\(\int \frac {1}{(3+3 \sin (e+f x))^{3/2}} \, dx\) [553]

Optimal result

Integrand size = 14, antiderivative size = 69 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)}}\right )}{6 \sqrt {6} f}-\frac {\cos (e+f x)}{2 f (3+3 \sin (e+f x))^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2729, 2728, 212} \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{2 \sqrt {2} a^{3/2} f}-\frac {\cos (e+f x)}{2 f (a \sin (e+f x)+a)^{3/2}} \]

[In]

Int[(a + a*Sin[e + f*x])^(-3/2),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d
*(2*n + 1))), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 a} \\ & = -\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{2 a f} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}-\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )+(1+i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) (1+\sin (e+f x))\right )}{6 \sqrt {3} f (1+\sin (e+f x))^{3/2}} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(-3/2),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-Cos[(e + f*x)/2] + Sin[(e + f*x)/2] + (1 + I)*(-1)^(3/4)*ArcTanh[(1/2
+ I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]*(1 + Sin[e + f*x])))/(6*Sqrt[3]*f*(1 + Sin[e + f*x])^(3/2))

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.81

method	result	size
default	\(-\frac {\left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (f x +e \right )+2 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{4 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(125\)

[In]

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

[Out]

-1/4/a^(7/2)*(2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*sin(f*x+e)+2*(a-a*sin(f*x+e))^(1
/2)*a^(3/2)+2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2)*(-a*(sin(f*x+e)-1))^(1/2)/cos(f*x
+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (62) = 124\).

Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.65 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{8 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

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

[Out]

1/8*(sqrt(2)*(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)*sqrt(a)*log(-(a*cos(f*x + e
)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos
(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) +
4*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a
^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))

Sympy [F]

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

[In]

integrate(1/(a+a*sin(f*x+e))**(3/2),x)

[Out]

Integral((a*sin(e + f*x) + a)**(-3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*sin(f*x + e) + a)^(-3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (62) = 124\).

Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.93 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (\frac {\log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{8 \, \sqrt {a} f} \]

[In]

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

[Out]

1/8*sqrt(2)*(log(sin(-1/4*pi + 1/2*f*x + 1/2*e) + 1)/(a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - log(-sin(-1/4*p
i + 1/2*f*x + 1/2*e) + 1)/(a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - 2*sin(-1/4*pi + 1/2*f*x + 1/2*e)/((sin(-1/
4*pi + 1/2*f*x + 1/2*e)^2 - 1)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))/(sqrt(a)*f)

Mupad [F(-1)]

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

[In]

int(1/(a + a*sin(e + f*x))^(3/2),x)

[Out]

int(1/(a + a*sin(e + f*x))^(3/2), x)