3.3.92 \(\int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx\) [292]

Optimal. Leaf size=239 \[ \frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}-\frac {5 a^2 \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (1+\cos (c+d x)+\sin (c+d x))}-\frac {5 a^2 \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (1+\cos (c+d x)+\sin (c+d x))}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 239, 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 = {2755, 2757, 2763, 2854, 209, 2912, 65, 221} \begin {gather*} \frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a \sin (c+d x)+a}}-\frac {5 a^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 )}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {5 a^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 )}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac {4 a (a \sin (c+d x)+a)^{3/2}}{d e \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(3/2),x]

[Out]

(5*a^3*(e*Cos[c + d*x])^(3/2))/(d*e^3*Sqrt[a + a*Sin[c + d*x]]) - (5*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]
*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) - (5*a^2*ArcTa
n[(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]])/(d*e^(3/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) + (4*a*(a + a*Sin[c + d*x])^(3/2))/(d*e*Sqrt[e*Cos[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 2755

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

Rule 2757

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[a*((2*m + p - 1)/(m + p)), Int
[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2,
0] && GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2763

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[g*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[g*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(b + b*Cos[e + f*x] + a
*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 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 {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^2\right ) \int \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)} \, dx}{e^2}\\ &=\frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^3\right ) \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx}{2 e^2}\\ &=\frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^3 \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}{2 e (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (5 a^3 \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}{2 e (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^3 \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 )}{2 d e (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (5 a^3 \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 )}{d e (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {5 a^3 \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)}}{d e^{3/2} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (5 a^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 )}{d e^2 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt {a+a \sin (c+d x)}}+\frac {4 a (a+a \sin (c+d x))^{3/2}}{d e \sqrt {e \cos (c+d x)}}-\frac {5 a^3 \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)}}{d e^{3/2} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {5 a^3 \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)}}{d e^{3/2} (a+a \cos (c+d x)+a \sin (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.15, size = 75, normalized size = 0.31 \begin {gather*} \frac {8 \sqrt [4]{2} \, _2F_1\left (-\frac {5}{4},-\frac {1}{4};\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{5/2}}{d e \sqrt {e \cos (c+d x)} (1+\sin (c+d x))^{9/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(3/2),x]

[Out]

(8*2^(1/4)*Hypergeometric2F1[-5/4, -1/4, 3/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(5/2))/(d*e*Sqrt[e*
Cos[c + d*x]]*(1 + Sin[c + d*x])^(9/4))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(442\) vs. \(2(209)=418\).
time = 0.16, size = 443, normalized size = 1.85

method	result	size
default	\(-\frac {\left (5 \sqrt {2}\, \cos \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+5 \sqrt {2}\, \cos \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right )-5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )-5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )+5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right )-4 \cos \left (d x +c \right ) \sin \left (d x +c \right )+36 \cos \left (d x +c \right )\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}}}{4 d \left (\cos ^{2}\left (d x +c \right )-2 \sin \left (d x +c \right )-2\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}\)	\(443\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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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((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

