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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.399834, 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} \[ -\frac{13 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{15 \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{9 \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 1.41112, size = 226, normalized size = 1.23 \[ \frac{\cos ^3\left (\frac{1}{2} (c+d x)\right ) \left (200 \sin \left (\frac{c}{2}\right ) \cos \left (\frac{d x}{2}\right )-20 \sin \left (\frac{3 c}{2}\right ) \cos \left (\frac{3 d x}{2}\right )+4 \sin \left (\frac{5 c}{2}\right ) \cos \left (\frac{5 d x}{2}\right )+200 \cos \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )-20 \cos \left (\frac{3 c}{2}\right ) \sin \left (\frac{3 d x}{2}\right )+4 \cos \left (\frac{5 c}{2}\right ) \sin \left (\frac{5 d x}{2}\right )+\frac{5}{\left (\cos \left (\frac{1}{4} (c+d x)\right )-\sin \left (\frac{1}{4} (c+d x)\right )\right )^2}-\frac{5}{\left (\sin \left (\frac{1}{4} (c+d x)\right )+\cos \left (\frac{1}{4} (c+d x)\right )\right )^2}+150 \log \left (\cos \left (\frac{1}{4} (c+d x)\right )-\sin \left (\frac{1}{4} (c+d x)\right )\right )-150 \log \left (\sin \left (\frac{1}{4} (c+d x)\right )+\cos \left (\frac{1}{4} (c+d x)\right )\right )\right )}{10 d (a (\cos (c+d x)+1))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(c + d*x)/2]^3*(150*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] - 150*Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4
]] + 200*Cos[(d*x)/2]*Sin[c/2] - 20*Cos[(3*d*x)/2]*Sin[(3*c)/2] + 4*Cos[(5*d*x)/2]*Sin[(5*c)/2] + 200*Cos[c/2]
*Sin[(d*x)/2] - 20*Cos[(3*c)/2]*Sin[(3*d*x)/2] + 4*Cos[(5*c)/2]*Sin[(5*d*x)/2] + 5/(Cos[(c + d*x)/4] - Sin[(c
+ d*x)/4])^2 - 5/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2))/(10*d*(a*(1 + Cos[c + d*x]))^(3/2))

________________________________________________________________________________________

Maple [A]  time = 1.523, size = 265, normalized size = 1.5 \begin{align*}{\frac{\sqrt{2}}{20\,d}\sqrt{a \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -32\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+32\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+75\,\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-80\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-75\,\ln \left ( 4\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+a}{\cos \left ( 1/2\,dx+c/2 \right ) }} \right ) a+85\,\sqrt{a}\sqrt{a \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \right ) \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{a}^{-{\frac{5}{2}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4/(a+cos(d*x+c)*a)^(3/2),x)

[Out]

1/20/cos(1/2*d*x+1/2*c)*2^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-32*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*sin
(1/2*d*x+1/2*c)^6+32*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4+75*ln(4/cos(1/2*d*x+1/2*c)*(a
^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+a))*a*sin(1/2*d*x+1/2*c)^2-80*a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)*sin
(1/2*d*x+1/2*c)^2-75*ln(4/cos(1/2*d*x+1/2*c)*(a^(1/2)*(a*sin(1/2*d*x+1/2*c)^2)^(1/2)+a))*a+85*a^(1/2)*(a*sin(1
/2*d*x+1/2*c)^2)^(1/2))/a^(5/2)/sin(1/2*d*x+1/2*c)/(cos(1/2*d*x+1/2*c)^2*a)^(1/2)/d

________________________________________________________________________________________

Maxima [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(cos(d*x+c)^4/(a+a*cos(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 1.64834, size = 497, normalized size = 2.72 \begin{align*} \frac{75 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} + 2 \, \sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \,{\left (4 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )^{2} + 36 \, \cos \left (d x + c\right ) + 49\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{40 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(75*sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*cos(d
*x + c) + a)*sqrt(a)*sin(d*x + c) - 2*a*cos(d*x + c) - 3*a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 4*(4*cos(
d*x + c)^3 - 4*cos(d*x + c)^2 + 36*cos(d*x + c) + 49)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c))/(a^2*d*cos(d*x +
c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.07941, size = 185, normalized size = 1.01 \begin{align*} \frac{\frac{75 \, \sqrt{2} \log \left ({\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} \right |}\right )}{a^{\frac{3}{2}}} + \frac{{\left ({\left ({\left (5 \, \sqrt{2} a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 127 \, \sqrt{2} a\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 175 \, \sqrt{2} a\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 85 \, \sqrt{2} a\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{5}{2}}}}{20 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(75*sqrt(2)*log(abs(-sqrt(a)*tan(1/2*d*x + 1/2*c) + sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a)))/a^(3/2) + (((5*s
qrt(2)*a*tan(1/2*d*x + 1/2*c)^2 + 127*sqrt(2)*a)*tan(1/2*d*x + 1/2*c)^2 + 175*sqrt(2)*a)*tan(1/2*d*x + 1/2*c)^
2 + 85*sqrt(2)*a)*tan(1/2*d*x + 1/2*c)/(a*tan(1/2*d*x + 1/2*c)^2 + a)^(5/2))/d