3.3.95 \(\int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx\) [295]

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

[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 = 0.10, 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 = {2751, 2750} \begin {gather*} \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}} \end {gather*}

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 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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 = 0.16, size = 54, normalized size = 0.71 \begin {gather*} -\frac {2 \sqrt {e \cos (c+d x)} \sec ^4(c+d x) (a (1+\sin (c+d x)))^{5/2} (-5+2 \sin (c+d x))}{21 d e^5} \end {gather*}

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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Maple [A]
time = 0.18, size = 44, normalized size = 0.58

method	result	size
default	\(-\frac {2 \left (2 \sin \left (d x +c \right )-5\right ) \cos \left (d x +c \right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}}}{21 d \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}\)	\(44\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (58) = 116\).
time = 0.55, size = 189, normalized size = 2.49 \begin {gather*} \frac {2 \, {\left (5 \, a^{\frac {5}{2}} - \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {5 \, a^{\frac {5}{2}} \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} e^{\left (-\frac {9}{2}\right )}}{21 \, d {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}} {\left (\frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}} \end {gather*}

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

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Fricas [A]
time = 0.35, size = 70, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (2 \, a^{2} \sin \left (d x + c\right ) - 5 \, a^{2}\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{21 \, {\left (d \cos \left (d x + c\right )^{2} e^{\frac {9}{2}} + 2 \, d e^{\frac {9}{2}} \sin \left (d x + c\right ) - 2 \, d e^{\frac {9}{2}}\right )}} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Mupad [B]
time = 6.34, size = 96, normalized size = 1.26 \begin {gather*} \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 )} \end {gather*}

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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