e^(-3/2)*integrate((a*sin(d*x + c) + a)^(5/2)/cos(d*x + c)^(3/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3371 vs. \(2 (190) = 380\).
time = 179.78, size = 3371, normalized size = 14.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/8*(20*sqrt(2)*(a^10/d^4)^(1/4)*d*arctan(-1/4*(2*sqrt(1/2)*((sqrt(2)*d^3*cos(d*x + c)^6*e^(9/2) - 3*sqrt(2)*d
^3*cos(d*x + c)^5*e^(9/2) - 8*sqrt(2)*d^3*cos(d*x + c)^4*e^(9/2) + 4*sqrt(2)*d^3*cos(d*x + c)^3*e^(9/2) + 8*sq
rt(2)*d^3*cos(d*x + c)^2*e^(9/2) - (sqrt(2)*d^3*cos(d*x + c)^5*e^(9/2) + 4*sqrt(2)*d^3*cos(d*x + c)^4*e^(9/2)
- 4*sqrt(2)*d^3*cos(d*x + c)^3*e^(9/2) - 8*sqrt(2)*d^3*cos(d*x + c)^2*e^(9/2))*sin(d*x + c))*(a^10/d^4)^(3/4)*
e^(-9/2) + (sqrt(2)*a^5*d*cos(d*x + c)^6*e^(3/2) + 5*sqrt(2)*a^5*d*cos(d*x + c)^5*e^(3/2) - 8*sqrt(2)*a^5*d*co
s(d*x + c)^4*e^(3/2) - 20*sqrt(2)*a^5*d*cos(d*x + c)^3*e^(3/2) + 8*sqrt(2)*a^5*d*cos(d*x + c)^2*e^(3/2) + 16*s
qrt(2)*a^5*d*cos(d*x + c)*e^(3/2) + (sqrt(2)*a^5*d*cos(d*x + c)^5*e^(3/2) - 4*sqrt(2)*a^5*d*cos(d*x + c)^4*e^(
3/2) - 12*sqrt(2)*a^5*d*cos(d*x + c)^3*e^(3/2) + 8*sqrt(2)*a^5*d*cos(d*x + c)^2*e^(3/2) + 16*sqrt(2)*a^5*d*cos
(d*x + c)*e^(3/2))*sin(d*x + c))*(a^10/d^4)^(1/4)*e^(-3/2) - (a^7*cos(d*x + c)^4 - 3*a^7*cos(d*x + c)^3 - 8*a^
7*cos(d*x + c)^2 + 4*a^7*cos(d*x + c) + 8*a^7 + (2*a^2*d^2*cos(d*x + c)^5*e^3 - 5*a^2*d^2*cos(d*x + c)^4*e^3 -
19*a^2*d^2*cos(d*x + c)^3*e^3 + 20*a^2*d^2*cos(d*x + c)*e^3 + 8*a^2*d^2*e^3 - (2*a^2*d^2*cos(d*x + c)^4*e^3 +
9*a^2*d^2*cos(d*x + c)^3*e^3 - 4*a^2*d^2*cos(d*x + c)^2*e^3 - 20*a^2*d^2*cos(d*x + c)*e^3 - 8*a^2*d^2*e^3)*si
n(d*x + c))*sqrt(a^10/d^4)*e^(-3) - (a^7*cos(d*x + c)^3 + 4*a^7*cos(d*x + c)^2 - 4*a^7*cos(d*x + c) - 8*a^7)*s
in(d*x + c))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))*sqrt((2*a^15*cos(d*x + c)*sin(d*x + c) + 2*a^15*cos(
d*x + c) + (a^10*d^2*e^3*sin(d*x + c) + a^10*d^2*e^3)*sqrt(a^10/d^4)*e^(-3) + (sqrt(2)*(a^10/d^4)^(1/4)*a^12*d
*cos(d*x + c) + (sqrt(2)*a^7*d^3*e^(9/2)*sin(d*x + c) + sqrt(2)*a^7*d^3*e^(9/2))*(a^10/d^4)^(3/4)*e^(-9/2))*sq
rt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(sin(d*x + c) + 1)) + ((2*sqrt(2)*a^7*d^3*cos(d*x + c)^5*e^(9/2) +
sqrt(2)*a^7*d^3*cos(d*x + c)^4*e^(9/2) - 13*sqrt(2)*a^7*d^3*cos(d*x + c)^3*e^(9/2) - 8*sqrt(2)*a^7*d^3*cos(d*x
+ c)^2*e^(9/2) + 12*sqrt(2)*a^7*d^3*cos(d*x + c)*e^(9/2) + 8*sqrt(2)*a^7*d^3*e^(9/2) - (7*sqrt(2)*a^7*d^3*cos
(d*x + c)^3*e^(9/2) + 4*sqrt(2)*a^7*d^3*cos(d*x + c)^2*e^(9/2) - 12*sqrt(2)*a^7*d^3*cos(d*x + c)*e^(9/2) - 8*s
qrt(2)*a^7*d^3*e^(9/2))*sin(d*x + c))*(a^10/d^4)^(3/4)*e^(-9/2) + (7*sqrt(2)*a^12*d*cos(d*x + c)^4*e^(3/2) + 3
*sqrt(2)*a^12*d*cos(d*x + c)^3*e^(3/2) - 16*sqrt(2)*a^12*d*cos(d*x + c)^2*e^(3/2) - 4*sqrt(2)*a^12*d*cos(d*x +
c)*e^(3/2) + 8*sqrt(2)*a^12*d*e^(3/2) + (2*sqrt(2)*a^12*d*cos(d*x + c)^4*e^(3/2) + sqrt(2)*a^12*d*cos(d*x + c
)^3*e^(3/2) - 12*sqrt(2)*a^12*d*cos(d*x + c)^2*e^(3/2) - 4*sqrt(2)*a^12*d*cos(d*x + c)*e^(3/2) + 8*sqrt(2)*a^1
