3.285 \(\int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2671} \[ \frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 36, normalized size = 1.00 \[ \frac {2 (a (\sin (c+d x)+1))^{3/2}}{3 d e (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 45, normalized size = 1.25 \[ -\frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} a}{3 \, {\left (d e^{3} \sin \left (d x + c\right ) - d e^{3}\right )}} \]

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))^(5/2),x, algorithm="fricas")

[Out]

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

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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))^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.18, size = 34, normalized size = 0.94 \[ \frac {2 \cos \left (d x +c \right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}}}{3 d \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}} \]

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))^(5/2),x)

[Out]

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

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maxima [B]  time = 1.75, size = 131, normalized size = 3.64 \[ \frac {2 \, {\left (a^{\frac {3}{2}} \sqrt {e} - \frac {a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{3 \, {\left (e^{3} + \frac {e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \]

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))^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.62, size = 47, normalized size = 1.31 \[ -\frac {2\,a\,\cos \left (c+d\,x\right )\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}}{3\,d\,e^2\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (\sin \left (c+d\,x\right )-1\right )} \]

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))^(5/2),x)

[Out]

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

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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))**(5/2),x)

[Out]

Timed out

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