3.3.91 \(\int \frac {(a+a \sin (c+d x))^{5/2}}{\sqrt {e \cos (c+d x)}} \, dx\) [291]

Optimal. Leaf size=247 \[ -\frac {7 a^2 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {21 a^2 \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d \sqrt {e} (1+\cos (c+d x)+\sin (c+d x))}+\frac {21 a^2 \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d \sqrt {e} (1+\cos (c+d x)+\sin (c+d x))}-\frac {a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e} \]

[Out]

-1/2*a*(a+a*sin(d*x+c))^(3/2)*(e*cos(d*x+c))^(1/2)/d/e-7/4*a^2*(e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/e
-21/4*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/(1+cos(d*x+c)+si
n(d*x+c))/e^(1/2)+21/4*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/(1+cos(d*x+c)+sin(d*x+c))/e^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2757, 2756, 2854, 209, 2912, 65, 221} \begin {gather*} \frac {21 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 )}{4 d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {7 a^2 \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{4 d e}-\frac {21 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 )}{4 d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {a (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}}{2 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*a^2*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*e) - (21*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]
*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (21*a^2*Ar
cTan[(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*S
in[c + d*x]])/(4*d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) - (a*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(
3/2))/(2*d*e)

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 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 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}}{\sqrt {e \cos (c+d x)}} \, dx &=-\frac {a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}+\frac {1}{4} (7 a) \int \frac {(a+a \sin (c+d x))^{3/2}}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {7 a^2 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}+\frac {1}{8} \left (21 a^2\right ) \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {7 a^2 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}+\frac {\left (21 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}{8 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (21 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}{8 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {7 a^2 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}-\frac {\left (21 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 )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (21 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 )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {7 a^2 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}+\frac {21 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)}}{4 d \sqrt {e} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (21 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 )}{4 d e (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {7 a^2 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d e}-\frac {a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}{2 d e}-\frac {21 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)}}{4 d \sqrt {e} (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {21 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)}}{4 d \sqrt {e} (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 = 0.09, size = 76, normalized size = 0.31 \begin {gather*} -\frac {8\ 2^{3/4} a \sqrt {e \cos (c+d x)} \, _2F_1\left (-\frac {7}{4},\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{3/2}}{d e (1+\sin (c+d x))^{7/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*2^(3/4)*a*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[-7/4, 1/4, 5/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x
]))^(3/2))/(d*e*(1 + Sin[c + d*x])^(7/4))

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Maple [A]
time = 0.23, size = 283, normalized size = 1.15

method	result	size
default	\(\frac {\left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}} \left (21 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )-21 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )+4 \left (\cos ^{3}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+18 \left (\cos ^{2}\left (d x +c \right )\right )-22 \cos \left (d x +c \right ) \sin \left (d x +c \right )-22 \cos \left (d x +c \right )\right )}{8 d \left (\cos ^{3}\left (d x +c \right )-\left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-3 \left (\cos ^{2}\left (d x +c \right )\right )-2 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2 \cos \left (d x +c \right )+4 \sin \left (d x +c \right )+4\right ) \sqrt {e \cos \left (d x +c \right )}}\)	\(283\)

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))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/8/d*(a*(1+sin(d*x+c)))^(5/2)*(21*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*2^(1/2))*sin(d*x+c)-21*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)+4*cos(d*x+c)^3+4*cos(d*x+c)^2*sin(d*x+c)+18
*cos(d*x+c)^2-22*cos(d*x+c)*sin(d*x+c)-22*cos(d*x+c))/(cos(d*x+c)^3-cos(d*x+c)^2*sin(d*x+c)-3*cos(d*x+c)^2-2*c
os(d*x+c)*sin(d*x+c)-2*cos(d*x+c)+4*sin(d*x+c)+4)/(e*cos(d*x+c))^(1/2)

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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))^(1/2),x, algorithm="maxima")

[Out]

e^(-1/2)*integrate((a*sin(d*x + c) + a)^(5/2)/sqrt(cos(d*x + c)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3330 vs. \(2 (190) = 380\).
time = 190.06, size = 3330, normalized size = 13.48 \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))^(1/2),x, algorithm="fricas")

[Out]

