3.61 \(\int \frac{\sin ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}-\frac{28 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{\sqrt{a} d} \]

[Out]

(Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - (28*Cos[c + d*x])/(
15*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d
*x]*Sqrt[a + a*Sin[c + d*x]])/(15*a*d)

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Rubi [A]  time = 0.232121, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2778, 2968, 3023, 2751, 2649, 206} \[ -\frac{2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 a d}-\frac{28 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - (28*Cos[c + d*x])/(
15*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d
*x]*Sqrt[a + a*Sin[c + d*x]])/(15*a*d)

Rule 2778

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp
[(-2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Sin[e + f*x]]), x] - Dist[1/(b*(2*n
- 1)), Int[((c + d*Sin[e + f*x])^(n - 2)*Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*n -
3))*Sin[e + f*x], x])/Sqrt[a + b*Sin[e + f*x]], 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] && GtQ[n, 1] && IntegerQ[2*n]

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{\sin ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{\int \frac{\sin (c+d x) (-4 a+a \sin (c+d x))}{\sqrt{a+a \sin (c+d x)}} \, dx}{5 a}\\ &=-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{\int \frac{-4 a \sin (c+d x)+a \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{5 a}\\ &=-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 a d}-\frac{2 \int \frac{\frac{a^2}{2}-7 a^2 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{15 a^2}\\ &=-\frac{28 \cos (c+d x)}{15 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 a d}-\int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{28 \cos (c+d x)}{15 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 a d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}-\frac{28 \cos (c+d x)}{15 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 a d}\\ \end{align*}

Mathematica [C]  time = 0.233201, size = 150, normalized size = 1.08 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (60 \sin \left (\frac{1}{2} (c+d x)\right )+5 \sin \left (\frac{3}{2} (c+d x)\right )-3 \sin \left (\frac{5}{2} (c+d x)\right )-60 \cos \left (\frac{1}{2} (c+d x)\right )+5 \cos \left (\frac{3}{2} (c+d x)\right )+3 \cos \left (\frac{5}{2} (c+d x)\right )+(-60-60 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{30 d \sqrt{a (\sin (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*((-60 - 60*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c +
d*x)/4])] - 60*Cos[(c + d*x)/2] + 5*Cos[(3*(c + d*x))/2] + 3*Cos[(5*(c + d*x))/2] + 60*Sin[(c + d*x)/2] + 5*Si
n[(3*(c + d*x))/2] - 3*Sin[(5*(c + d*x))/2]))/(30*d*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]  time = 0.589, size = 130, normalized size = 0.9 \begin{align*}{\frac{1+\sin \left ( dx+c \right ) }{15\,{a}^{3}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 15\,{a}^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) -6\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}+10\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}a-30\,{a}^{2}\sqrt{a-a\sin \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x)

[Out]

1/15*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(15*a^(5/2)*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a
^(1/2))-6*(a-a*sin(d*x+c))^(5/2)+10*(a-a*sin(d*x+c))^(3/2)*a-30*a^2*(a-a*sin(d*x+c))^(1/2))/a^3/cos(d*x+c)/(a+
a*sin(d*x+c))^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.90647, size = 648, normalized size = 4.66 \begin{align*} \frac{\frac{15 \, \sqrt{2}{\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac{2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt{a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt{a}} + 4 \,{\left (3 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} -{\left (3 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 17\right )} \sin \left (d x + c\right ) - 16 \, \cos \left (d x + c\right ) - 17\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{30 \,{\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/30*(15*sqrt(2)*(a*cos(d*x + c) + a*sin(d*x + c) + a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c)
+ 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x
+ c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) + 4*(3*cos(d*x + c)^3 + 4*cos(d*x + c)^2
- (3*cos(d*x + c)^2 - cos(d*x + c) - 17)*sin(d*x + c) - 16*cos(d*x + c) - 17)*sqrt(a*sin(d*x + c) + a))/(a*d*
cos(d*x + c) + a*d*sin(d*x + c) + a*d)

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

[Out]

Timed out

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Giac [B]  time = 3.41578, size = 410, normalized size = 2.95 \begin{align*} -\frac{\frac{120 \, \sqrt{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} + \sqrt{a}\right )}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )} - \frac{{\left ({\left ({\left ({\left (\frac{13 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{7}} - \frac{15 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{a^{7}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{40 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{a^{7}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{40 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{a^{7}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{15 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{a^{7}}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{13 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{a^{7}}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{5}{2}}} - \frac{{\left (120 \, \sqrt{2} a^{\frac{21}{2}} \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + 17 \, \sqrt{2} \sqrt{-a} a\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{\sqrt{-a} a^{\frac{21}{2}}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

-1/60*(120*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a) + sq
rt(a))/sqrt(-a))/(sqrt(-a)*sgn(tan(1/2*d*x + 1/2*c) + 1)) - (((((13*sgn(tan(1/2*d*x + 1/2*c) + 1)*tan(1/2*d*x
+ 1/2*c)/a^7 - 15*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^7)*tan(1/2*d*x + 1/2*c) + 40*sgn(tan(1/2*d*x + 1/2*c) + 1)/a
^7)*tan(1/2*d*x + 1/2*c) - 40*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^7)*tan(1/2*d*x + 1/2*c) + 15*sgn(tan(1/2*d*x + 1
/2*c) + 1)/a^7)*tan(1/2*d*x + 1/2*c) - 13*sgn(tan(1/2*d*x + 1/2*c) + 1)/a^7)/(a*tan(1/2*d*x + 1/2*c)^2 + a)^(5
/2) - (120*sqrt(2)*a^(21/2)*arctan(sqrt(a)/sqrt(-a)) + 17*sqrt(2)*sqrt(-a)*a)*sgn(tan(1/2*d*x + 1/2*c) + 1)/(s
qrt(-a)*a^(21/2)))/d