3.4.5 \(\int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx\) [305]

Optimal. Leaf size=247 \[ \frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {5 e^3 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d}-\frac {5 e^{7/2} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {5 e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d (1+\cos (c+d x)+\sin (c+d x))} \]

[Out]

1/2*e*(e*cos(d*x+c))^(5/2)/a/d/(a+a*sin(d*x+c))^(1/2)+5/4*e^3*(e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/a^2/
d-5/4*e^(7/2)*arcsinh((e*cos(d*x+c))^(1/2)/e^(1/2))*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/a^2/d/(1+cos(d
*x+c)+sin(d*x+c))+5/4*e^(7/2)*arctan(sin(d*x+c)*e^(1/2)/(e*cos(d*x+c))^(1/2)/(1+cos(d*x+c))^(1/2))*(1+cos(d*x+
c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/a^2/d/(1+cos(d*x+c)+sin(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2758, 2764, 2756, 2854, 209, 2912, 65, 221} \begin {gather*} \frac {5 e^{7/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {ArcTan}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{4 a^2 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {5 e^{7/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{4 a^2 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {5 e^3 \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{4 a^2 d}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(e*Cos[c + d*x])^(5/2))/(2*a*d*Sqrt[a + a*Sin[c + d*x]]) + (5*e^3*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d
*x]])/(4*a^2*d) - (5*e^(7/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d
*x]])/(4*a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (5*e^(7/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d
*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d*(1 + Cos[c + d*x] + Si
n[c + d*x]))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 2756

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)], x_Symbol] :> Dist[a*Sqrt[1
+ Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])), Int[Sqrt[1 + Cos[e + f*x]]/
Sqrt[g*Cos[e + f*x]], x], x] + Dist[b*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b
*Sin[e + f*x])), Int[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f,
g}, x] && EqQ[a^2 - b^2, 0]

Rule 2758

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[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 2764

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[g*Sqrt
[g*Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(b*f)), x] + Dist[g^2/(2*a), Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Co
s[e + f*x]], x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]

Rule 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {\left (5 e^2\right ) \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx}{4 a}\\ &=\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {5 e^3 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d}+\frac {\left (5 e^4\right ) \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{8 a^2}\\ &=\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {5 e^3 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d}+\frac {\left (5 e^4 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{8 a (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (5 e^4 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{8 a (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {5 e^3 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d}-\frac {\left (5 e^4 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{8 a d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (5 e^4 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+e x^2} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right )}{4 a d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {5 e^3 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d}+\frac {5 e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d \left (a^2+a^2 \cos (c+d x)+a^2 \sin (c+d x)\right )}-\frac {\left (5 e^3 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{e}}} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{4 a d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {5 e^3 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d}-\frac {5 e^{7/2} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d \left (a^2+a^2 \cos (c+d x)+a^2 \sin (c+d x)\right )}+\frac {5 e^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d \left (a^2+a^2 \cos (c+d x)+a^2 \sin (c+d x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.14, size = 80, normalized size = 0.32 \begin {gather*} -\frac {2\ 2^{3/4} (e \cos (c+d x))^{9/2} \, _2F_1\left (\frac {1}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt {a (1+\sin (c+d x))}}{9 a^2 d e (1+\sin (c+d x))^{11/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*2^(3/4)*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[1/4, 9/4, 13/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c +
d*x])])/(9*a^2*d*e*(1 + Sin[c + d*x])^(11/4))

