3.312 \(\int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=193 \[ -\frac{256 (a \sin (c+d x)+a)^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}+\frac{128 (a \sin (c+d x)+a)^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{16}{77 a d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}} \]

[Out]

-2/(11*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2)) - 16/(77*a*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a
*Sin[c + d*x]]) - (32*Sqrt[a + a*Sin[c + d*x]])/(77*a^2*d*e*(e*Cos[c + d*x])^(5/2)) + (128*(a + a*Sin[c + d*x]
)^(3/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(5/2)) - (256*(a + a*Sin[c + d*x])^(5/2))/(385*a^4*d*e*(e*Cos[c + d*x])^
(5/2))

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Rubi [A]  time = 0.371601, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ -\frac{256 (a \sin (c+d x)+a)^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}+\frac{128 (a \sin (c+d x)+a)^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{16}{77 a d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^(3/2)),x]

[Out]

-2/(11*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2)) - 16/(77*a*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a
*Sin[c + d*x]]) - (32*Sqrt[a + a*Sin[c + d*x]])/(77*a^2*d*e*(e*Cos[c + d*x])^(5/2)) + (128*(a + a*Sin[c + d*x]
)^(3/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(5/2)) - (256*(a + a*Sin[c + d*x])^(5/2))/(385*a^4*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} (a+a \sin (c+d x))^{3/2}} \, dx &=-\frac{2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}+\frac{8 \int \frac{1}{(e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}} \, dx}{11 a}\\ &=-\frac{2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac{16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}}+\frac{48 \int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^2}\\ &=-\frac{2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac{16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}}-\frac{32 \sqrt{a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{64 \int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^3}\\ &=-\frac{2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac{16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}}-\frac{32 \sqrt{a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{128 (a+a \sin (c+d x))^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac{128 \int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{77 a^4}\\ &=-\frac{2}{11 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}-\frac{16}{77 a d e (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}}-\frac{32 \sqrt{a+a \sin (c+d x)}}{77 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{128 (a+a \sin (c+d x))^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac{256 (a+a \sin (c+d x))^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.26944, size = 76, normalized size = 0.39 \[ \frac{2 (104 \sin (c+d x)+48 \sin (3 (c+d x))+8 \cos (2 (c+d x))-16 \cos (4 (c+d x))+45)}{385 d e (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^(3/2)),x]

[Out]

(2*(45 + 8*Cos[2*(c + d*x)] - 16*Cos[4*(c + d*x)] + 104*Sin[c + d*x] + 48*Sin[3*(c + d*x)]))/(385*d*e*(e*Cos[c
+ d*x])^(5/2)*(a*(1 + Sin[c + d*x]))^(3/2))

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Maple [A]  time = 0.114, size = 80, normalized size = 0.4 \begin{align*}{\frac{ \left ( -256\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+384\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +288\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+112\,\sin \left ( dx+c \right ) +42 \right ) \cos \left ( dx+c \right ) }{385\,d} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \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))^(3/2),x)

[Out]

2/385/d*(-128*cos(d*x+c)^4+192*cos(d*x+c)^2*sin(d*x+c)+144*cos(d*x+c)^2+56*sin(d*x+c)+21)*cos(d*x+c)/(e*cos(d*
x+c))^(7/2)/(a*(1+sin(d*x+c)))^(3/2)

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Maxima [B]  time = 1.69086, size = 609, normalized size = 3.16 \begin{align*} \frac{2 \,{\left (37 \, \sqrt{a} \sqrt{e} + \frac{496 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{559 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{544 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{1526 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1526 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{544 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{559 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{496 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{37 \, \sqrt{a} \sqrt{e} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5}}{385 \,{\left (a^{2} e^{4} + \frac{5 \, a^{2} e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{10 \, a^{2} e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{10 \, a^{2} e^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{5 \, a^{2} e^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{a^{2} e^{4} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{13}{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))^(3/2),x, algorithm="maxima")

[Out]

2/385*(37*sqrt(a)*sqrt(e) + 496*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 559*sqrt(a)*sqrt(e)*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 - 544*sqrt(a)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1526*sqrt(a)*sqrt(e)*s
in(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1526*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 544*sqrt(a)*sq
rt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 559*sqrt(a)*sqrt(e)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 496*sqrt
(a)*sqrt(e)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 37*sqrt(a)*sqrt(e)*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)*(s
in(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5/((a^2*e^4 + 5*a^2*e^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a^2*e
^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*a^2*e^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*a^2*e^4*sin(d*x +
c)^8/(cos(d*x + c) + 1)^8 + a^2*e^4*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)*d*(sin(d*x + c)/(cos(d*x + c) + 1)
+ 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))

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Fricas [A]  time = 2.66741, size = 323, normalized size = 1.67 \begin{align*} \frac{2 \,{\left (128 \, \cos \left (d x + c\right )^{4} - 144 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (24 \, \cos \left (d x + c\right )^{2} + 7\right )} \sin \left (d x + c\right ) - 21\right )} \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{385 \,{\left (a^{2} d e^{4} \cos \left (d x + c\right )^{5} - 2 \, a^{2} d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} 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))^(3/2),x, algorithm="fricas")

[Out]

2/385*(128*cos(d*x + c)^4 - 144*cos(d*x + c)^2 - 8*(24*cos(d*x + c)^2 + 7)*sin(d*x + c) - 21)*sqrt(e*cos(d*x +
c))*sqrt(a*sin(d*x + c) + a)/(a^2*d*e^4*cos(d*x + c)^5 - 2*a^2*d*e^4*cos(d*x + c)^3*sin(d*x + c) - 2*a^2*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))**(3/2),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}}{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{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))^(3/2),x, algorithm="giac")

[Out]

integrate(1/((e*cos(d*x + c))^(7/2)*(a*sin(d*x + c) + a)^(3/2)), x)