3.4.2 \(\int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}} \, dx\) [302]

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

[Out]

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

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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 {4 \sqrt {a \sin (c+d x)+a}}{3 a d e \sqrt {e \cos (c+d x)}}-\frac {2}{3 d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.11, size = 46, normalized size = 0.61 \begin {gather*} \frac {2 (1+2 \sin (c+d x))}{3 d e \sqrt {e \cos (c+d x)} \sqrt {a (1+\sin (c+d x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 44, normalized size = 0.58

method	result	size
default	\(\frac {2 \left (2 \sin \left (d x +c \right )+1\right ) \cos \left (d x +c \right )}{3 d \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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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 = 0.54, size = 189, normalized size = 2.49 \begin {gather*} \frac {2 \, {\left (\sqrt {a} + \frac {4 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {\sqrt {a} \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} e^{\left (-\frac {3}{2}\right )}}{3 \, {\left (a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \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 {5}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 63, normalized size = 0.83 \begin {gather*} \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (2 \, \sin \left (d x + c\right ) + 1\right )} \sqrt {\cos \left (d x + c\right )}}{3 \, {\left (a d \cos \left (d x + c\right ) e^{\frac {3}{2}} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right ) e^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))**(3/2)/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Integral(1/(sqrt(a*(sin(c + d*x) + 1))*(e*cos(c + d*x))**(3/2)), x)

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

[Out]

Timed out

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Mupad [B]
time = 6.01, size = 77, normalized size = 1.01 \begin {gather*} \frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (3\,\sin \left (c+d\,x\right )-\cos \left (2\,c+2\,d\,x\right )+2\right )}{3\,a\,d\,e\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\sin \left (c+d\,x\right )-\cos \left (2\,c+2\,d\,x\right )+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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