\(\int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx\) [319]

Optimal result

Integrand size = 27, antiderivative size = 154 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a^2 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^3 d e \sqrt {e \cos (c+d x)}} \]

[Out]

-2/11/d/e/(a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(1/2)-12/77/a/d/e/(a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(1/2)-
16/77/a^2/d/e/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(1/2)+32/77*(a+a*sin(d*x+c))^(1/2)/a^3/d/e/(e*cos(d*x+c))^
(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {32 \sqrt {a \sin (c+d x)+a}}{77 a^3 d e \sqrt {e \cos (c+d x)}}-\frac {16}{77 a^2 d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}-\frac {12}{77 a d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}}-\frac {2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}} \]

[In]

Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)),x]

[Out]

-2/(11*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)) - 12/(77*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c +
d*x])^(3/2)) - 16/(77*a^2*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (32*Sqrt[a + a*Sin[c + d*x]])/
(77*a^3*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*} \text {integral}& = -\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}+\frac {6 \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx}{11 a} \\ & = -\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}+\frac {24 \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}} \, dx}{77 a^2} \\ & = -\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a^2 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {16 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx}{77 a^3} \\ & = -\frac {2}{11 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}}-\frac {12}{77 a d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {16}{77 a^2 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {32 \sqrt {a+a \sin (c+d x)}}{77 a^3 d e \sqrt {e \cos (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {-10+52 \sin (c+d x)+80 \sin ^2(c+d x)+32 \sin ^3(c+d x)}{77 d e \sqrt {e \cos (c+d x)} (a (1+\sin (c+d x)))^{5/2}} \]

[In]

Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)),x]

[Out]

(-10 + 52*Sin[c + d*x] + 80*Sin[c + d*x]^2 + 32*Sin[c + d*x]^3)/(77*d*e*Sqrt[e*Cos[c + d*x]]*(a*(1 + Sin[c + d
*x]))^(5/2))

Maple [A] (verified)

Time = 2.86 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.58

method	result	size
default	\(\frac {\frac {32 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{77}+\frac {80 \left (\cos ^{2}\left (d x +c \right )\right )}{77}-\frac {12 \sin \left (d x +c \right )}{11}-\frac {10}{11}}{d \sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \left (\cos ^{2}\left (d x +c \right )-2 \sin \left (d x +c \right )-2\right ) e \,a^{2}}\)	\(90\)

[In]

int(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/77/d*(16*cos(d*x+c)^2*sin(d*x+c)+40*cos(d*x+c)^2-42*sin(d*x+c)-35)/(e*cos(d*x+c))^(1/2)/(a*(1+sin(d*x+c)))^(
1/2)/(cos(d*x+c)^2-2*sin(d*x+c)-2)/e/a^2

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (40 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 21\right )} \sin \left (d x + c\right ) - 35\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{77 \, {\left (3 \, a^{3} d e^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{3} d e^{2} \cos \left (d x + c\right ) + {\left (a^{3} d e^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{3} d e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]

[In]

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

[Out]

2/77*sqrt(e*cos(d*x + c))*(40*cos(d*x + c)^2 + 2*(8*cos(d*x + c)^2 - 21)*sin(d*x + c) - 35)*sqrt(a*sin(d*x + c
) + a)/(3*a^3*d*e^2*cos(d*x + c)^3 - 4*a^3*d*e^2*cos(d*x + c) + (a^3*d*e^2*cos(d*x + c)^3 - 4*a^3*d*e^2*cos(d*
x + c))*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (130) = 260\).

Time = 0.37 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (5 \, \sqrt {a} \sqrt {e} - \frac {52 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {150 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {180 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {180 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {150 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {52 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {5 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{77 \, {\left (a^{3} e^{2} + \frac {4 \, a^{3} e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{3} e^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{3} e^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{3} e^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

-2/77*(5*sqrt(a)*sqrt(e) - 52*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 150*sqrt(a)*sqrt(e)*sin(d*x +
c)^2/(cos(d*x + c) + 1)^2 - 180*sqrt(a)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 180*sqrt(a)*sqrt(e)*sin(
d*x + c)^5/(cos(d*x + c) + 1)^5 + 150*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 52*sqrt(a)*sqrt(e)
*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 5*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^3*e^2 + 4*a^3*e^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a^3*e^2*sin(d*x + c)
^4/(cos(d*x + c) + 1)^4 + 4*a^3*e^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a^3*e^2*sin(d*x + c)^8/(cos(d*x + c)
+ 1)^8)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(3/2))

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 10.85 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {76\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+30\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-40\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-8\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {385\,a^3\,d\,e\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}+\frac {1155\,a^3\,d\,e\,\sin \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}-\frac {231\,a^3\,d\,e\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}-\frac {77\,a^3\,d\,e\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}} \]

[In]

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

[Out]

(76*sin(c + d*x)*(a + a*sin(c + d*x))^(1/2) + 30*(a + a*sin(c + d*x))^(1/2) - 40*cos(2*c + 2*d*x)*(a + a*sin(c
+ d*x))^(1/2) - 8*sin(3*c + 3*d*x)*(a + a*sin(c + d*x))^(1/2))/((385*a^3*d*e*((e*exp(- c*1i - d*x*1i))/2 + (e
*exp(c*1i + d*x*1i))/2)^(1/2))/2 + (1155*a^3*d*e*sin(c + d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*
1i))/2)^(1/2))/4 - (231*a^3*d*e*cos(2*c + 2*d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2)
)/2 - (77*a^3*d*e*sin(3*c + 3*d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))/2)^(1/2))/4)