\(\int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx\) [320]

Optimal result

Integrand size = 27, antiderivative size = 193 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {256 (a+a \sin (c+d x))^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}} \]

[Out]

-2/13/d/e/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2)-16/117/a/d/e/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(3/2)
+256/585*(a+a*sin(d*x+c))^(3/2)/a^4/d/e/(e*cos(d*x+c))^(3/2)-32/195/a^2/d/e/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+
c))^(1/2)-128/195*(a+a*sin(d*x+c))^(1/2)/a^3/d/e/(e*cos(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {256 (a \sin (c+d x)+a)^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}-\frac {128 \sqrt {a \sin (c+d x)+a}}{195 a^3 d e (e \cos (c+d x))^{3/2}}-\frac {32}{195 a^2 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac {16}{117 a d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}-\frac {2}{13 d e (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}} \]

[In]

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

[Out]

-2/(13*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)) - 16/(117*a*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Si
n[c + d*x])^(3/2)) - 32/(195*a^2*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (128*Sqrt[a + a*Sin[c
+ d*x]])/(195*a^3*d*e*(e*Cos[c + d*x])^(3/2)) + (256*(a + a*Sin[c + d*x])^(3/2))/(585*a^4*d*e*(e*Cos[c + d*x])
^(3/2))

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}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}+\frac {8 \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx}{13 a} \\ & = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}+\frac {16 \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}} \, dx}{39 a^2} \\ & = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}+\frac {64 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{195 a^3} \\ & = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {128 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{195 a^4} \\ & = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {256 (a+a \sin (c+d x))^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 (77+136 \cos (2 (c+d x))-16 \cos (4 (c+d x))-40 \sin (c+d x)+80 \sin (3 (c+d x)))}{585 d e (e \cos (c+d x))^{3/2} (a (1+\sin (c+d x)))^{5/2}} \]

[In]

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

[Out]

(-2*(77 + 136*Cos[2*(c + d*x)] - 16*Cos[4*(c + d*x)] - 40*Sin[c + d*x] + 80*Sin[3*(c + d*x)]))/(585*d*e*(e*Cos
[c + d*x])^(3/2)*(a*(1 + Sin[c + d*x]))^(5/2))

Maple [A] (verified)

Time = 2.81 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54

method	result	size
default	\(-\frac {2 \left (-128 \left (\cos ^{3}\left (d x +c \right )\right )+320 \cos \left (d x +c \right ) \sin \left (d x +c \right )+400 \cos \left (d x +c \right )-120 \tan \left (d x +c \right )-75 \sec \left (d x +c \right )\right )}{585 d \,a^{2} e^{2} \left (2 \sin \left (d x +c \right )-\left (\cos ^{2}\left (d x +c \right )\right )+2\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {e \cos \left (d x +c \right )}}\)	\(105\)

[In]

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

[Out]

-2/585/d/a^2/e^2/(2*sin(d*x+c)-cos(d*x+c)^2+2)/(a*(1+sin(d*x+c)))^(1/2)/(e*cos(d*x+c))^(1/2)*(-128*cos(d*x+c)^
3+320*cos(d*x+c)*sin(d*x+c)+400*cos(d*x+c)-120*tan(d*x+c)-75*sec(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (128 \, \cos \left (d x + c\right )^{4} - 400 \, \cos \left (d x + c\right )^{2} - 40 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) + 75\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{585 \, {\left (3 \, a^{3} d e^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} d e^{3} \cos \left (d x + c\right )^{2} + {\left (a^{3} d e^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} d e^{3} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

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

[Out]

-2/585*(128*cos(d*x + c)^4 - 400*cos(d*x + c)^2 - 40*(8*cos(d*x + c)^2 - 3)*sin(d*x + c) + 75)*sqrt(e*cos(d*x
+ c))*sqrt(a*sin(d*x + c) + a)/(3*a^3*d*e^3*cos(d*x + c)^4 - 4*a^3*d*e^3*cos(d*x + c)^2 + (a^3*d*e^3*cos(d*x +
c)^4 - 4*a^3*d*e^3*cos(d*x + c)^2)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

integrate(1/(e*cos(d*x+c))**(5/2)/(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. 451 vs. \(2 (163) = 326\).

Time = 0.33 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (197 \, \sqrt {a} \sqrt {e} + \frac {400 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {15 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1760 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2230 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2230 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1760 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {15 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {400 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {197 \, \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}}{585 \, {\left (a^{3} e^{3} + \frac {5 \, a^{3} e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} e^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} e^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} e^{3} \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 {15}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \]

[In]

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

[Out]

-2/585*(197*sqrt(a)*sqrt(e) + 400*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 15*sqrt(a)*sqrt(e)*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 - 1760*sqrt(a)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 2230*sqrt(a)*sqrt(e)
*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2230*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1760*sqrt(a)
*sqrt(e)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 15*sqrt(a)*sqrt(e)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 400*sq
rt(a)*sqrt(e)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 197*sqrt(a)*sqrt(e)*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)
*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5/((a^3*e^3 + 5*a^3*e^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*a^
3*e^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*a^3*e^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*a^3*e^3*sin(d*x
+ c)^8/(cos(d*x + c) + 1)^8 + a^3*e^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)*d*(sin(d*x + c)/(cos(d*x + c) +
1) + 1)^(15/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(5/2))

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 11.51 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}\,\left (\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,2464{}\mathrm {i}}{585\,a^3\,d\,e^2}+\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )\,4352{}\mathrm {i}}{585\,a^3\,d\,e^2}-\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )\,512{}\mathrm {i}}{585\,a^3\,d\,e^2}+\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )\,512{}\mathrm {i}}{117\,a^3\,d\,e^2}-\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )\,256{}\mathrm {i}}{117\,a^3\,d\,e^2}\right )}{\cos \left (c+d\,x\right )\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,28{}\mathrm {i}-{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,12{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,28{}\mathrm {i}-{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (4\,c+4\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,2{}\mathrm {i}} \]

[In]

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

[Out]

-((a + a*sin(c + d*x))^(1/2)*((exp(c*4i + d*x*4i)*2464i)/(585*a^3*d*e^2) + (exp(c*4i + d*x*4i)*cos(2*c + 2*d*x
)*4352i)/(585*a^3*d*e^2) - (exp(c*4i + d*x*4i)*cos(4*c + 4*d*x)*512i)/(585*a^3*d*e^2) + (exp(c*4i + d*x*4i)*si
n(3*c + 3*d*x)*512i)/(117*a^3*d*e^2) - (exp(c*4i + d*x*4i)*sin(c + d*x)*256i)/(117*a^3*d*e^2)))/(cos(c + d*x)*
exp(c*4i + d*x*4i)*(e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*28i - exp(c*4i + d*x*4i)*cos(3*c
+ 3*d*x)*(e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*12i + exp(c*4i + d*x*4i)*sin(2*c + 2*d*x)*(
e*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*28i - exp(c*4i + d*x*4i)*sin(4*c + 4*d*x)*(e*(exp(- c
*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*2i)