3.6.96 \(\int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx\) [596]

Optimal. Leaf size=126 \[ -\frac {(c+d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} \sqrt {c-d} f}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a+a \sin (e+f x))^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2843, 12, 2861, 214} \begin {gather*} -\frac {(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 )}{2 \sqrt {2} a^{3/2} f \sqrt {c-d}}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a \sin (e+f x)+a)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((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]])])/(Sqrt[2]*a^(3/2)*Sqrt[c - d]*f) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*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 214

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

Rule 2843

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 2861

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx &=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {a (c+d)}{2 \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{2 a^2}\\ &=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {(c+d) \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{4 a}\\ &=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {(c+d) \text {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 )}{2 f}\\ &=-\frac {(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 )}{2 \sqrt {2} a^{3/2} \sqrt {c-d} f}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (a+a \sin (e+f x))^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(372\) vs. \(2(126)=252\).
time = 5.28, size = 372, normalized size = 2.95 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}+\frac {(c+d) \left (\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )}{\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2+2 \tan \left (\frac {1}{2} (e+f x)\right )}-\frac {-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{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 {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}}\right )}{4 f (a (1+\sin (e+f x)))^{3/2} \sqrt {c+d \sin (e+f x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*((-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos
[(e + f*x)/2] + Sin[(e + f*x)/2]) + ((c + d)*(Log[1 + Tan[(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]))))/(4*f*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c + d*Sin[e + f*x]])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1372\) vs. \(2(103)=206\).
time = 11.17, size = 1373, normalized size = 10.90

method	result	size
default	\(\text {Expression too large to display}\)	\(1373\)

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))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/8/f*((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)+cos(f*
x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*sin(f*x+e)*cos(f*x+e)*2^(1/2)*c+
(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)+cos(f*x+e)*c-
d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*sin(f*x+e)*cos(f*x+e)*2^(1/2)*d+(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)+cos(f*x+e)*c-d*cos(f
*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*cos(f*x+e)^2*2^(1/2)*c+(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)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+
e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*cos(f*x+e)^2*2^(1/2)*d-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)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e
)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*sin(f*x+e)*2^(1/2)*c-2*(2*c-2*d)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*s
in(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*
x+e)+sin(f*x+e)))*sin(f*x+e)*2^(1/2)*d+(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)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*
cos(f*x+e)*2^(1/2)*c+(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)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*cos(f*x+e)*2^(1/2)
*d-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)+cos(f*x+
e)*c-d*cos(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*c-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)+cos(f*x+e)*c-d*cos(f*x+e)+c*sin(f*x
+e)-d*sin(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*2^(1/2)*d-4*sin(f*x+e)*cos(f*x+e)*((c+d*sin(f*x+e))/(cos(f*x+
e)+1))^(1/2)*c+4*sin(f*x+e)*cos(f*x+e)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*d)*(c+d*sin(f*x+e))^(1/2)/sin(f
*x+e)/(a*(1+sin(f*x+e)))^(3/2)/((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)/(c-d)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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))^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (109) = 218\).
time = 0.53, size = 944, normalized size = 7.49 \begin {gather*} \left [\frac {{\left ({\left (c + d\right )} \cos \left (f x + e\right )^{2} - {\left (c + d\right )} \cos \left (f x + e\right ) - {\left ({\left (c + d\right )} \cos \left (f x + e\right ) + 2 \, c + 2 \, d\right )} \sin \left (f x + e\right ) - 2 \, c - 2 \, d\right )} \sqrt {2 \, a c - 2 \, a d} \log \left (\frac {{\left (a c^{2} - 14 \, a c d + 17 \, a d^{2}\right )} \cos \left (f x + e\right )^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left (13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (c - 3 \, d\right )} \cos \left (f x + e\right )^{2} - {\left (3 \, c - d\right )} \cos \left (f x + e\right ) + {\left ({\left (c - 3 \, d\right )} \cos \left (f x + e\right ) + 4 \, c - 4 \, d\right )} \sin \left (f x + e\right ) - 4 \, c + 4 \, d\right )} \sqrt {2 \, a c - 2 \, a d} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} - 2 \, {\left (9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right ) - {\left (4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left (a c^{2} - 14 \, a c d + 17 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 4}\right ) + 8 \, {\left ({\left (c - d\right )} \cos \left (f x + e\right ) - {\left (c - d\right )} \sin \left (f x + e\right ) + c - d\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{16 \, {\left ({\left (a^{2} c - a^{2} d\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{2} c - a^{2} d\right )} f \cos \left (f x + e\right ) - 2 \, {\left (a^{2} c - a^{2} d\right )} f - {\left ({\left (a^{2} c - a^{2} d\right )} f \cos \left (f x + e\right ) + 2 \, {\left (a^{2} c - a^{2} d\right )} f\right )} \sin \left (f x + e\right )\right )}}, -\frac {{\left ({\left (c + d\right )} \cos \left (f x + e\right )^{2} - {\left (c + d\right )} \cos \left (f x + e\right ) - {\left ({\left (c + d\right )} \cos \left (f x + e\right ) + 2 \, c + 2 \, d\right )} \sin \left (f x + e\right ) - 2 \, c - 2 \, d\right )} \sqrt {-2 \, a c + 2 \, a d} \arctan \left (\frac {\sqrt {-2 \, a c + 2 \, a d} \sqrt {a \sin \left (f x + e\right ) + a} {\left ({\left (c - 3 \, d\right )} \sin \left (f x + e\right ) - 3 \, c + d\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{4 \, {\left ({\left (a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a c^{2} - a c d\right )} \cos \left (f x + e\right )\right )}}\right ) - 4 \, {\left ({\left (c - d\right )} \cos \left (f x + e\right ) - {\left (c - d\right )} \sin \left (f x + e\right ) + c - d\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{8 \, {\left ({\left (a^{2} c - a^{2} d\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{2} c - a^{2} d\right )} f \cos \left (f x + e\right ) - 2 \, {\left (a^{2} c - a^{2} d\right )} f - {\left ({\left (a^{2} c - a^{2} d\right )} f \cos \left (f x + e\right ) + 2 \, {\left (a^{2} c - a^{2} d\right )} f\right )} \sin \left (f x + e\right )\right )}}\right ] \end {gather*}

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))^(3/2),x, algorithm="fricas")

[Out]

[1/16*(((c + d)*cos(f*x + e)^2 - (c + d)*cos(f*x + e) - ((c + d)*cos(f*x + e) + 2*c + 2*d)*sin(f*x + e) - 2*c
- 2*d)*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*si
n(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*((c - d)*c
os(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c - a^2*d
)*f*cos(f*x + e)^2 - (a^2*c - a^2*d)*f*cos(f*x + e) - 2*(a^2*c - a^2*d)*f - ((a^2*c - a^2*d)*f*cos(f*x + e) +
2*(a^2*c - a^2*d)*f)*sin(f*x + e)), -1/8*(((c + d)*cos(f*x + e)^2 - (c + d)*cos(f*x + e) - ((c + d)*cos(f*x +
e) + 2*c + 2*d)*sin(f*x + e) - 2*c - 2*d)*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*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x +
e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c - a^2*d)*f*cos(f*x + e)^2 - (a^2*c - a^2*d)*f*cos(f*x + e) - 2*(a^2
*c - a^2*d)*f - ((a^2*c - a^2*d)*f*cos(f*x + e) + 2*(a^2*c - a^2*d)*f)*sin(f*x + e))]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

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))**(3/2),x)

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}

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))^(3/2),x)

[Out]

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

________________________________________________________________________________________