3.282 \(\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=243 \[ -\frac{5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt{a \sin (c+d x)+a}}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)} \]

[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]))

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Rubi [A]  time = 0.358337, antiderivative size = 243, normalized size of antiderivative = 1., 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 = {2678, 2684, 2775, 203, 2833, 63, 215} \[ -\frac{5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt{a \sin (c+d x)+a}}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)} \]

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 2678

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 2684

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 2775

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 203

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

Rule 2833

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*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 63

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 215

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]

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [C]  time = 0.117161, size = 77, normalized size = 0.32 \[ -\frac{8 \sqrt [4]{2} (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac{5}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e (\sin (c+d x)+1)^{9/4}} \]

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 = 0.166, size = 262, normalized size = 1.1 \begin{align*}{\frac{1}{8\,d \left ( \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) + \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) -2 \right ) \cos \left ( dx+c \right ) } \left ( 5\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \sqrt{2}\sin \left ( dx+c \right ) +5\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \sqrt{2}\sin \left ( dx+c \right ) +4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+10\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +14\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-10\,\cos \left ( dx+c \right ) \right ) \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{3}{2}}}\sqrt{e\cos \left ( dx+c \right ) }} \end{align*}

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)

[Out]

1/8/d*(5*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*2^(1/2)
*sin(d*x+c)+5*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*2^(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)^2*sin(d*x+c)-4*cos(d*x+c)^3+10*cos(d*x+c)*sin(d*x+c)+14*c
os(d*x+c)^2-10*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., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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