1/32*(84*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^(3/2) + 5*sqrt(2)*
d^3*cos(d*x + c)^5*e^(3/2) - 8*sqrt(2)*d^3*cos(d*x + c)^4*e^(3/2) - 20*sqrt(2)*d^3*cos(d*x + c)^3*e^(3/2) + 8*
sqrt(2)*d^3*cos(d*x + c)^2*e^(3/2) + 16*sqrt(2)*d^3*cos(d*x + c)*e^(3/2) + (sqrt(2)*d^3*cos(d*x + c)^5*e^(3/2)
- 4*sqrt(2)*d^3*cos(d*x + c)^4*e^(3/2) - 12*sqrt(2)*d^3*cos(d*x + c)^3*e^(3/2) + 8*sqrt(2)*d^3*cos(d*x + c)^2
*e^(3/2) + 16*sqrt(2)*d^3*cos(d*x + c)*e^(3/2))*sin(d*x + c))*(a^10/d^4)^(3/4)*e^(-3/2) + (sqrt(2)*a^5*d*cos(d
*x + c)^6*e^(1/2) - 3*sqrt(2)*a^5*d*cos(d*x + c)^5*e^(1/2) - 8*sqrt(2)*a^5*d*cos(d*x + c)^4*e^(1/2) + 4*sqrt(2
)*a^5*d*cos(d*x + c)^3*e^(1/2) + 8*sqrt(2)*a^5*d*cos(d*x + c)^2*e^(1/2) - (sqrt(2)*a^5*d*cos(d*x + c)^5*e^(1/2
) + 4*sqrt(2)*a^5*d*cos(d*x + c)^4*e^(1/2) - 4*sqrt(2)*a^5*d*cos(d*x + c)^3*e^(1/2) - 8*sqrt(2)*a^5*d*cos(d*x
+ c)^2*e^(1/2))*sin(d*x + c))*(a^10/d^4)^(1/4)*e^(-1/2) - (a^7*cos(d*x + c)^4 - 3*a^7*cos(d*x + c)^3 - 8*a^7*c
os(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 - 5*a^2*d^2*cos(d*x + c)^4*e - 19*a^2
*d^2*cos(d*x + c)^3*e + 20*a^2*d^2*cos(d*x + c)*e + 8*a^2*d^2*e - (2*a^2*d^2*cos(d*x + c)^4*e + 9*a^2*d^2*cos(
d*x + c)^3*e - 4*a^2*d^2*cos(d*x + c)^2*e - 20*a^2*d^2*cos(d*x + c)*e - 8*a^2*d^2*e)*sin(d*x + c))*sqrt(a^10/d
^4)*e^(-1) - (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)*sin(d*x + c) + 2*a^15*cos(d*x + c) + (a^10*d^2*e*
sin(d*x + c) + a^10*d^2*e)*sqrt(a^10/d^4)*e^(-1) + (sqrt(2)*(a^10/d^4)^(3/4)*a^7*d^3*cos(d*x + c) + (sqrt(2)*a
^12*d*e^(1/2)*sin(d*x + c) + sqrt(2)*a^12*d*e^(1/2))*(a^10/d^4)^(1/4)*e^(-1/2))*sqrt(a*sin(d*x + c) + a)*sqrt(
cos(d*x + c)))/(sin(d*x + c) + 1)) - ((7*sqrt(2)*a^7*d^3*cos(d*x + c)^4*e^(3/2) + 3*sqrt(2)*a^7*d^3*cos(d*x +
c)^3*e^(3/2) - 16*sqrt(2)*a^7*d^3*cos(d*x + c)^2*e^(3/2) - 4*sqrt(2)*a^7*d^3*cos(d*x + c)*e^(3/2) + 8*sqrt(2)*
a^7*d^3*e^(3/2) + (2*sqrt(2)*a^7*d^3*cos(d*x + c)^4*e^(3/2) + sqrt(2)*a^7*d^3*cos(d*x + c)^3*e^(3/2) - 12*sqrt
(2)*a^7*d^3*cos(d*x + c)^2*e^(3/2) - 4*sqrt(2)*a^7*d^3*cos(d*x + c)*e^(3/2) + 8*sqrt(2)*a^7*d^3*e^(3/2))*sin(d
*x + c))*(a^10/d^4)^(3/4)*e^(-3/2) + (2*sqrt(2)*a^12*d*cos(d*x + c)^5*e^(1/2) + sqrt(2)*a^12*d*cos(d*x + c)^4*
e^(1/2) - 13*sqrt(2)*a^12*d*cos(d*x + c)^3*e^(1/2) - 8*sqrt(2)*a^12*d*cos(d*x + c)^2*e^(1/2) + 12*sqrt(2)*a^12
*d*cos(d*x + c)*e^(1/2) + 8*sqrt(2)*a^12*d*e^(1/2) - (7*sqrt(2)*a^12*d*cos(d*x + c)^3*e^(1/2) + 4*sqrt(2)*a^12
*d*cos(d*x + c)^2*e^(1/2) - 12*sqrt(2)*a^12*d*cos(d*x + c)*e^(1/2) - 8*sqrt(2)*a^12*d*e^(1/2))*sin(d*x + c))*(