________________________________________________________________________________________

Maple [A]
time = 0.23, size = 266, normalized size = 1.08

method	result	size
default	\(-\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \left (-5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )+5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )+4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 \left (\cos ^{3}\left (d x +c \right )\right )+10 \cos \left (d x +c \right ) \sin \left (d x +c \right )-14 \left (\cos ^{2}\left (d x +c \right )\right )+10 \cos \left (d x +c \right )\right )}{8 d \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )-\left (\cos ^{2}\left (d x +c \right )\right )-2 \sin \left (d x +c \right )-\cos \left (d x +c \right )+2\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}} \cos \left (d x +c \right )}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/d*(e*cos(d*x+c))^(7/2)*(-5*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*sin(d*x+c)+5*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arcta
n(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))*sin(d*x+c)+4*cos(d*x+c)^2*sin(d*x+c)+4*cos(d*x+c)^3+10*cos
(d*x+c)*sin(d*x+c)-14*cos(d*x+c)^2+10*cos(d*x+c))/(cos(d*x+c)*sin(d*x+c)-cos(d*x+c)^2-2*sin(d*x+c)-cos(d*x+c)+
2)/(a*(1+sin(d*x+c)))^(3/2)/cos(d*x+c)

________________________________________________________________________________________

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+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3361 vs. \(2 (192) = 384\).
time = 190.26, size = 3361, normalized size = 13.61 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/32*(20*sqrt(2)*a^2*d*(1/(a^6*d^4))^(1/4)*arctan(-1/4*(2*sqrt(1/2)*((sqrt(2)*a^5*d^3*cos(d*x + c)^6 + 5*sqrt(
2)*a^5*d^3*cos(d*x + c)^5 - 8*sqrt(2)*a^5*d^3*cos(d*x + c)^4 - 20*sqrt(2)*a^5*d^3*cos(d*x + c)^3 + 8*sqrt(2)*a
^5*d^3*cos(d*x + c)^2 + 16*sqrt(2)*a^5*d^3*cos(d*x + c) + (sqrt(2)*a^5*d^3*cos(d*x + c)^5 - 4*sqrt(2)*a^5*d^3*
cos(d*x + c)^4 - 12*sqrt(2)*a^5*d^3*cos(d*x + c)^3 + 8*sqrt(2)*a^5*d^3*cos(d*x + c)^2 + 16*sqrt(2)*a^5*d^3*cos
(d*x + c))*sin(d*x + c))*(1/(a^6*d^4))^(3/4)*e^(21/2) + (sqrt(2)*a^2*d*cos(d*x + c)^6*e^7 - 3*sqrt(2)*a^2*d*co
