3.68 \(\int \frac{\sin ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{13 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{10 a^2 d}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sin ^2(c+d x) \cos (c+d x)}{10 a d \sqrt{a \sin (c+d x)+a}}-\frac{31 \cos (c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(15*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (Cos[c + d*x]*
Sin[c + d*x]^3)/(2*d*(a + a*Sin[c + d*x])^(3/2)) - (31*Cos[c + d*x])/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (9*Cos
[c + d*x]*Sin[c + d*x]^2)/(10*a*d*Sqrt[a + a*Sin[c + d*x]]) + (13*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(10*a
^2*d)

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Rubi [A]  time = 0.382127, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2765, 2983, 2968, 3023, 2751, 2649, 206} \[ \frac{13 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{10 a^2 d}+\frac{15 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sin ^2(c+d x) \cos (c+d x)}{10 a d \sqrt{a \sin (c+d x)+a}}-\frac{31 \cos (c+d x)}{5 a d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]

[Out]

(15*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (Cos[c + d*x]*
Sin[c + d*x]^3)/(2*d*(a + a*Sin[c + d*x])^(3/2)) - (31*Cos[c + d*x])/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (9*Cos
[c + d*x]*Sin[c + d*x]^2)/(10*a*d*Sqrt[a + a*Sin[c + d*x]]) + (13*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(10*a
^2*d)

Rule 2765

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], 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] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2983

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(B*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(
m + n + 1)), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*b*c*(
m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (I
ntegerQ[n] || EqQ[m + 1/2, 0])

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 ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{\cos (c+d x) \sin ^3(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{\int \frac{\sin ^2(c+d x) \left (3 a-\frac{9}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{2 a^2}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{9 \cos (c+d x) \sin ^2(c+d x)}{10 a d \sqrt{a+a \sin (c+d x)}}-\frac{\int \frac{\sin (c+d x) \left (-9 a^2+\frac{39}{4} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{5 a^3}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{9 \cos (c+d x) \sin ^2(c+d x)}{10 a d \sqrt{a+a \sin (c+d x)}}-\frac{\int \frac{-9 a^2 \sin (c+d x)+\frac{39}{4} a^2 \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{5 a^3}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{9 \cos (c+d x) \sin ^2(c+d x)}{10 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{10 a^2 d}-\frac{2 \int \frac{\frac{39 a^3}{8}-\frac{93}{4} a^3 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{15 a^4}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{31 \cos (c+d x)}{5 a d \sqrt{a+a \sin (c+d x)}}-\frac{9 \cos (c+d x) \sin ^2(c+d x)}{10 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{10 a^2 d}-\frac{15 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{4 a}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{31 \cos (c+d x)}{5 a d \sqrt{a+a \sin (c+d x)}}-\frac{9 \cos (c+d x) \sin ^2(c+d x)}{10 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{10 a^2 d}+\frac{15 \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 )}{2 a d}\\ &=\frac{15 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac{31 \cos (c+d x)}{5 a d \sqrt{a+a \sin (c+d x)}}-\frac{9 \cos (c+d x) \sin ^2(c+d x)}{10 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{10 a^2 d}\\ \end{align*}

Mathematica [C]  time = 0.453447, size = 178, normalized size = 0.97 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (55 \sin \left (\frac{1}{2} (c+d x)\right )-41 \sin \left (\frac{3}{2} (c+d x)\right )+3 \sin \left (\frac{5}{2} (c+d x)\right )+\sin \left (\frac{7}{2} (c+d x)\right )-55 \cos \left (\frac{1}{2} (c+d x)\right )-41 \cos \left (\frac{3}{2} (c+d x)\right )-3 \cos \left (\frac{5}{2} (c+d x)\right )+\cos \left (\frac{7}{2} (c+d x)\right )-(150+150 i) (-1)^{3/4} (\sin (c+d x)+1) \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{20 d (a (\sin (c+d x)+1))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-55*Cos[(c + d*x)/2] - 41*Cos[(3*(c + d*x))/2] - 3*Cos[(5*(c + d*x))/2
] + Cos[(7*(c + d*x))/2] + 55*Sin[(c + d*x)/2] - (150 + 150*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 +
Tan[(c + d*x)/4])]*(1 + Sin[c + d*x]) - 41*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2] + Sin[(7*(c + d*x))/
2]))/(20*d*(a*(1 + Sin[c + d*x]))^(3/2))

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Maple [A]  time = 0.533, size = 183, normalized size = 1. \begin{align*}{\frac{1}{20\,d\cos \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \left ( -80\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{5/2}-8\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{a}+75\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) -90\,\sqrt{a-a\sin \left ( dx+c \right ) }{a}^{5/2}-8\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{a}+75\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \right ) \sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{-{\frac{9}{2}}}{\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)^4/(a+a*sin(d*x+c))^(3/2),x)

[Out]

1/20*(sin(d*x+c)*(-80*(a-a*sin(d*x+c))^(1/2)*a^(5/2)-8*(a-a*sin(d*x+c))^(5/2)*a^(1/2)+75*2^(1/2)*arctanh(1/2*(
a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^3)-90*(a-a*sin(d*x+c))^(1/2)*a^(5/2)-8*(a-a*sin(d*x+c))^(5/2)*a^(1/2)
+75*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^3)*(-a*(sin(d*x+c)-1))^(1/2)/a^(9/2)/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 )^{4}}{{\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(sin(d*x+c)^4/(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)^4/(a*sin(d*x + c) + a)^(3/2), x)

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Fricas [B]  time = 1.82819, size = 834, normalized size = 4.56 \begin{align*} \frac{75 \, \sqrt{2}{\left (\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} \log \left (-\frac{a \cos \left (d x + c\right )^{2} + 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\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 ) - 4 \,{\left (4 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - 48 \, \cos \left (d x + c\right )^{2} +{\left (4 \, \cos \left (d x + c\right )^{3} + 8 \, \cos \left (d x + c\right )^{2} - 40 \, \cos \left (d x + c\right ) + 5\right )} \sin \left (d x + c\right ) - 45 \, \cos \left (d x + c\right ) - 5\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{40 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d -{\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4/(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/40*(75*sqrt(2)*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x
+ c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a
*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)
) - 4*(4*cos(d*x + c)^4 - 4*cos(d*x + c)^3 - 48*cos(d*x + c)^2 + (4*cos(d*x + c)^3 + 8*cos(d*x + c)^2 - 40*cos
(d*x + c) + 5)*sin(d*x + c) - 45*cos(d*x + c) - 5)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^2 - a^2*d*cos
(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4/(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

sage2