a^10/d^4)^(1/4)*e^(-1/2))*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*co
s(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))) - 84*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^(3/2) + 5*sqrt(2)*d^3*cos(d*x + c)^5
*e^(3/2) - 8*sqrt(2)*d^3*cos(d*x + c)^4*e^(3/2) - 20*sqrt(2)*d^3*cos(d*x + c)^3*e^(3/2) + 8*sqrt(2)*d^3*cos(d*
x + c)^2*e^(3/2) + 16*sqrt(2)*d^3*cos(d*x + c)*e^(3/2) + (sqrt(2)*d^3*cos(d*x + c)^5*e^(3/2) - 4*sqrt(2)*d^3*c
os(d*x + c)^4*e^(3/2) - 12*sqrt(2)*d^3*cos(d*x + c)^3*e^(3/2) + 8*sqrt(2)*d^3*cos(d*x + c)^2*e^(3/2) + 16*sqrt
(2)*d^3*cos(d*x + c)*e^(3/2))*sin(d*x + c))*(a^10/d^4)^(3/4)*e^(-3/2) + (sqrt(2)*a^5*d*cos(d*x + c)^6*e^(1/2)
- 3*sqrt(2)*a^5*d*cos(d*x + c)^5*e^(1/2) - 8*sqrt(2)*a^5*d*cos(d*x + c)^4*e^(1/2) + 4*sqrt(2)*a^5*d*cos(d*x +
c)^3*e^(1/2) + 8*sqrt(2)*a^5*d*cos(d*x + c)^2*e^(1/2) - (sqrt(2)*a^5*d*cos(d*x + c)^5*e^(1/2) + 4*sqrt(2)*a^5*
d*cos(d*x + c)^4*e^(1/2) - 4*sqrt(2)*a^5*d*cos(d*x + c)^3*e^(1/2) - 8*sqrt(2)*a^5*d*cos(d*x + c)^2*e^(1/2))*si
n(d*x + c))*(a^10/d^4)^(1/4)*e^(-1/2) + (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 - 5*a^2*d^2*cos(d*x + c)^4*e - 19*a^2*d^2*cos(d*x + c)^
3*e + 20*a^2*d^2*cos(d*x + c)*e + 8*a^2*d^2*e - (2*a^2*d^2*cos(d*x + c)^4*e + 9*a^2*d^2*cos(d*x + c)^3*e - 4*a
^2*d^2*cos(d*x + c)^2*e - 20*a^2*d^2*cos(d*x + c)*e - 8*a^2*d^2*e)*sin(d*x + c))*sqrt(a^10/d^4)*e^(-1) - (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)*sqr
t(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*sin(d*x + c) + a^1
0*d^2*e)*sqrt(a^10/d^4)*e^(-1) - (sqrt(2)*(a^10/d^4)^(3/4)*a^7*d^3*cos(d*x + c) + (sqrt(2)*a^12*d*e^(1/2)*sin(
d*x + c) + sqrt(2)*a^12*d*e^(1/2))*(a^10/d^4)^(1/4)*e^(-1/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(si
n(d*x + c) + 1)) - ((7*sqrt(2)*a^7*d^3*cos(d*x + c)^4*e^(3/2) + 3*sqrt(2)*a^7*d^3*cos(d*x + c)^3*e^(3/2) - 16*
sqrt(2)*a^7*d^3*cos(d*x + c)^2*e^(3/2) - 4*sqrt(2)*a^7*d^3*cos(d*x + c)*e^(3/2) + 8*sqrt(2)*a^7*d^3*e^(3/2) +
(2*sqrt(2)*a^7*d^3*cos(d*x + c)^4*e^(3/2) + sqr...

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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))**(1/2),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))^(1/2),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}}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,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))^(1/2),x)

[Out]

int((a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(1/2), x)

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