s(d*x + c)^5*e^7 - 8*sqrt(2)*a^2*d*cos(d*x + c)^4*e^7 + 4*sqrt(2)*a^2*d*cos(d*x + c)^3*e^7 + 8*sqrt(2)*a^2*d*c
os(d*x + c)^2*e^7 - (sqrt(2)*a^2*d*cos(d*x + c)^5*e^7 + 4*sqrt(2)*a^2*d*cos(d*x + c)^4*e^7 - 4*sqrt(2)*a^2*d*c
os(d*x + c)^3*e^7 - 8*sqrt(2)*a^2*d*cos(d*x + c)^2*e^7)*sin(d*x + c))*(1/(a^6*d^4))^(1/4)*e^(7/2) - (cos(d*x +
c)^4*e^(21/2) - 3*cos(d*x + c)^3*e^(21/2) - 8*cos(d*x + c)^2*e^(21/2) + (2*a^3*d^2*cos(d*x + c)^5*e^(7/2) - 5
*a^3*d^2*cos(d*x + c)^4*e^(7/2) - 19*a^3*d^2*cos(d*x + c)^3*e^(7/2) + 20*a^3*d^2*cos(d*x + c)*e^(7/2) + 8*a^3*
d^2*e^(7/2) - (2*a^3*d^2*cos(d*x + c)^4*e^(7/2) + 9*a^3*d^2*cos(d*x + c)^3*e^(7/2) - 4*a^3*d^2*cos(d*x + c)^2*
e^(7/2) - 20*a^3*d^2*cos(d*x + c)*e^(7/2) - 8*a^3*d^2*e^(7/2))*sin(d*x + c))*sqrt(1/(a^6*d^4))*e^7 + 4*cos(d*x
+ c)*e^(21/2) - (cos(d*x + c)^3*e^(21/2) + 4*cos(d*x + c)^2*e^(21/2) - 4*cos(d*x + c)*e^(21/2) - 8*e^(21/2))*
sin(d*x + c) + 8*e^(21/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))*sqrt((2*a*cos(d*x + c)*e^21*sin(d*x +
c) + 2*a*cos(d*x + c)*e^21 + (a^4*d^2*e^14*sin(d*x + c) + a^4*d^2*e^14)*sqrt(1/(a^6*d^4))*e^7 + (sqrt(2)*a^5*d
^3*(1/(a^6*d^4))^(3/4)*cos(d*x + c)*e^21 + (sqrt(2)*a^2*d*e^(35/2)*sin(d*x + c) + sqrt(2)*a^2*d*e^(35/2))*(1/(
a^6*d^4))^(1/4)*e^(7/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(sin(d*x + c) + 1)) - ((7*sqrt(2)*a^5*d^
3*cos(d*x + c)^4*e^(21/2) + 3*sqrt(2)*a^5*d^3*cos(d*x + c)^3*e^(21/2) - 16*sqrt(2)*a^5*d^3*cos(d*x + c)^2*e^(2
1/2) - 4*sqrt(2)*a^5*d^3*cos(d*x + c)*e^(21/2) + 8*sqrt(2)*a^5*d^3*e^(21/2) + (2*sqrt(2)*a^5*d^3*cos(d*x + c)^
4*e^(21/2) + sqrt(2)*a^5*d^3*cos(d*x + c)^3*e^(21/2) - 12*sqrt(2)*a^5*d^3*cos(d*x + c)^2*e^(21/2) - 4*sqrt(2)*
a^5*d^3*cos(d*x + c)*e^(21/2) + 8*sqrt(2)*a^5*d^3*e^(21/2))*sin(d*x + c))*(1/(a^6*d^4))^(3/4)*e^(21/2) + (2*sq
rt(2)*a^2*d*cos(d*x + c)^5*e^(35/2) + sqrt(2)*a^2*d*cos(d*x + c)^4*e^(35/2) - 13*sqrt(2)*a^2*d*cos(d*x + c)^3*
e^(35/2) - 8*sqrt(2)*a^2*d*cos(d*x + c)^2*e^(35/2) + 12*sqrt(2)*a^2*d*cos(d*x + c)*e^(35/2) + 8*sqrt(2)*a^2*d*
e^(35/2) - (7*sqrt(2)*a^2*d*cos(d*x + c)^3*e^(35/2) + 4*sqrt(2)*a^2*d*cos(d*x + c)^2*e^(35/2) - 12*sqrt(2)*a^2
*d*cos(d*x + c)*e^(35/2) - 8*sqrt(2)*a^2*d*e^(35/2))*sin(d*x + c))*(1/(a^6*d^4))^(1/4)*e^(7/2))*sqrt(a*sin(d*x
+ c) + a)*sqrt(cos(d*x + c)))/(a*cos(d*x + c)^6*e^21 + a*cos(d*x + c)^5*e^21 - 8*a*cos(d*x + c)^4*e^21 - 8*a*
