3.213 \(\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=250 \[ \frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{3 (121 A-21 B) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}} \]

[Out]

(-3*(121*A - 21*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*S
qrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)
*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sin[c + d*x])/(192*a^2
*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x
]]*Sqrt[a + a*Cos[c + d*x]])

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Rubi [A]  time = 0.803849, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2978, 2984, 12, 2782, 205} \[ \frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{3 (121 A-21 B) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)),x]

[Out]

(-3*(121*A - 21*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*S
qrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)
*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sin[c + d*x])/(192*a^2
*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x
]]*Sqrt[a + a*Cos[c + d*x]])

Rule 2978

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*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*
x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,
0])

Rule 2984

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*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^(n + 1))/(f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin
[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*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] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2782

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[(-2*a)/f, Subst[Int[1/(2*b^2 - (a*c - 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 205

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

Rubi steps

\begin{align*} \int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx &=-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}+\frac{\int \frac{\frac{1}{2} a (13 A-B)-3 a (A-B) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}+\frac{\int \frac{\frac{3}{4} a^2 (41 A-5 B)-a^2 (19 A-7 B) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\frac{1}{8} a^3 (691 A-103 B)-\frac{1}{4} a^3 (199 A-43 B) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{\int -\frac{9 a^4 (121 A-21 B)}{16 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{24 a^7}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}-\frac{(3 (121 A-21 B)) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{128 a^3}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{(3 (121 A-21 B)) \operatorname{Subst}\left (\int \frac{1}{2 a^2+a x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{64 a^2 d}\\ &=-\frac{3 (121 A-21 B) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 2.73043, size = 240, normalized size = 0.96 \[ \frac{\cos ^7\left (\frac{1}{2} (c+d x)\right ) \left (\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) (9 (941 A-121 B) \cos (c+d x)+4 (937 A-133 B) \cos (2 (c+d x))+691 A \cos (3 (c+d x))+5284 A-103 B \cos (3 (c+d x))-532 B)}{16 \sqrt{\cos (c+d x)}}-\frac{9 i (121 A-21 B) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )}{\sqrt{1+e^{2 i (c+d x)}}}\right )}{24 d (a (\cos (c+d x)+1))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)),x]

[Out]

(Cos[(c + d*x)/2]^7*(((-9*I)*(121*A - 21*B)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))
]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[1 + E^((2*I)*(c + d*x))] + ((52
84*A - 532*B + 9*(941*A - 121*B)*Cos[c + d*x] + 4*(937*A - 133*B)*Cos[2*(c + d*x)] + 691*A*Cos[3*(c + d*x)] -
103*B*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/(16*Sqrt[Cos[c + d*x]])))/(24*d*(a*(1 + Cos[c + d
*x]))^(7/2))

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Maple [B]  time = 0.625, size = 581, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c))/cos(d*x+c)^(3/2)/(a+cos(d*x+c)*a)^(7/2),x)

[Out]

1/384/d*(a*(1+cos(d*x+c)))^(1/2)*(-1+cos(d*x+c))^2*(1089*A*sin(d*x+c)*2^(1/2)*cos(d*x+c)^3*(cos(d*x+c)/(1+cos(
d*x+c)))^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))-189*B*sin(d*x+c)*2^(1/2)*cos(d*x+c)^3*(cos(d*x+c)/(1+cos(d*x
+c)))^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))+3267*A*cos(d*x+c)^2*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2
)*2^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))-567*B*cos(d*x+c)^2*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*2
^(1/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))+3267*A*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)*arcsi
n((-1+cos(d*x+c))/sin(d*x+c))*2^(1/2)-567*B*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)*arcsin((-1
+cos(d*x+c))/sin(d*x+c))*2^(1/2)+1089*A*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arcsin((-1+cos(d*x+c))/si
n(d*x+c))*2^(1/2)-1382*A*cos(d*x+c)^4-189*B*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arcsin((-1+cos(d*x+c)
)/sin(d*x+c))*2^(1/2)+206*B*cos(d*x+c)^4-2366*A*cos(d*x+c)^3+326*B*cos(d*x+c)^3+550*A*cos(d*x+c)^2-142*B*cos(d
*x+c)^2+2430*A*cos(d*x+c)-390*B*cos(d*x+c)+768*A)/a^4/sin(d*x+c)^5/(1+cos(d*x+c))/cos(d*x+c)^(1/2)

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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((A+B*cos(d*x+c))/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.73288, size = 810, normalized size = 3.24 \begin{align*} -\frac{9 \, \sqrt{2}{\left ({\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 4 \,{\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 6 \,{\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt{a} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \,{\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) - 2 \,{\left ({\left (691 \, A - 103 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (937 \, A - 133 \, B\right )} \cos \left (d x + c\right )^{2} + 39 \,{\left (41 \, A - 5 \, B\right )} \cos \left (d x + c\right ) + 384 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{384 \,{\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

-1/384*(9*sqrt(2)*((121*A - 21*B)*cos(d*x + c)^5 + 4*(121*A - 21*B)*cos(d*x + c)^4 + 6*(121*A - 21*B)*cos(d*x
+ c)^3 + 4*(121*A - 21*B)*cos(d*x + c)^2 + (121*A - 21*B)*cos(d*x + c))*sqrt(a)*arctan(1/2*sqrt(2)*sqrt(a*cos(
d*x + c) + a)*sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*cos(d*x + c))) - 2*((691*A - 103*B
)*cos(d*x + c)^3 + 2*(937*A - 133*B)*cos(d*x + c)^2 + 39*(41*A - 5*B)*cos(d*x + c) + 384*A)*sqrt(a*cos(d*x + c
) + a)*sqrt(cos(d*x + c))*sin(d*x + c))/(a^4*d*cos(d*x + c)^5 + 4*a^4*d*cos(d*x + c)^4 + 6*a^4*d*cos(d*x + c)^
3 + 4*a^4*d*cos(d*x + c)^2 + a^4*d*cos(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((A+B*cos(d*x+c))/cos(d*x+c)**(3/2)/(a+a*cos(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((B*cos(d*x + c) + A)/((a*cos(d*x + c) + a)^(7/2)*cos(d*x + c)^(3/2)), x)