3.3.84 \(\int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{3/2}} \, dx\) [284]

Optimal. Leaf size=210 \[ \frac {4 a \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}}-\frac {2 a^2 \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d e^{3/2} (a+a \cos (c+d x)+a \sin (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2755, 2763, 2854, 209, 2912, 65, 221} \begin {gather*} -\frac {2 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} (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac {2 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} (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {4 a \sqrt {a \sin (c+d x)+a}}{d e \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*Sqrt[a + a*Sin[c + d*x]])/(d*e*Sqrt[e*Cos[c + d*x]]) - (2*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)*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) - (2*a^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]])/(d*e^(3/2)*(a + a*Cos[c + d*x] + a*Sin[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 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))^{3/2}}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {4 a \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}}-\frac {a^2 \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx}{e^2}\\ &=\frac {4 a \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (a^2 \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}{e (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (a^2 \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}{e (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {4 a \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (a^2 \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 )}{d e (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (2 a^2 \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 {4 a \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}}-\frac {2 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} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (2 a^2 \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 {4 a \sqrt {a+a \sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}}-\frac {2 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} (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {2 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} (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 = 0.12, size = 75, normalized size = 0.36 \begin {gather*} \frac {4 \sqrt [4]{2} \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{3/2}}{d e \sqrt {e \cos (c+d x)} (1+\sin (c+d x))^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.17, size = 323, normalized size = 1.54

method	result	size
default	\(-\frac {2 \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (-1+\cos \left (d x +c \right )\right ) \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )+\sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )-\sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )-\sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right )-2 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+2 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+2 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )}{d \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin \left (d x +c \right )-\cos \left (d x +c \right )+1\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}\)	\(323\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d*(a*(1+sin(d*x+c)))^(3/2)*(-1+cos(d*x+c))*(2^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2)
)*sin(d*x+c)+2^(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
)-2^(1/2)*arctan(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2^(1/2))-2^(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))-2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+2*(-2*cos(d*x+c
)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)+2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))/sin(d*x+c)/(-2*cos(d*x+c)/(1+cos(d*
x+c)))^(1/2)/(sin(d*x+c)-cos(d*x+c)+1)/(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))^(3/2)/(e*cos(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

e^(-3/2)*integrate((a*sin(d*x + c) + a)^(3/2)/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. 3313 vs. \(2 (168) = 336\).
time = 176.02, size = 3313, normalized size = 15.78 \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))^(3/2)/(e*cos(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(sin(c + d*x) + 1))**(3/2)/(e*cos(c + d*x))**(3/2), x)

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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))^(3/2)/(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 )}^{3/2}}{{\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))^(3/2)/(e*cos(c + d*x))^(3/2),x)

[Out]

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

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