∫ ∘ _________ c+d sin(e+fx) _(a+asin (e+fx))5∕2 dx

3.602 \(\int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac {(3 c-5 d) (c+d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f (c-d)^{3/2}}-\frac {(3 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (c-d) (a \sin (e+f x)+a)^{3/2}}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}} \]

[Out]

-1/32*(3*c-5*d)*(c+d)*arctanh(1/2*cos(f*x+e)*a^(1/2)*(c-d)^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e

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

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

________________________________________________________________________________________

Rubi [A] time = 0.49, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2764, 2978, 12, 2782, 208} \[ -\frac {(3 c-5 d) (c+d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f (c-d)^{3/2}}-\frac {(3 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a f (c-d) (a \sin (e+f x)+a)^{3/2}}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a \sin (e+f x)+a)^{5/2}} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((3*c - 5*d)*(c + d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*

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

n[e + f*x])^(5/2)) - ((3*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*(c - d)*f*(a + a*Sin[e + f*x])^(3

/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,

b}, x] && NegQ[a/b]

Rule 2764

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim

p[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)),

Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a*d*n - b*c*(m + 1) - b*d*(m + n + 1)*Sin[

e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^

2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2782

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D

ist[(-2*a)/f, Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c

+ d*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 -

d^2, 0]

Rule 2978

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_

.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*

x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +

1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*

(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2

- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,

0])

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {\int \frac {\frac {1}{2} a (3 c+d)+a d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{4 a^2}\\ &=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a (c-d) f (a+a \sin (e+f x))^{3/2}}-\frac {\int -\frac {a^2 (3 c-5 d) (c+d)}{4 \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{8 a^4 (c-d)}\\ &=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a (c-d) f (a+a \sin (e+f x))^{3/2}}+\frac {((3 c-5 d) (c+d)) \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{32 a^2 (c-d)}\\ &=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a (c-d) f (a+a \sin (e+f x))^{3/2}}-\frac {((3 c-5 d) (c+d)) \operatorname {Subst}\left (\int \frac {1}{2 a^2-(a c-a d) x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{16 a (c-d) f}\\ &=-\frac {(3 c-5 d) (c+d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} (c-d)^{3/2} f}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c-d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{16 a (c-d) f (a+a \sin (e+f x))^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] time = 7.60, size = 412, normalized size = 2.16 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (\frac {\left (3 c^2-2 c d-5 d^2\right ) \left (\log \left (\tan \left (\frac {1}{2} (e+f x)\right )+1\right )-\log \left ((d-c) \tan \left (\frac {1}{2} (e+f x)\right )+2 \sqrt {c-d} \sqrt {\frac {1}{\cos (e+f x)+1}} \sqrt {c+d \sin (e+f x)}+c-d\right )\right )}{\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2 \tan \left (\frac {1}{2} (e+f x)\right )+2}-\frac {\frac {\sqrt {c-d} \left (\frac {1}{\cos (e+f x)+1}\right )^{3/2} (c \sin (e+f x)+d \cos (e+f x)+d)}{\sqrt {c+d \sin (e+f x)}}-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{(d-c) \tan \left (\frac {1}{2} (e+f x)\right )+2 \sqrt {c-d} \sqrt {\frac {1}{\cos (e+f x)+1}} \sqrt {c+d \sin (e+f x)}+c-d}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) ((3 c-d) \sin (e+f x)+7 c-5 d) (c+d \sin (e+f x))}{\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}\right )}{32 f (c-d) (a (\sin (e+f x)+1))^{5/2} \sqrt {c+d \sin (e+f x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(7*c - 5*d + (3*c - d)*Sin

[e + f*x])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + ((3*c^2 - 2*c*d - 5*d^2)*(Log[1 + T

an[(e + f*x)/2]] - Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)

*Tan[(e + f*x)/2]]))/(Sec[(e + f*x)/2]^2/(2 + 2*Tan[(e + f*x)/2]) - (-1/2*((c - d)*Sec[(e + f*x)/2]^2) + (Sqrt

[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*(d + d*Cos[e + f*x] + c*Sin[e + f*x]))/Sqrt[c + d*Sin[e + f*x]])/(c -

d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]))))/(32*(

c - d)*f*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

fricas [B] time = 0.90, size = 1474, normalized size = 7.72 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/128*(((3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^3 + 3*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^2 - 12*c^2 + 8*c*d +

