3.304 \(\int \frac{1}{(e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=154 \[ -\frac{32 (a \sin (c+d x)+a)^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}+\frac{16 (a \sin (c+d x)+a)^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{4 \sqrt{a \sin (c+d x)+a}}{7 a d e (e \cos (c+d x))^{5/2}}-\frac{2}{7 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}} \]

[Out]

-2/(7*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]) - (4*Sqrt[a + a*Sin[c + d*x]])/(7*a*d*e*(e*Cos[c +
d*x])^(5/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(5/2)) - (32*(a + a*Sin[c + d*x])^(
5/2))/(35*a^3*d*e*(e*Cos[c + d*x])^(5/2))

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Rubi [A]  time = 0.287667, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ -\frac{32 (a \sin (c+d x)+a)^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}+\frac{16 (a \sin (c+d x)+a)^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{4 \sqrt{a \sin (c+d x)+a}}{7 a d e (e \cos (c+d x))^{5/2}}-\frac{2}{7 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + a*Sin[c + d*x]]),x]

[Out]

-2/(7*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]) - (4*Sqrt[a + a*Sin[c + d*x]])/(7*a*d*e*(e*Cos[c +
d*x])^(5/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(5/2)) - (32*(a + a*Sin[c + d*x])^(
5/2))/(35*a^3*d*e*(e*Cos[c + d*x])^(5/2))

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]

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{1}{(e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{2}{7 d e (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}}+\frac{6 \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx}{7 a}\\ &=-\frac{2}{7 d e (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}}-\frac{4 \sqrt{a+a \sin (c+d x)}}{7 a d e (e \cos (c+d x))^{5/2}}+\frac{8 \int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx}{7 a^2}\\ &=-\frac{2}{7 d e (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}}-\frac{4 \sqrt{a+a \sin (c+d x)}}{7 a d e (e \cos (c+d x))^{5/2}}+\frac{16 (a+a \sin (c+d x))^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{16 \int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{7 a^3}\\ &=-\frac{2}{7 d e (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}}-\frac{4 \sqrt{a+a \sin (c+d x)}}{7 a d e (e \cos (c+d x))^{5/2}}+\frac{16 (a+a \sin (c+d x))^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{32 (a+a \sin (c+d x))^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.178332, size = 66, normalized size = 0.43 \[ \frac{2 (10 \sin (c+d x)+4 \sin (3 (c+d x))+4 \cos (2 (c+d x))+5)}{35 d e \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + a*Sin[c + d*x]]),x]

[Out]

(2*(5 + 4*Cos[2*(c + d*x)] + 10*Sin[c + d*x] + 4*Sin[3*(c + d*x)]))/(35*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a*(1 +
Sin[c + d*x])])

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Maple [A]  time = 0.121, size = 70, normalized size = 0.5 \begin{align*}{\frac{ \left ( 32\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +16\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+12\,\sin \left ( dx+c \right ) +2 \right ) \cos \left ( dx+c \right ) }{35\,d} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a \left ( 1+\sin \left ( dx+c \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(1/2),x)

[Out]

2/35/d*(16*cos(d*x+c)^2*sin(d*x+c)+8*cos(d*x+c)^2+6*sin(d*x+c)+1)*cos(d*x+c)/(e*cos(d*x+c))^(7/2)/(a*(1+sin(d*
x+c)))^(1/2)

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Maxima [B]  time = 1.65325, size = 490, normalized size = 3.18 \begin{align*} \frac{2 \,{\left (9 \, \sqrt{a} \sqrt{e} + \frac{44 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{14 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{84 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{84 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{14 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{44 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{9 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{35 \,{\left (a e^{4} + \frac{4 \, a e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a e^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a e^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{9}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

2/35*(9*sqrt(a)*sqrt(e) + 44*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 14*sqrt(a)*sqrt(e)*sin(d*x + c)
^2/(cos(d*x + c) + 1)^2 - 84*sqrt(a)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 84*sqrt(a)*sqrt(e)*sin(d*x
+ c)^5/(cos(d*x + c) + 1)^5 + 14*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 44*sqrt(a)*sqrt(e)*sin(
d*x + c)^7/(cos(d*x + c) + 1)^7 - 9*sqrt(a)*sqrt(e)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*(sin(d*x + c)^2/(cos(
d*x + c) + 1)^2 + 1)^4/((a*e^4 + 4*a*e^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a*e^4*sin(d*x + c)^4/(cos(d*x
+ c) + 1)^4 + 4*a*e^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*e^4*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d*(sin
(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))

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Fricas [A]  time = 2.06038, size = 240, normalized size = 1.56 \begin{align*} \frac{2 \, \sqrt{e \cos \left (d x + c\right )}{\left (8 \, \cos \left (d x + c\right )^{2} + 2 \,{\left (8 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{35 \,{\left (a d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d e^{4} \cos \left (d x + c\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/35*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 + 2*(8*cos(d*x + c)^2 + 3)*sin(d*x + c) + 1)*sqrt(a*sin(d*x + c) +
a)/(a*d*e^4*cos(d*x + c)^3*sin(d*x + c) + a*d*e^4*cos(d*x + c)^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))**(7/2)/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{7}{2}} \sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/((e*cos(d*x + c))^(7/2)*sqrt(a*sin(d*x + c) + a)), x)