/2)*(-154*b*(78*a^2 + 11*b^2)*Sqrt[Cos[c + d*x]] + 2080*(11*a^3 + 6*a*b^2)*EllipticF[(c +
d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(-77*b*(624*a^2 + 73*b^2)*Cos[2*(c + d*x)] + 154*b*(-78*a^2 + b^2)*Cos[4*(c +
d*x)] + 693*b^3*Cos[6*(c + d*x)] + 156*a*(506*a^2 + 213*b^2)*Sin[c + d*x] + 234*a*(44*a^2 - 39*b^2)*Sin[3*(c
+ d*x)] - 4914*a*b^2*Sin[5*(c + d*x)]))/3))/(48048*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. \(617\) vs. \(2(237)=474\).
time = 11.38, size = 618, normalized size = 2.61

method	result	size
default	\(-\frac {2 e^{4} \left (-240240 a^{2} b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20592 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3003 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-96096 a^{2} b \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-157248 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+393120 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6864 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2145 \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^{3}-381888 a \,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 )+179712 a \,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 )-36036 a \,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 )+1170 a \,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 )-30888 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+308 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-30030 a^{2} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240240 a^{2} b \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-308000 b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+113960 b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-18172 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-308 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+88704 b^{3} \left (\sin ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+433664 b^{3} \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-310464 b^{3} \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120120 a^{2} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24024 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1170 \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 \,b^{2}\right )}{9009 \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}\)	\(618\)

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))^3,x,method=_RETURNVERBOSE)

[Out]

-2/9009/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^4*(-6864*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1
/2*c)^2-96096*a^2*b*sin(1/2*d*x+1/2*c)^11+240240*a^2*b*sin(1/2*d*x+1/2*c)^9-240240*a^2*b*sin(1/2*d*x+1/2*c)^7+
120120*a^2*b*sin(1/2*d*x+1/2*c)^5-30030*a^2*b*sin(1/2*d*x+1/2*c)^3+3003*a^2*b*sin(1/2*d*x+1/2*c)+20592*a^3*cos
(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-30888*a^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+24024*a^3*cos(1/2*d*x+1
/2*c)*sin(1/2*d*x+1/2*c)^4+1170*(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*b^2-310464*b^3*sin(1/2*d*x+1/2*c)^13+433664*b^3*sin(1/2*d*x+1/2*c)^11+88704*b^3*sin(1/
2*d*x+1/2*c)^15-308000*b^3*sin(1/2*d*x+1/2*c)^9+113960*b^3*sin(1/2*d*x+1/2*c)^7-18172*b^3*sin(1/2*d*x+1/2*c)^5
-308*b^3*sin(1/2*d*x+1/2*c)^3+308*b^3*sin(1/2*d*x+1/2*c)-381888*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+
179712*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-36036*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+1170*
a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2145*(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^3-157248*a*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+393120*a*b^
2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10)/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))^3,x, algorithm="maxima")

[Out]

e^(7/2)*integrate((b*sin(d*x + c) + a)^3*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 = 0.14, size = 193, normalized size = 0.81 \begin {gather*} \frac {-195 i \, \sqrt {2} {\left (11 \, a^{3} + 6 \, a b^{2}\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 (11 \, a^{3} + 6 \, a b^{2}\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 (693 \, b^{3} \cos \left (d x + c\right )^{6} e^{\frac {7}{2}} - 1001 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} - 39 \, {\left (63 \, a b^{2} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} - 3 \, {\left (11 \, a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} - 5 \, {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{9009 \, 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))^3,x, algorithm="fricas")

[Out]

1/9009*(-195*I*sqrt(2)*(11*a^3 + 6*a*b^2)*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) +
195*I*sqrt(2)*(11*a^3 + 6*a*b^2)*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(693*b^
3*cos(d*x + c)^6*e^(7/2) - 1001*(3*a^2*b + b^3)*cos(d*x + c)^4*e^(7/2) - 39*(63*a*b^2*cos(d*x + c)^4*e^(7/2) -
3*(11*a^3 + 6*a*b^2)*cos(d*x + c)^2*e^(7/2) - 5*(11*a^3 + 6*a*b^2)*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))**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7316 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))^3,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^3*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 )}^3 \,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))^3,x)

[Out]

int((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^3, x)

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