20*d^2 - 2*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e) + ((3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^2 - 12*c^2 + 8*c*d + 2

0*d^2 - 2*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*

a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c -

3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)

*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(

f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d

+ 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e)

- 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*((3*c^2 - 4*c*d + d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (

7*c^2 - 12*c*d + 5*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e))*sin(f*x +

e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^3 + 3*(

a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^2 - 2*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e) - 4*(a^3*c^

2 - 2*a^3*c*d + a^3*d^2)*f + ((a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^2 - 2*(a^3*c^2 - 2*a^3*c*d + a^3*

d^2)*f*cos(f*x + e) - 4*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f)*sin(f*x + e)), -1/64*(((3*c^2 - 2*c*d - 5*d^2)*cos(

f*x + e)^3 + 3*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^2 - 12*c^2 + 8*c*d + 20*d^2 - 2*(3*c^2 - 2*c*d - 5*d^2)*co

s(f*x + e) + ((3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^2 - 12*c^2 + 8*c*d + 20*d^2 - 2*(3*c^2 - 2*c*d - 5*d^2)*cos

(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3

*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c

*d)*cos(f*x + e))) - 4*((3*c^2 - 4*c*d + d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 12*c*d + 5*d^2

)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e

) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^3 + 3*(a^3*c^2 - 2*a^3*c*d +

a^3*d^2)*f*cos(f*x + e)^2 - 2*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e) - 4*(a^3*c^2 - 2*a^3*c*d + a^3*d^

2)*f + ((a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^2 - 2*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e) -

4*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f)*sin(f*x + e))]

________________________________________________________________________________________

giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*sin(f*x + e) + c)/(a*sin(f*x + e) + a)^(5/2), x)

________________________________________________________________________________________

maple [B] time = 0.36, size = 3050, normalized size = 15.97 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64/f*(12*cos(f*x+e)^3*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*c^2-12*cos(f*x+e)*((c+d*sin(f*x+e))/(cos(f*x+e

)+1))^(1/2)*c^2-28*sin(f*x+e)*cos(f*x+e)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*c^2-20*sin(f*x+e)*cos(f*x+e)*

((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*d^2-12*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))

/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f

*x+e)))*2^(1/2)*c^2+20*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*s

in(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*d^2+4*co

s(f*x+e)^3*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*d^2-5*sin(f*x+e)*cos(f*x+e)^2*(2*c-2*d)^(1/2)*ln(2*((2*c-2*

d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*c

os(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*d^2+3*sin(f*x+e)*cos(f*x+e)^2*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d

)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*co

s(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c^2-4*cos(f*x+e)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*d^2+

9*cos(f*x+e)^2*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e

)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c^2-15*cos(f*x+e

)^2*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x

+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*d^2-12*sin(f*x+e)*(2*c-2*d)

^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x

+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c^2+20*sin(f*x+e)*(2*c-2*d)^(1/2)*ln(2*(

(2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x

+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*d^2+6*cos(f*x+e)*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2

)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+

e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c^2-10*cos(f*x+e)*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c

+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-co

s(f*x+e)+sin(f*x+e)))*2^(1/2)*d^2+48*sin(f*x+e)*cos(f*x+e)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*c*d-16*cos(

f*x+e)^3*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*c*d+16*cos(f*x+e)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*c*d

+2*cos(f*x+e)^3*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+

e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c*d+6*sin(f*x+e

)*cos(f*x+e)*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+

c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c^2-10*sin(f*x+e)*

cos(f*x+e)*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*

sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*d^2-6*cos(f*x+e)^2*(

2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d

*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c*d+8*sin(f*x+e)*(2*c-2*d)^(1/2)

*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*

cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c*d-4*cos(f*x+e)*(2*c-2*d)^(1/2)*ln(2*((2*c-2*

d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*c

os(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c*d-2*sin(f*x+e)*cos(f*x+e)^2*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d

)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*co

s(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c*d-4*sin(f*x+e)*cos(f*x+e)*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(

1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f

*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c*d+8*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*

x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+

sin(f*x+e)))*2^(1/2)*c*d-3*cos(f*x+e)^3*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f

*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))

*2^(1/2)*c^2+5*cos(f*x+e)^3*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1

/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*d^2

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

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*sin(f*x + e) + c)/(a*sin(f*x + e) + a)^(5/2), x)

________________________________________________________________________________________

mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]