cos(d*x + c)^3*e^21 + 8*a*cos(d*x + c)^2*e^21 + 8*a*cos(d*x + c)*e^21 - 4*(a*cos(d*x + c)^4*e^21 + a*cos(d*x +
c)^3*e^21 - 2*a*cos(d*x + c)^2*e^21 - 2*a*cos(d*x + c)*e^21)*sin(d*x + c)))*e^(7/2) - 20*sqrt(2)*a^2*d*(1/(a^
6*d^4))^(1/4)*arctan(1/4*(2*sqrt(1/2)*((sqrt(2)*a^5*d^3*cos(d*x + c)^6 + 5*sqrt(2)*a^5*d^3*cos(d*x + c)^5 - 8*
sqrt(2)*a^5*d^3*cos(d*x + c)^4 - 20*sqrt(2)*a^5*d^3*cos(d*x + c)^3 + 8*sqrt(2)*a^5*d^3*cos(d*x + c)^2 + 16*sqr
t(2)*a^5*d^3*cos(d*x + c) + (sqrt(2)*a^5*d^3*cos(d*x + c)^5 - 4*sqrt(2)*a^5*d^3*cos(d*x + c)^4 - 12*sqrt(2)*a^
5*d^3*cos(d*x + c)^3 + 8*sqrt(2)*a^5*d^3*cos(d*x + c)^2 + 16*sqrt(2)*a^5*d^3*cos(d*x + c))*sin(d*x + c))*(1/(a
^6*d^4))^(3/4)*e^(21/2) + (sqrt(2)*a^2*d*cos(d*x + c)^6*e^7 - 3*sqrt(2)*a^2*d*cos(d*x + c)^5*e^7 - 8*sqrt(2)*a
^2*d*cos(d*x + c)^4*e^7 + 4*sqrt(2)*a^2*d*cos(d*x + c)^3*e^7 + 8*sqrt(2)*a^2*d*cos(d*x + c)^2*e^7 - (sqrt(2)*a
^2*d*cos(d*x + c)^5*e^7 + 4*sqrt(2)*a^2*d*cos(d*x + c)^4*e^7 - 4*sqrt(2)*a^2*d*cos(d*x + c)^3*e^7 - 8*sqrt(2)*
a^2*d*cos(d*x + c)^2*e^7)*sin(d*x + c))*(1/(a^6*d^4))^(1/4)*e^(7/2) + (cos(d*x + c)^4*e^(21/2) - 3*cos(d*x + c
)^3*e^(21/2) - 8*cos(d*x + c)^2*e^(21/2) + (2*a^3*d^2*cos(d*x + c)^5*e^(7/2) - 5*a^3*d^2*cos(d*x + c)^4*e^(7/2
) - 19*a^3*d^2*cos(d*x + c)^3*e^(7/2) + 20*a^3*d^2*cos(d*x + c)*e^(7/2) + 8*a^3*d^2*e^(7/2) - (2*a^3*d^2*cos(d
*x + c)^4*e^(7/2) + 9*a^3*d^2*cos(d*x + c)^3*e^(7/2) - 4*a^3*d^2*cos(d*x + c)^2*e^(7/2) - 20*a^3*d^2*cos(d*x +
c)*e^(7/2) - 8*a^3*d^2*e^(7/2))*sin(d*x + c))*sqrt(1/(a^6*d^4))*e^7 + 4*cos(d*x + c)*e^(21/2) - (cos(d*x + c)
^3*e^(21/2) + 4*cos(d*x + c)^2*e^(21/2) - 4*cos(d*x + c)*e^(21/2) - 8*e^(21/2))*sin(d*x + c) + 8*e^(21/2))*sqr
t(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))*sqrt((2*a*cos(d*x + c)*e^21*sin(d*x + c) + 2*a*cos(d*x + c)*e^21 + (
a^4*d^2*e^14*sin(d*x + c) + a^4*d^2*e^14)*sqrt(1/(a^6*d^4))*e^7 - (sqrt(2)*a^5*d^3*(1/(a^6*d^4))^(3/4)*cos(d*x
+ c)*e^21 + (sqrt(2)*a^2*d*e^(35/2)*sin(d*x + c) + sqrt(2)*a^2*d*e^(35/2))*(1/(a^6*d^4))^(1/4)*e^(7/2))*sqrt(
a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(sin(d*x + c) + 1)) - ((7*sqrt(2)*a^5*d^3*cos(d*x + c)^4*e^(21/2) + 3*
sqrt(2)*a^5*d^3*cos(d*x + c)^3*e^(21/2) - 16*sq...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________