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

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

[Out]

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

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Rubi [A]  time = 0.0706675, 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} $-\frac{256 a^4 \cos (c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{24 a^2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + a*Sin[c + d*x])^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.654412, size = 154, normalized size = 1.29 $-\frac{a^3 (\sin (c+d x)+1)^3 \sqrt{a (\sin (c+d x)+1)} \left (-1225 \sin \left (\frac{1}{2} (c+d x)\right )+245 \sin \left (\frac{3}{2} (c+d x)\right )+49 \sin \left (\frac{5}{2} (c+d x)\right )-5 \sin \left (\frac{7}{2} (c+d x)\right )+1225 \cos \left (\frac{1}{2} (c+d x)\right )+245 \cos \left (\frac{3}{2} (c+d x)\right )-49 \cos \left (\frac{5}{2} (c+d x)\right )-5 \cos \left (\frac{7}{2} (c+d x)\right )\right )}{140 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.092, size = 75, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{4} \left ( \sin \left ( dx+c \right ) -1 \right ) \left ( 5\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+27\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+71\,\sin \left ( dx+c \right ) +177 \right ) }{35\,d\cos \left ( dx+c \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((a+a*sin(d*x+c))^(7/2),x)

[Out]

2/35*(1+sin(d*x+c))*a^4*(sin(d*x+c)-1)*(5*sin(d*x+c)^3+27*sin(d*x+c)^2+71*sin(d*x+c)+177)/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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(d*x + c) + a)^(7/2), x)

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Fricas [A]  time = 1.63007, size = 360, normalized size = 3.03 \begin{align*} \frac{2 \,{\left (5 \, a^{3} \cos \left (d x + c\right )^{4} + 27 \, a^{3} \cos \left (d x + c\right )^{3} - 54 \, a^{3} \cos \left (d x + c\right )^{2} - 204 \, a^{3} \cos \left (d x + c\right ) - 128 \, a^{3} +{\left (5 \, a^{3} \cos \left (d x + c\right )^{3} - 22 \, a^{3} \cos \left (d x + c\right )^{2} - 76 \, a^{3} \cos \left (d x + c\right ) + 128 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{35 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*a^3*cos(d*x + c)^4 + 27*a^3*cos(d*x + c)^3 - 54*a^3*cos(d*x + c)^2 - 204*a^3*cos(d*x + c) - 128*a^3 +
(5*a^3*cos(d*x + c)^3 - 22*a^3*cos(d*x + c)^2 - 76*a^3*cos(d*x + c) + 128*a^3)*sin(d*x + c))*sqrt(a*sin(d*x +
c) + a)/(d*cos(d*x + c) + d*sin(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*sin(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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(d*x + c) + a)^(7/2), x)