3.3.85 \(\int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx\) [285]

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

[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 = 0.05, 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 = {2750} \begin {gather*} \frac {2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}} \end {gather*}

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 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]

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 \begin {gather*} \frac {2 (a (1+\sin (c+d x)))^{3/2}}{3 d e (e \cos (c+d x))^{3/2}} \end {gather*}

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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Maple [A]
time = 0.14, size = 34, normalized size = 0.94

method	result	size
default	\(\frac {2 \cos \left (d x +c \right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}}}{3 d \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}\)	\(34\)

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,method=_RETURNVERBOSE)

[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] Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (27) = 54\).
time = 0.54, size = 77, normalized size = 2.14 \begin {gather*} \frac {2 \, {\left (a^{\frac {3}{2}} - \frac {a^{\frac {3}{2}} \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} e^{\left (-\frac {5}{2}\right )}}{3 \, d {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \end {gather*}

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

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Fricas [A]
time = 0.36, size = 41, normalized size = 1.14 \begin {gather*} -\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} a \sqrt {\cos \left (d x + c\right )}}{3 \, {\left (d e^{\frac {5}{2}} \sin \left (d x + c\right ) - d e^{\frac {5}{2}}\right )}} \end {gather*}

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

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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))**(3/2)/(e*cos(d*x+c))**(5/2),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))^(3/2)/(e*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 5.62, size = 47, normalized size = 1.31 \begin {gather*} -\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 )} \end {gather*}

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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