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

Optimal. Leaf size=74 \[ \frac {4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 74, 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} \[ \frac {4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + a*Sin[c + d*x]])/(d*e*(e*Cos[c + d*x])^(3/2)) + (4*(a + a*Sin[c + d*x])^(3/2))/(3*a*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]

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

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Mathematica [A]  time = 0.25, size = 46, normalized size = 0.62 \[ \frac {2 (2 \sin (c+d x)-1) \sqrt {a (\sin (c+d x)+1)}}{3 d e (e \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.89, size = 48, normalized size = 0.65 \[ \frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (2 \, \sin \left (d x + c\right ) - 1\right )}}{3 \, d e^{3} \cos \left (d x + c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(1/2)/(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)*(2*sin(d*x + c) - 1)/(d*e^3*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))^(1/2)/(e*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.21, size = 44, normalized size = 0.59 \[ \frac {2 \left (2 \sin \left (d x +c \right )-1\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \cos \left (d x +c \right )}{3 d \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.97, size = 206, normalized size = 2.78 \[ -\frac {2 \, {\left (\sqrt {a} \sqrt {e} - \frac {4 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {\sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{3 \, {\left (e^{3} + \frac {2 \, e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(1/2)/(e*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.67, size = 61, normalized size = 0.82 \[ -\frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\cos \left (c+d\,x\right )-\sin \left (2\,c+2\,d\,x\right )\right )}{3\,d\,e^2\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )\,\sqrt {e\,\cos \left (c+d\,x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))**(1/2)/(e*cos(d*x+c))**(5/2),x)

[Out]

Integral(sqrt(a*(sin(c + d*x) + 1))/(e*cos(c + d*x))**(5/2), x)

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