3.286 \(\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}} \]

[Out]

(2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(5/2)) - (4*(a + a*Sin[c + d*x])^(5/2))/(5*a*d*e*(e*Cos[c
+ d*x])^(5/2))

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Rubi [A]  time = 0.148083, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(5/2)) - (4*(a + a*Sin[c + d*x])^(5/2))/(5*a*d*e*(e*Cos[c
+ d*x])^(5/2))

Rule 2672

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 2671

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx &=\frac{2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac{2 \int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{a}\\ &=\frac{2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac{4 (a+a \sin (c+d x))^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.128922, size = 72, normalized size = 0.97 \[ -\frac{2 a (2 \sin (c+d x)-3) \sqrt{a (\sin (c+d x)+1)}}{5 d e^3 \sqrt{e \cos (c+d x)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Sqrt[a*(1 + Sin[c + d*x])]*(-3 + 2*Sin[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]]*(Cos[(c + d*x)/2] - Sin[
(c + d*x)/2])^2)

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Maple [A]  time = 0.096, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 4\,\sin \left ( dx+c \right ) -6 \right ) \cos \left ( dx+c \right ) }{5\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5/d*(2*sin(d*x+c)-3)*(a*(1+sin(d*x+c)))^(3/2)*cos(d*x+c)/(e*cos(d*x+c))^(7/2)

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Maxima [B]  time = 1.5794, size = 279, normalized size = 3.77 \begin{align*} \frac{2 \,{\left (3 \, a^{\frac{3}{2}} \sqrt{e} - \frac{4 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{4 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{3 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{5 \,{\left (e^{4} + \frac{2 \, e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

2/5*(3*a^(3/2)*sqrt(e) - 4*a^(3/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 4*a^(3/2)*sqrt(e)*sin(d*x + c)^3/
(cos(d*x + c) + 1)^3 - 3*a^(3/2)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*(sin(d*x + c)^2/(cos(d*x + c) +
1)^2 + 1)^2/((e^4 + 2*e^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + e^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*d*sqr
t(sin(d*x + c)/(cos(d*x + c) + 1) + 1)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))

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Fricas [A]  time = 2.54234, size = 177, normalized size = 2.39 \begin{align*} \frac{2 \, \sqrt{e \cos \left (d x + c\right )}{\left (2 \, a \sin \left (d x + c\right ) - 3 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{5 \,{\left (d e^{4} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d e^{4} \cos \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*sqrt(e*cos(d*x + c))*(2*a*sin(d*x + c) - 3*a)*sqrt(a*sin(d*x + c) + a)/(d*e^4*cos(d*x + c)*sin(d*x + c) -
d*e^4*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+a*sin(d*x+c))**(3/2)/(e*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*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(7/2),x, algorithm="giac")

[Out]

Timed out