3.3.82 \(\int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx\) [282]
Optimal. Leaf size=243 \[ -\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {5 a \sqrt {e} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {5 a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (1+\cos (c+d x)+\sin (c+d x))} \]
[Out]
-5/4*a^2*(e*cos(d*x+c))^(3/2)/d/e/(a+a*sin(d*x+c))^(1/2)-1/2*a*(e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^(1/2)/d/e
+5/4*a*arcsinh((e*cos(d*x+c))^(1/2)/e^(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)+sin(d*x+c))+5/4*a*arctan(sin(d*x+c)*e^(1/2)/(e*cos(d*x+c))^(1/2)/(1+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)+sin(d*x+c))
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Rubi [A]
time = 0.24, antiderivative size = 243, 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, 2763,
2854, 209, 2912, 65, 221} \begin {gather*} -\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a \sin (c+d x)+a}}+\frac {5 a \sqrt {e} \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 (\sin (c+d x)+\cos (c+d x)+1)}-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}+\frac {5 a \sqrt {e} \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 (\sin (c+d x)+\cos (c+d x)+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2),x]
[Out]
(-5*a^2*(e*Cos[c + d*x])^(3/2))/(4*d*e*Sqrt[a + a*Sin[c + d*x]]) - (a*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c
+ d*x]])/(2*d*e) + (5*a*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c
+ d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (5*a*Sqrt[e]*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*d*(1 + Cos[c + d*x] + 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 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 \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx &=-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {1}{4} (5 a) \int \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {1}{8} \left (5 a^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {\left (5 a^2 e \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 (5 a^2 e \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 {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {\left (5 a^2 e \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 (5 a^2 e \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 {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {5 a^2 \sqrt {e} \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 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (5 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 )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {5 a^2 \sqrt {e} \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 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {5 a^2 \sqrt {e} \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 (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 = 77, normalized size = 0.32 \begin {gather*} -\frac {8 \sqrt [4]{2} (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac {5}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{3/2}}{3 d e (1+\sin (c+d x))^{9/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2),x]
[Out]
(-8*2^(1/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-5/4, 3/4, 7/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x
]))^(3/2))/(3*d*e*(1 + Sin[c + d*x])^(9/4))
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Maple [A]
time = 0.20, size = 262, normalized size = 1.08
