3.295 \(\int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx\)

Optimal. Leaf size=76 \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}} \]

[Out]

2/3*(a+a*sin(d*x+c))^(5/2)/d/e/(e*cos(d*x+c))^(7/2)-4/21*(a+a*sin(d*x+c))^(7/2)/a/d/e/(e*cos(d*x+c))^(7/2)

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Rubi [A]  time = 0.15, antiderivative size = 76, normalized size of antiderivative = 1.00, 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} \[ \frac {2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2),x]

[Out]

(2*(a + a*Sin[c + d*x])^(5/2))/(3*d*e*(e*Cos[c + d*x])^(7/2)) - (4*(a + a*Sin[c + d*x])^(7/2))/(21*a*d*e*(e*Co
s[c + d*x])^(7/2))

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]

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]

Rubi steps

\begin {align*} \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx &=\frac {2 (a+a \sin (c+d x))^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {2 \int \frac {(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{9/2}} \, dx}{3 a}\\ &=\frac {2 (a+a \sin (c+d x))^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac {4 (a+a \sin (c+d x))^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 54, normalized size = 0.71 \[ -\frac {2 (2 \sin (c+d x)-5) \sec ^4(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt {e \cos (c+d x)}}{21 d e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2),x]

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*(a*(1 + Sin[c + d*x]))^(5/2)*(-5 + 2*Sin[c + d*x]))/(21*d*e^5)

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fricas [A]  time = 1.16, size = 75, normalized size = 0.99 \[ \frac {2 \, {\left (2 \, a^{2} \sin \left (d x + c\right ) - 5 \, a^{2}\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{21 \, {\left (d e^{5} \cos \left (d x + c\right )^{2} + 2 \, d e^{5} \sin \left (d x + c\right ) - 2 \, d e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(9/2),x, algorithm="fricas")

[Out]

2/21*(2*a^2*sin(d*x + c) - 5*a^2)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*e^5*cos(d*x + c)^2 + 2*d*e^
5*sin(d*x + c) - 2*d*e^5)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(9/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.20, size = 44, normalized size = 0.58 \[ -\frac {2 \left (2 \sin \left (d x +c \right )-5\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}} \cos \left (d x +c \right )}{21 d \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(9/2),x)

[Out]

-2/21/d*(2*sin(d*x+c)-5)*(a*(1+sin(d*x+c)))^(5/2)*cos(d*x+c)/(e*cos(d*x+c))^(9/2)

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maxima [B]  time = 1.00, size = 207, normalized size = 2.72 \[ \frac {2 \, {\left (5 \, a^{\frac {5}{2}} \sqrt {e} - \frac {4 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5 \, a^{\frac {5}{2}} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{21 \, {\left (e^{5} + \frac {2 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {e^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

2/21*(5*a^(5/2)*sqrt(e) - 4*a^(5/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 4*a^(5/2)*sqrt(e)*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 - 5*a^(5/2)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*sqrt(sin(d*x + c)/(cos(d*x + c)
+ 1) + 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^2/((e^5 + 2*e^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + e^5
*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)*d*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2))

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mupad [B]  time = 6.34, size = 96, normalized size = 1.26 \[ \frac {4\,a^2\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\cos \left (3\,c+3\,d\,x\right )-11\,\cos \left (c+d\,x\right )+7\,\sin \left (2\,c+2\,d\,x\right )\right )}{21\,d\,e^4\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (15\,\sin \left (c+d\,x\right )+6\,\cos \left (2\,c+2\,d\,x\right )-\sin \left (3\,c+3\,d\,x\right )-10\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(5/2)/(e*cos(c + d*x))^(9/2),x)

[Out]

(4*a^2*(a*(sin(c + d*x) + 1))^(1/2)*(cos(3*c + 3*d*x) - 11*cos(c + d*x) + 7*sin(2*c + 2*d*x)))/(21*d*e^4*(e*co
s(c + d*x))^(1/2)*(15*sin(c + d*x) + 6*cos(2*c + 2*d*x) - sin(3*c + 3*d*x) - 10))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))**(5/2)/(e*cos(d*x+c))**(9/2),x)

[Out]

Timed out

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