3.4.18 \(\int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx\) [318]

Optimal. Leaf size=115 \[ -\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a+a \sin (c+d x))^{3/2}}-\frac {16 \sqrt {e \cos (c+d x)}}{45 a^2 d e \sqrt {a+a \sin (c+d x)}} \]

[Out]

-2/9*(e*cos(d*x+c))^(1/2)/d/e/(a+a*sin(d*x+c))^(5/2)-8/45*(e*cos(d*x+c))^(1/2)/a/d/e/(a+a*sin(d*x+c))^(3/2)-16
/45*(e*cos(d*x+c))^(1/2)/a^2/d/e/(a+a*sin(d*x+c))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \begin {gather*} -\frac {16 \sqrt {e \cos (c+d x)}}{45 a^2 d e \sqrt {a \sin (c+d x)+a}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a \sin (c+d x)+a)^{3/2}}-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[e*Cos[c + d*x]])/(9*d*e*(a + a*Sin[c + d*x])^(5/2)) - (8*Sqrt[e*Cos[c + d*x]])/(45*a*d*e*(a + a*Sin[c
+ d*x])^(3/2)) - (16*Sqrt[e*Cos[c + d*x]])/(45*a^2*d*e*Sqrt[a + a*Sin[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*} \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}+\frac {4 \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}} \, dx}{9 a}\\ &=-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a+a \sin (c+d x))^{3/2}}+\frac {8 \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}} \, dx}{45 a^2}\\ &=-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a+a \sin (c+d x))^{3/2}}-\frac {16 \sqrt {e \cos (c+d x)}}{45 a^2 d e \sqrt {a+a \sin (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.15, size = 69, normalized size = 0.60 \begin {gather*} -\frac {2 \sqrt {e \cos (c+d x)} \sqrt {a (1+\sin (c+d x))} \left (17+20 \sin (c+d x)+8 \sin ^2(c+d x)\right )}{45 a^3 d e (1+\sin (c+d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*Sqrt[a*(1 + Sin[c + d*x])]*(17 + 20*Sin[c + d*x] + 8*Sin[c + d*x]^2))/(45*a^3*d*e*(1
+ Sin[c + d*x])^3)

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 54, normalized size = 0.47

method	result	size
default	\(-\frac {2 \left (-8 \left (\cos ^{2}\left (d x +c \right )\right )+20 \sin \left (d x +c \right )+25\right ) \cos \left (d x +c \right )}{45 d \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}} \sqrt {e \cos \left (d x +c \right )}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45/d*(-8*cos(d*x+c)^2+20*sin(d*x+c)+25)*cos(d*x+c)/(a*(1+sin(d*x+c)))^(5/2)/(e*cos(d*x+c))^(1/2)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (88) = 176\).
time = 0.55, size = 266, normalized size = 2.31 \begin {gather*} -\frac {2 \, {\left (17 \, \sqrt {a} + \frac {40 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {49 \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {49 \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {40 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {17 \, \sqrt {a} \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} e^{\left (-\frac {1}{2}\right )}}{45 \, {\left (a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \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 {11}{2}} \sqrt {-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(17*sqrt(a) + 40*sqrt(a)*sin(d*x + c)/(cos(d*x + c) + 1) + 49*sqrt(a)*sin(d*x + c)^2/(cos(d*x + c) + 1)^
2 - 49*sqrt(a)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 40*sqrt(a)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 17*sqrt(
a)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^3*e^(-1/2)/((a^3 + 3*a^3*sin
(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a^3*sin(d*x + c)^6/(cos(d*x + c
) + 1)^6)*d*(sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*sqrt(-sin(d*x + c)/(cos(d*x + c) + 1) + 1))

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 100, normalized size = 0.87 \begin {gather*} -\frac {2 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 20 \, \sin \left (d x + c\right ) - 25\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{45 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 4 \, a^{3} d e^{\frac {1}{2}} + {\left (a^{3} d \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 4 \, a^{3} d e^{\frac {1}{2}}\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(8*cos(d*x + c)^2 - 20*sin(d*x + c) - 25)*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c))/(3*a^3*d*cos(d*x +
c)^2*e^(1/2) - 4*a^3*d*e^(1/2) + (a^3*d*cos(d*x + c)^2*e^(1/2) - 4*a^3*d*e^(1/2))*sin(d*x + c))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \sqrt {e \cos {\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Mupad [B]
time = 7.66, size = 137, normalized size = 1.19 \begin {gather*} -\frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (137\,\cos \left (c+d\,x\right )-71\,\cos \left (3\,c+3\,d\,x\right )+2\,\cos \left (5\,c+5\,d\,x\right )+144\,\sin \left (2\,c+2\,d\,x\right )-18\,\sin \left (4\,c+4\,d\,x\right )\right )}{45\,a^3\,d\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (210\,\sin \left (c+d\,x\right )-120\,\cos \left (2\,c+2\,d\,x\right )+10\,\cos \left (4\,c+4\,d\,x\right )-45\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (5\,c+5\,d\,x\right )+126\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*(a*(sin(c + d*x) + 1))^(1/2)*(137*cos(c + d*x) - 71*cos(3*c + 3*d*x) + 2*cos(5*c + 5*d*x) + 144*sin(2*c +
2*d*x) - 18*sin(4*c + 4*d*x)))/(45*a^3*d*(e*cos(c + d*x))^(1/2)*(210*sin(c + d*x) - 120*cos(2*c + 2*d*x) + 10*
cos(4*c + 4*d*x) - 45*sin(3*c + 3*d*x) + sin(5*c + 5*d*x) + 126))

________________________________________________________________________________________