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\left (5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )+5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )-4 \left (\cos ^{3}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+14 \left (\cos ^{2}\left (d x +c \right )\right )+10 \cos \left (d x +c \right ) \sin \left (d x +c \right )-10 \cos \left (d x +c \right )\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}} \sqrt {e \cos \left (d x +c \right )}}{8 d \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos ^{2}\left (d x +c \right )-2 \sin \left (d x +c \right )+\cos \left (d x +c \right )-2\right ) \cos \left (d x +c \right )}\) |
\(262\) |
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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))^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/8/d*(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*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*si
n(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)+14*cos(d*x+c)^2+10*cos(d*x+c)
*sin(d*x+c)-10*cos(d*x+c))*(a*(1+sin(d*x+c)))^(3/2)*(e*cos(d*x+c))^(1/2)/(cos(d*x+c)*sin(d*x+c)+cos(d*x+c)^2-2
*sin(d*x+c)+cos(d*x+c)-2)/cos(d*x+c)
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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))^(1/2),x, algorithm="maxima")
[Out]
e^(1/2)*integrate((a*sin(d*x + c) + a)^(3/2)*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. 3443 vs.
\(2 (186) = 372\).
time = 195.04, size = 3443, normalized size = 14.17 \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))^(1/2),x, algorithm="fricas")
[Out]
-1/32*(20*(sqrt(2)*d*sin(d*x + c) + sqrt(2)*d)*(a^6/d^4)^(1/4)*arctan(-1/4*(2*sqrt(1/2)*((sqrt(2)*d^3*cos(d*x
+ c)^6 - 3*sqrt(2)*d^3*cos(d*x + c)^5 - 8*sqrt(2)*d^3*cos(d*x + c)^4 + 4*sqrt(2)*d^3*cos(d*x + c)^3 + 8*sqrt(2
)*d^3*cos(d*x + c)^2 - (sqrt(2)*d^3*cos(d*x + c)^5 + 4*sqrt(2)*d^3*cos(d*x + c)^4 - 4*sqrt(2)*d^3*cos(d*x + c)
^3 - 8*sqrt(2)*d^3*cos(d*x + c)^2)*sin(d*x + c))*(a^6/d^4)^(3/4)*e^(3/2) + (sqrt(2)*a^3*d*cos(d*x + c)^6*e + 5
*sqrt(2)*a^3*d*cos(d*x + c)^5*e - 8*sqrt(2)*a^3*d*cos(d*x + c)^4*e - 20*sqrt(2)*a^3*d*cos(d*x + c)^3*e + 8*sqr
t(2)*a^3*d*cos(d*x + c)^2*e + 16*sqrt(2)*a^3*d*cos(d*x + c)*e + (sqrt(2)*a^3*d*cos(d*x + c)^5*e - 4*sqrt(2)*a^
3*d*cos(d*x + c)^4*e - 12*sqrt(2)*a^3*d*cos(d*x + c)^3*e + 8*sqrt(2)*a^3*d*cos(d*x + c)^2*e + 16*sqrt(2)*a^3*d
*cos(d*x + c)*e)*sin(d*x + c))*(a^6/d^4)^(1/4)*e^(1/2) - (a^4*cos(d*x + c)^4*e^(3/2) - 3*a^4*cos(d*x + c)^3*e^
(3/2) - 8*a^4*cos(d*x + c)^2*e^(3/2) + 4*a^4*cos(d*x + c)*e^(3/2) + 8*a^4*e^(3/2) + (2*a*d^2*cos(d*x + c)^5*e^
(1/2) - 5*a*d^2*cos(d*x + c)^4*e^(1/2) - 19*a*d^2*cos(d*x + c)^3*e^(1/2) + 20*a*d^2*cos(d*x + c)*e^(1/2) + 8*a
*d^2*e^(1/2) - (2*a*d^2*cos(d*x + c)^4*e^(1/2) + 9*a*d^2*cos(d*x + c)^3*e^(1/2) - 4*a*d^2*cos(d*x + c)^2*e^(1/
2) - 20*a*d^2*cos(d*x + c)*e^(1/2) - 8*a*d^2*e^(1/2))*sin(d*x + c))*sqrt(a^6/d^4)*e - (a^4*cos(d*x + c)^3*e^(3
/2) + 4*a^4*cos(d*x + c)^2*e^(3/2) - 4*a^4*cos(d*x + c)*e^(3/2) - 8*a^4*e^(3/2))*sin(d*x + c))*sqrt(a*sin(d*x
+ c) + a)*sqrt(cos(d*x + c)))*sqrt((2*a^9*cos(d*x + c)*e^3*sin(d*x + c) + 2*a^9*cos(d*x + c)*e^3 + (a^6*d^2*e^
2*sin(d*x + c) + a^6*d^2*e^2)*sqrt(a^6/d^4)*e + (sqrt(2)*(a^6/d^4)^(1/4)*a^7*d*cos(d*x + c)*e^3 + (sqrt(2)*a^4
*d^3*e^(3/2)*sin(d*x + c) + sqrt(2)*a^4*d^3*e^(3/2))*(a^6/d^4)^(3/4)*e^(3/2))*sqrt(a*sin(d*x + c) + a)*sqrt(co
s(d*x + c)))/(sin(d*x + c) + 1)) + ((2*sqrt(2)*a^4*d^3*cos(d*x + c)^5*e^(3/2) + sqrt(2)*a^4*d^3*cos(d*x + c)^4
*e^(3/2) - 13*sqrt(2)*a^4*d^3*cos(d*x + c)^3*e^(3/2) - 8*sqrt(2)*a^4*d^3*cos(d*x + c)^2*e^(3/2) + 12*sqrt(2)*a
^4*d^3*cos(d*x + c)*e^(3/2) + 8*sqrt(2)*a^4*d^3*e^(3/2) - (7*sqrt(2)*a^4*d^3*cos(d*x + c)^3*e^(3/2) + 4*sqrt(2
