3.287 \(\int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac {16 (a \sin (c+d x)+a)^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}+\frac {8 (a \sin (c+d x)+a)^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}} \]

[Out]

-2*(a+a*sin(d*x+c))^(3/2)/d/e/(e*cos(d*x+c))^(7/2)+8/3*(a+a*sin(d*x+c))^(5/2)/a/d/e/(e*cos(d*x+c))^(7/2)-16/21
*(a+a*sin(d*x+c))^(7/2)/a^2/d/e/(e*cos(d*x+c))^(7/2)

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Rubi [A]  time = 0.23, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ -\frac {16 (a \sin (c+d x)+a)^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}+\frac {8 (a \sin (c+d x)+a)^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2),x]

[Out]

(-2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(7/2)) + (8*(a + a*Sin[c + d*x])^(5/2))/(3*a*d*e*(e*Cos[
c + d*x])^(7/2)) - (16*(a + a*Sin[c + d*x])^(7/2))/(21*a^2*d*e*(e*Cos[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))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx &=-\frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{7/2}}+\frac {4 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx}{a}\\ &=-\frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{7/2}}+\frac {8 (a+a \sin (c+d x))^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac {8 \int \frac {(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{9/2}} \, dx}{3 a^2}\\ &=-\frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{7/2}}+\frac {8 (a+a \sin (c+d x))^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac {16 (a+a \sin (c+d x))^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 105, normalized size = 0.93 \[ \frac {2 a \sqrt {a (\sin (c+d x)+1)} (12 \sin (c+d x)+4 \cos (2 (c+d x))-5)}{21 d e^4 \sqrt {e \cos (c+d x)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2),x]

[Out]

(2*a*Sqrt[a*(1 + Sin[c + d*x])]*(-5 + 4*Cos[2*(c + d*x)] + 12*Sin[c + d*x]))/(21*d*e^4*Sqrt[e*Cos[c + d*x]]*(C
os[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

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fricas [A]  time = 0.66, size = 84, normalized size = 0.74 \[ -\frac {2 \, {\left (8 \, a \cos \left (d x + c\right )^{2} + 12 \, a \sin \left (d x + c\right ) - 9 \, a\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{21 \, {\left (d e^{5} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d e^{5} \cos \left (d x + c\right )^{2}\right )}} \]

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))^(9/2),x, algorithm="fricas")

[Out]

-2/21*(8*a*cos(d*x + c)^2 + 12*a*sin(d*x + c) - 9*a)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*e^5*cos(
d*x + c)^2*sin(d*x + c) - d*e^5*cos(d*x + c)^2)

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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))^(3/2)/(e*cos(d*x+c))^(9/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.20, size = 54, normalized size = 0.48 \[ \frac {2 \left (8 \left (\cos ^{2}\left (d x +c \right )\right )+12 \sin \left (d x +c \right )-9\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}} \cos \left (d x +c \right )}{21 d \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(9/2),x)

[Out]

2/21/d*(8*cos(d*x+c)^2+12*sin(d*x+c)-9)*(a*(1+sin(d*x+c)))^(3/2)*cos(d*x+c)/(e*cos(d*x+c))^(9/2)

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maxima [B]  time = 1.13, size = 281, normalized size = 2.49 \[ -\frac {2 \, {\left (a^{\frac {3}{2}} \sqrt {e} - \frac {24 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {33 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {33 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {24 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{21 \, {\left (e^{5} + \frac {3 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, e^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {e^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

-2/21*(a^(3/2)*sqrt(e) - 24*a^(3/2)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 33*a^(3/2)*sqrt(e)*sin(d*x + c)^
2/(cos(d*x + c) + 1)^2 - 33*a^(3/2)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 24*a^(3/2)*sqrt(e)*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5 - a^(3/2)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*(sin(d*x + c)^2/(cos(d*x + c
) + 1)^2 + 1)^3/((e^5 + 3*e^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*e^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
+ e^5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(3/2)*(-sin(d*x + c)/(cos(d
*x + c) + 1) + 1)^(9/2))

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mupad [B]  time = 6.81, size = 116, normalized size = 1.03 \[ \frac {8\,a\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (12\,\cos \left (c+d\,x\right )-10\,\cos \left (3\,c+3\,d\,x\right )-17\,\sin \left (2\,c+2\,d\,x\right )+2\,\sin \left (4\,c+4\,d\,x\right )\right )}{21\,d\,e^4\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\sin \left (c+d\,x\right )-4\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )+4\,\sin \left (3\,c+3\,d\,x\right )-5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(3/2)/(e*cos(c + d*x))^(9/2),x)

[Out]

(8*a*(a*(sin(c + d*x) + 1))^(1/2)*(12*cos(c + d*x) - 10*cos(3*c + 3*d*x) - 17*sin(2*c + 2*d*x) + 2*sin(4*c + 4
*d*x)))/(21*d*e^4*(e*cos(c + d*x))^(1/2)*(4*sin(c + d*x) - 4*cos(2*c + 2*d*x) + cos(4*c + 4*d*x) + 4*sin(3*c +
3*d*x) - 5))

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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))**(3/2)/(e*cos(d*x+c))**(9/2),x)

[Out]

Timed out

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