3.1 \(\int (a+a \cos (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=119 \[ \frac{256 a^4 \sin (c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{64 a^3 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{24 a^2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]

[Out]

(256*a^4*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (64*a^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d)
+ (24*a^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*a*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*
d)

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Rubi [A]  time = 0.0696602, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac{256 a^4 \sin (c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{64 a^3 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{24 a^2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(7/2),x]

[Out]

(256*a^4*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (64*a^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d)
+ (24*a^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*a*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*
d)

Rule 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rule 2646

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

Rubi steps

\begin{align*} \int (a+a \cos (c+d x))^{7/2} \, dx &=\frac{2 a (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{7} (12 a) \int (a+a \cos (c+d x))^{5/2} \, dx\\ &=\frac{24 a^2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 a (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{35} \left (96 a^2\right ) \int (a+a \cos (c+d x))^{3/2} \, dx\\ &=\frac{64 a^3 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{35 d}+\frac{24 a^2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 a (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac{1}{35} \left (128 a^3\right ) \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{256 a^4 \sin (c+d x)}{35 d \sqrt{a+a \cos (c+d x)}}+\frac{64 a^3 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{35 d}+\frac{24 a^2 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 a (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.269067, size = 83, normalized size = 0.7 \[ \frac{a^3 \left (1225 \sin \left (\frac{1}{2} (c+d x)\right )+245 \sin \left (\frac{3}{2} (c+d x)\right )+49 \sin \left (\frac{5}{2} (c+d x)\right )+5 \sin \left (\frac{7}{2} (c+d x)\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{140 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(1225*Sin[(c + d*x)/2] + 245*Sin[(3*(c + d*x))/2] + 49*Sin[(5
*(c + d*x))/2] + 5*Sin[(7*(c + d*x))/2]))/(140*d)

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Maple [A]  time = 0.754, size = 86, normalized size = 0.7 \begin{align*}{\frac{16\,{a}^{4}\sqrt{2}}{35\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 5\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}+6\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+16 \right ){\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((a+cos(d*x+c)*a)^(7/2),x)

[Out]

16/35*a^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)*(5*cos(1/2*d*x+1/2*c)^6+6*cos(1/2*d*x+1/2*c)^4+8*cos(1/2*d*x+1
/2*c)^2+16)*2^(1/2)/(cos(1/2*d*x+1/2*c)^2*a)^(1/2)/d

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Maxima [A]  time = 1.94754, size = 104, normalized size = 0.87 \begin{align*} \frac{{\left (5 \, \sqrt{2} a^{3} \sin \left (\frac{7}{2} \, d x + \frac{7}{2} \, c\right ) + 49 \, \sqrt{2} a^{3} \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 245 \, \sqrt{2} a^{3} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 1225 \, \sqrt{2} a^{3} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}}{140 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/140*(5*sqrt(2)*a^3*sin(7/2*d*x + 7/2*c) + 49*sqrt(2)*a^3*sin(5/2*d*x + 5/2*c) + 245*sqrt(2)*a^3*sin(3/2*d*x
+ 3/2*c) + 1225*sqrt(2)*a^3*sin(1/2*d*x + 1/2*c))*sqrt(a)/d

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Fricas [A]  time = 1.57748, size = 194, normalized size = 1.63 \begin{align*} \frac{2 \,{\left (5 \, a^{3} \cos \left (d x + c\right )^{3} + 27 \, a^{3} \cos \left (d x + c\right )^{2} + 71 \, a^{3} \cos \left (d x + c\right ) + 177 \, a^{3}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{35 \,{\left (d \cos \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*a^3*cos(d*x + c)^3 + 27*a^3*cos(d*x + c)^2 + 71*a^3*cos(d*x + c) + 177*a^3)*sqrt(a*cos(d*x + c) + a)*s
in(d*x + c)/(d*cos(d*x + c) + 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((a+a*cos(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [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*cos(d*x+c))^(7/2),x, algorithm="giac")

[Out]

Timed out