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

Optimal. Leaf size=154 \[ -\frac {32 (a \sin (c+d x)+a)^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}+\frac {16 (a \sin (c+d x)+a)^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {4 \sqrt {a \sin (c+d x)+a}}{7 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}} \]

[Out]

16/7*(a+a*sin(d*x+c))^(3/2)/a^2/d/e/(e*cos(d*x+c))^(5/2)-32/35*(a+a*sin(d*x+c))^(5/2)/a^3/d/e/(e*cos(d*x+c))^(
5/2)-2/7/d/e/(e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^(1/2)-4/7*(a+a*sin(d*x+c))^(1/2)/a/d/e/(e*cos(d*x+c))^(5/2)

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Rubi [A]  time = 0.29, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ -\frac {32 (a \sin (c+d x)+a)^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}+\frac {16 (a \sin (c+d x)+a)^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {4 \sqrt {a \sin (c+d x)+a}}{7 a d e (e \cos (c+d x))^{5/2}}-\frac {2}{7 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(7*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]) - (4*Sqrt[a + a*Sin[c + d*x]])/(7*a*d*e*(e*Cos[c +
d*x])^(5/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(5/2)) - (32*(a + a*Sin[c + d*x])^(
5/2))/(35*a^3*d*e*(e*Cos[c + d*x])^(5/2))

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]

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]

Rubi steps

\begin {align*} \int \frac {1}{(e \cos (c+d x))^{7/2} \sqrt {a+a \sin (c+d x)}} \, dx &=-\frac {2}{7 d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}+\frac {6 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx}{7 a}\\ &=-\frac {2}{7 d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {4 \sqrt {a+a \sin (c+d x)}}{7 a d e (e \cos (c+d x))^{5/2}}+\frac {8 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx}{7 a^2}\\ &=-\frac {2}{7 d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {4 \sqrt {a+a \sin (c+d x)}}{7 a d e (e \cos (c+d x))^{5/2}}+\frac {16 (a+a \sin (c+d x))^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {16 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{7 a^3}\\ &=-\frac {2}{7 d e (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}-\frac {4 \sqrt {a+a \sin (c+d x)}}{7 a d e (e \cos (c+d x))^{5/2}}+\frac {16 (a+a \sin (c+d x))^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac {32 (a+a \sin (c+d x))^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 66, normalized size = 0.43 \[ \frac {2 (10 \sin (c+d x)+4 \sin (3 (c+d x))+4 \cos (2 (c+d x))+5)}{35 d e \sqrt {a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(5 + 4*Cos[2*(c + d*x)] + 10*Sin[c + d*x] + 4*Sin[3*(c + d*x)]))/(35*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a*(1 +
Sin[c + d*x])])

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fricas [A]  time = 0.90, size = 93, normalized size = 0.60 \[ \frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35 \, {\left (a d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d e^{4} \cos \left (d x + c\right )^{3}\right )}} \]

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))^(1/2),x, algorithm="fricas")

[Out]

2/35*sqrt(e*cos(d*x + c))*(8*cos(d*x + c)^2 + 2*(8*cos(d*x + c)^2 + 3)*sin(d*x + c) + 1)*sqrt(a*sin(d*x + c) +
a)/(a*d*e^4*cos(d*x + c)^3*sin(d*x + c) + a*d*e^4*cos(d*x + c)^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} \sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]

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))^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.19, size = 70, normalized size = 0.45 \[ \frac {2 \left (16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+8 \left (\cos ^{2}\left (d x +c \right )\right )+6 \sin \left (d x +c \right )+1\right ) \cos \left (d x +c \right )}{35 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sqrt {a \left (1+\sin \left (d x +c \right )\right )}} \]

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))^(1/2),x)

[Out]

2/35/d*(16*cos(d*x+c)^2*sin(d*x+c)+8*cos(d*x+c)^2+6*sin(d*x+c)+1)*cos(d*x+c)/(e*cos(d*x+c))^(7/2)/(a*(1+sin(d*
x+c)))^(1/2)

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maxima [B]  time = 0.87, size = 363, normalized size = 2.36 \[ \frac {2 \, {\left (9 \, \sqrt {a} \sqrt {e} + \frac {44 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {84 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {84 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {44 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {9 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{35 \, {\left (a e^{4} + \frac {4 \, a e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a e^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a e^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}}} \]

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))^(1/2),x, algorithm="maxima")

[Out]

2/35*(9*sqrt(a)*sqrt(e) + 44*sqrt(a)*sqrt(e)*sin(d*x + c)/(cos(d*x + c) + 1) - 14*sqrt(a)*sqrt(e)*sin(d*x + c)
^2/(cos(d*x + c) + 1)^2 - 84*sqrt(a)*sqrt(e)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 84*sqrt(a)*sqrt(e)*sin(d*x
+ c)^5/(cos(d*x + c) + 1)^5 + 14*sqrt(a)*sqrt(e)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 44*sqrt(a)*sqrt(e)*sin(
d*x + c)^7/(cos(d*x + c) + 1)^7 - 9*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*e^4 + 4*a*e^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a*e^4*sin(d*x + c)^4/(cos(d*x
+ c) + 1)^4 + 4*a*e^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*e^4*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)*d*(sin
(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))

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mupad [B]  time = 11.01, size = 261, normalized size = 1.69 \[ \frac {20\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+10\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+8\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+8\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {35\,a\,d\,e^3\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}+\frac {35\,a\,d\,e^3\,\sin \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}+\frac {35\,a\,d\,e^3\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{2}+\frac {35\,a\,d\,e^3\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(20*sin(c + d*x)*(a + a*sin(c + d*x))^(1/2) + 10*(a + a*sin(c + d*x))^(1/2) + 8*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))/((35*a*d*e^3*((e*exp(- c*1i - d*x*1i))/2 + (e*e
xp(c*1i + d*x*1i))/2)^(1/2))/2 + (35*a*d*e^3*sin(c + d*x)*((e*exp(- c*1i - d*x*1i))/2 + (e*exp(c*1i + d*x*1i))
/2)^(1/2))/4 + (35*a*d*e^3*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 +
(35*a*d*e^3*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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**(1/2),x)

[Out]

Timed out

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