)*a^4*d^3*cos(d*x + c)^2*e^(3/2) - 12*sqrt(2)*a^4*d^3*cos(d*x + c)*e^(3/2) - 8*sqrt(2)*a^4*d^3*e^(3/2))*sin(d*
x + c))*(a^6/d^4)^(3/4)*e^(3/2) + (7*sqrt(2)*a^7*d*cos(d*x + c)^4*e^(5/2) + 3*sqrt(2)*a^7*d*cos(d*x + c)^3*e^(
5/2) - 16*sqrt(2)*a^7*d*cos(d*x + c)^2*e^(5/2) - 4*sqrt(2)*a^7*d*cos(d*x + c)*e^(5/2) + 8*sqrt(2)*a^7*d*e^(5/2
) + (2*sqrt(2)*a^7*d*cos(d*x + c)^4*e^(5/2) + sqrt(2)*a^7*d*cos(d*x + c)^3*e^(5/2) - 12*sqrt(2)*a^7*d*cos(d*x
+ c)^2*e^(5/2) - 4*sqrt(2)*a^7*d*cos(d*x + c)*e^(5/2) + 8*sqrt(2)*a^7*d*e^(5/2))*sin(d*x + c))*(a^6/d^4)^(1/4)
*e^(1/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(a^9*cos(d*x + c)^6*e^3 + a^9*cos(d*x + c)^5*e^3 - 8*a^
9*cos(d*x + c)^4*e^3 - 8*a^9*cos(d*x + c)^3*e^3 + 8*a^9*cos(d*x + c)^2*e^3 + 8*a^9*cos(d*x + c)*e^3 - 4*(a^9*c
os(d*x + c)^4*e^3 + a^9*cos(d*x + c)^3*e^3 - 2*a^9*cos(d*x + c)^2*e^3 - 2*a^9*cos(d*x + c)*e^3)*sin(d*x + c)))
*e^(1/2) - 20*(sqrt(2)*d*sin(d*x + c) + sqrt(2)*d)*(a^6/d^4)^(1/4)*arctan(1/4*(2*sqrt(1/2)*((sqrt(2)*d^3*cos(d
*x + c)^6 - 3*sqrt(2)*d^3*cos(d*x + c)^5 - 8*sqrt(2)*d^3*cos(d*x + c)^4 + 4*sqrt(2)*d^3*cos(d*x + c)^3 + 8*sqr
t(2)*d^3*cos(d*x + c)^2 - (sqrt(2)*d^3*cos(d*x + c)^5 + 4*sqrt(2)*d^3*cos(d*x + c)^4 - 4*sqrt(2)*d^3*cos(d*x +
c)^3 - 8*sqrt(2)*d^3*cos(d*x + c)^2)*sin(d*x + c))*(a^6/d^4)^(3/4)*e^(3/2) + (sqrt(2)*a^3*d*cos(d*x + c)^6*e
+ 5*sqrt(2)*a^3*d*cos(d*x + c)^5*e - 8*sqrt(2)*a^3*d*cos(d*x + c)^4*e - 20*sqrt(2)*a^3*d*cos(d*x + c)^3*e + 8*
sqrt(2)*a^3*d*cos(d*x + c)^2*e + 16*sqrt(2)*a^3*d*cos(d*x + c)*e + (sqrt(2)*a^3*d*cos(d*x + c)^5*e - 4*sqrt(2)
*a^3*d*cos(d*x + c)^4*e - 12*sqrt(2)*a^3*d*cos(d*x + c)^3*e + 8*sqrt(2)*a^3*d*cos(d*x + c)^2*e + 16*sqrt(2)*a^
3*d*cos(d*x + c)*e)*sin(d*x + c))*(a^6/d^4)^(1/4)*e^(1/2) + (a^4*cos(d*x + c)^4*e^(3/2) - 3*a^4*cos(d*x + c)^3
*e^(3/2) - 8*a^4*cos(d*x + c)^2*e^(3/2) + 4*a^4*cos(d*x + c)*e^(3/2) + 8*a^4*e^(3/2) + (2*a*d^2*cos(d*x + c)^5
*e^(1/2) - 5*a*d^2*cos(d*x + c)^4*e^(1/2) - 19*a*d^2*cos(d*x + c)^3*e^(1/2) + 20*a*d^2*cos(d*x + c)*e^(1/2) +
8*a*d^2*e^(1/2) - (2*a*d^2*cos(d*x + c)^4*e^(1/2) + 9*a*d^2*cos(d*x + c)^3*e^(1/2) - 4*a*d^2*cos(d*x + c)^2*e^
(1/2) - 20*a*d^2*cos(d*x + c)*e^(1/2) - 8*a*d^2*e^(1/2))*sin(d*x + c))*sqrt(a^6/d^4)*e - (a^4*cos(d*x + c)^3*e
^(3/2) + 4*a^4*cos(d*x + c)^2*e^(3/2) - 4*a^4*cos(d*x + c)*e^(3/2) - 8*a^4*e^(3/2))*sin(d*x + c))*sqrt(a*sin(d
*x + c) + a)*sqrt(cos(d*x + c)))*sqrt((2*a^9*cos(d*x + c)*e^3*sin(d*x + c) + 2*a^9*cos(d*x + c)*e^3 + (a^6*d^2
*e^2*sin(d*x + c) + a^6*d^2*e^2)*sqrt(a^6/d^4)*e - (sqrt(2)*(a^6/d^4)^(1/4)*a^7*d*cos(d*x + c)*e^3 + (sqrt(2)*
a^4*d^3*e^(3/2)*sin(d*x + c) + sqrt(2)*a^4*d^3*e^(3/2))*(a^6/d^4)^(3/4)*e^(3/2))*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^(3/2) + sqrt(2)*a^4*d^3*cos(d*x + c
)^4*e^(3/2) - 13*sqrt(2)*a^4*d^3*cos(d*x + c)^3*e^(3/2) - 8*sqrt(2)*a^4*d^3*cos(d*x + c)^2*e^(3/2) + 12*sqrt(2
)*a^4*d^3*cos(d*x + c)*e^(3/2) + 8*sqrt(2)*a^4*...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {e \cos {\left (c + d x \right )}}\, 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))**(1/2),x)
[Out]
Integral((a*(sin(c + d*x) + 1))**(3/2)*sqrt(e*cos(c + d*x)), x)
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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((a+a*sin(d*x+c))^(3/2)*(e*cos(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate((a*sin(d*x + c) + a)^(3/2)*sqrt(cos(d*x + c))*e^(1/2), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\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))^(1/2)*(a + a*sin(c + d*x))^(3/2),x)
[Out]
int((e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^(3/2), x)
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