\(\int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx\) [310]

Optimal result

Integrand size = 27, antiderivative size = 115 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2}{7 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {8}{21 a d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {16 \sqrt {a+a \sin (c+d x)}}{21 a^2 d e \sqrt {e \cos (c+d x)}} \]

[Out]

-2/7/d/e/(a+a*sin(d*x+c))^(3/2)/(e*cos(d*x+c))^(1/2)-8/21/a/d/e/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^(1/2)+16
/21*(a+a*sin(d*x+c))^(1/2)/a^2/d/e/(e*cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {16 \sqrt {a \sin (c+d x)+a}}{21 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {8}{21 a d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}-\frac {2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}} \]

[In]

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

[Out]

-2/(7*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)) - 8/(21*a*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c
+ d*x]]) + (16*Sqrt[a + a*Sin[c + d*x]])/(21*a^2*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}{7 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}+\frac {4 \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}} \, dx}{7 a} \\ & = -\frac {2}{7 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {8}{21 a d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {8 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx}{21 a^2} \\ & = -\frac {2}{7 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {8}{21 a d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {16 \sqrt {a+a \sin (c+d x)}}{21 a^2 d e \sqrt {e \cos (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {2+24 \sin (c+d x)+16 \sin ^2(c+d x)}{21 d e \sqrt {e \cos (c+d x)} (a (1+\sin (c+d x)))^{3/2}} \]

[In]

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

[Out]

(2 + 24*Sin[c + d*x] + 16*Sin[c + d*x]^2)/(21*d*e*Sqrt[e*Cos[c + d*x]]*(a*(1 + Sin[c + d*x]))^(3/2))

Maple [A] (verified)

Time = 2.69 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.56

method	result	size
default	\(-\frac {2 \left (8 \left (\cos ^{2}\left (d x +c \right )\right )-12 \sin \left (d x +c \right )-9\right )}{21 d e \sqrt {e \cos \left (d x +c \right )}\, a \left (1+\sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}\)	\(64\)

[In]

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

[Out]

-2/21/d*(8*cos(d*x+c)^2-12*sin(d*x+c)-9)/e/(e*cos(d*x+c))^(1/2)/a/(1+sin(d*x+c))/(a*(1+sin(d*x+c)))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) - 9\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{21 \, {\left (a^{2} d e^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} d e^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d e^{2} \cos \left (d x + c\right )\right )}} \]

[In]

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

[Out]

2/21*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 - 12*sin(d*x + c) - 9)*sqrt(a*sin(d*x + c) + a)/(a^2*d*e^2*cos(d*x
+ c)^3 - 2*a^2*d*e^2*cos(d*x + c)*sin(d*x + c) - 2*a^2*d*e^2*cos(d*x + c))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (97) = 194\).

Time = 0.32 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.56 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (\sqrt {a} \sqrt {e} + \frac {24 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {33 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {33 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {24 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {\sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{21 \, {\left (a^{2} e^{2} + \frac {3 \, a^{2} e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} e^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} e^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

2/21*(sqrt(a)*sqrt(e) + 24*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) + 33*sqrt(a)*sqrt(e)*sin(d*x + c)^2
/(cos(d*x + c) + 1)^2 - 33*sqrt(a)*sqrt(e)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 24*sqrt(a)*sqrt(e)*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5 - sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*(sin(d*x + c)^2/(cos(d*x + c)
+ 1)^2 + 1)^3/((a^2*e^2 + 3*a^2*e^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2*e^2*sin(d*x + c)^4/(cos(d*x +
c) + 1)^4 + a^2*e^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(-sin(
d*x + c)/(cos(d*x + c) + 1) + 1)^(3/2))

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 6.52 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (70\,\sin \left (c+d\,x\right )-41\,\cos \left (2\,c+2\,d\,x\right )+2\,\cos \left (4\,c+4\,d\,x\right )-14\,\sin \left (3\,c+3\,d\,x\right )+41\right )}{21\,a^2\,d\,e\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (56\,\sin \left (c+d\,x\right )-28\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )-8\,\sin \left (3\,c+3\,d\,x\right )+35\right )} \]

[In]

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

[Out]

(8*(a*(sin(c + d*x) + 1))^(1/2)*(70*sin(c + d*x) - 41*cos(2*c + 2*d*x) + 2*cos(4*c + 4*d*x) - 14*sin(3*c + 3*d
*x) + 41))/(21*a^2*d*e*(e*cos(c + d*x))^(1/2)*(56*sin(c + d*x) - 28*cos(2*c + 2*d*x) + cos(4*c + 4*d*x) - 8*si
n(3*c + 3*d*x) + 35))