2*d*e^(3/2))*sin(d*x + c))*(a^10/d^4)^(1/4)*e^(-3/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(a^15*cos(d
*x + c)^6 + a^15*cos(d*x + c)^5 - 8*a^15*cos(d*x + c)^4 - 8*a^15*cos(d*x + c)^3 + 8*a^15*cos(d*x + c)^2 + 8*a^
15*cos(d*x + c) - 4*(a^15*cos(d*x + c)^4 + a^15*cos(d*x + c)^3 - 2*a^15*cos(d*x + c)^2 - 2*a^15*cos(d*x + c))*
sin(d*x + c)))*cos(d*x + c) - 20*sqrt(2)*(a^10/d^4)^(1/4)*d*arctan(1/4*(2*sqrt(1/2)*((sqrt(2)*d^3*cos(d*x + c)
^6*e^(9/2) - 3*sqrt(2)*d^3*cos(d*x + c)^5*e^(9/2) - 8*sqrt(2)*d^3*cos(d*x + c)^4*e^(9/2) + 4*sqrt(2)*d^3*cos(d
*x + c)^3*e^(9/2) + 8*sqrt(2)*d^3*cos(d*x + c)^2*e^(9/2) - (sqrt(2)*d^3*cos(d*x + c)^5*e^(9/2) + 4*sqrt(2)*d^3
*cos(d*x + c)^4*e^(9/2) - 4*sqrt(2)*d^3*cos(d*x + c)^3*e^(9/2) - 8*sqrt(2)*d^3*cos(d*x + c)^2*e^(9/2))*sin(d*x
+ c))*(a^10/d^4)^(3/4)*e^(-9/2) + (sqrt(2)*a^5*d*cos(d*x + c)^6*e^(3/2) + 5*sqrt(2)*a^5*d*cos(d*x + c)^5*e^(3
/2) - 8*sqrt(2)*a^5*d*cos(d*x + c)^4*e^(3/2) - 20*sqrt(2)*a^5*d*cos(d*x + c)^3*e^(3/2) + 8*sqrt(2)*a^5*d*cos(d
*x + c)^2*e^(3/2) + 16*sqrt(2)*a^5*d*cos(d*x + c)*e^(3/2) + (sqrt(2)*a^5*d*cos(d*x + c)^5*e^(3/2) - 4*sqrt(2)*
a^5*d*cos(d*x + c)^4*e^(3/2) - 12*sqrt(2)*a^5*d*cos(d*x + c)^3*e^(3/2) + 8*sqrt(2)*a^5*d*cos(d*x + c)^2*e^(3/2
) + 16*sqrt(2)*a^5*d*cos(d*x + c)*e^(3/2))*sin(d*x + c))*(a^10/d^4)^(1/4)*e^(-3/2) + (a^7*cos(d*x + c)^4 - 3*a
^7*cos(d*x + c)^3 - 8*a^7*cos(d*x + c)^2 + 4*a^7*cos(d*x + c) + 8*a^7 + (2*a^2*d^2*cos(d*x + c)^5*e^3 - 5*a^2*
d^2*cos(d*x + c)^4*e^3 - 19*a^2*d^2*cos(d*x + c)^3*e^3 + 20*a^2*d^2*cos(d*x + c)*e^3 + 8*a^2*d^2*e^3 - (2*a^2*
d^2*cos(d*x + c)^4*e^3 + 9*a^2*d^2*cos(d*x + c)^3*e^3 - 4*a^2*d^2*cos(d*x + c)^2*e^3 - 20*a^2*d^2*cos(d*x + c)
*e^3 - 8*a^2*d^2*e^3)*sin(d*x + c))*sqrt(a^10/d^4)*e^(-3) - (a^7*cos(d*x + c)^3 + 4*a^7*cos(d*x + c)^2 - 4*a^7
*cos(d*x + c) - 8*a^7)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))*sqrt((2*a^15*cos(d*x + c)*si
n(d*x + c) + 2*a^15*cos(d*x + c) + (a^10*d^2*e^3*sin(d*x + c) + a^10*d^2*e^3)*sqrt(a^10/d^4)*e^(-3) - (sqrt(2)
*(a^10/d^4)^(1/4)*a^12*d*cos(d*x + c) + (sqrt(2)*a^7*d^3*e^(9/2)*sin(d*x + c) + sqrt(2)*a^7*d^3*e^(9/2))*(a^10
/d^4)^(3/4)*e^(-9/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(sin(d*x + c) + 1)) + ((2*sqrt(2)*a^7*d^3*c
os(d*x + c)^5*e^(9/2) + sqrt(2)*a^7*d^3*cos(d*x + c)^4*e^(9/2) - 13*sqrt(2)*a^7*d^3*cos(d*x + c)^3*e^(9/2) - 8
*sqrt(2)*a^7*d^3*cos(d*x + c)^2*e^(9/2) + 12*sq...

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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((a+a*sin(d*x+c))**(5/2)/(e*cos(d*x+c))**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

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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((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(3/2),x)

[Out]

int((a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(3/2), x)

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