\(\int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx\) [544]

Optimal result

Integrand size = 23, antiderivative size = 91 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\frac {2 b}{d e \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}} \]

[Out]

2*b/d/e/(e*cos(d*x+c))^(1/2)+2*a*sin(d*x+c)/d/e/(e*cos(d*x+c))^(1/2)-2*a*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*
d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/e^2/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2748, 2716, 2721, 2719} \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=-\frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}+\frac {2 b}{d e \sqrt {e \cos (c+d x)}} \]

[In]

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

[Out]

(2*b)/(d*e*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*
x]]) + (2*a*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]])

Rule 2716

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

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2721

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

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {2 b}{d e \sqrt {e \cos (c+d x)}}+a \int \frac {1}{(e \cos (c+d x))^{3/2}} \, dx \\ & = \frac {2 b}{d e \sqrt {e \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {a \int \sqrt {e \cos (c+d x)} \, dx}{e^2} \\ & = \frac {2 b}{d e \sqrt {e \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (a \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{e^2 \sqrt {\cos (c+d x)}} \\ & = \frac {2 b}{d e \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.59 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\frac {2 \left (b-a \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+a \sin (c+d x)\right )}{d e \sqrt {e \cos (c+d x)}} \]

[In]

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

[Out]

(2*(b - a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + a*Sin[c + d*x]))/(d*e*Sqrt[e*Cos[c + d*x]])

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}\)	\(119\)
parts	\(-\frac {2 a \left (-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{e \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 b}{d e \sqrt {e \cos \left (d x +c \right )}}\)	\(219\)

[In]

int((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/e/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/sin(1/2*d*x+1/2*c)*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a-(sin(1
/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a+b*sin(1/2*d*x+
1/2*c))/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\frac {-i \, \sqrt {2} a \sqrt {e} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + i \, \sqrt {2} a \sqrt {e} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + b\right )}}{d e^{2} \cos \left (d x + c\right )} \]

[In]

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

(-I*sqrt(2)*a*sqrt(e)*cos(d*x + c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x
+ c))) + I*sqrt(2)*a*sqrt(e)*cos(d*x + c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*s
in(d*x + c))) + 2*sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + b))/(d*e^2*cos(d*x + c))

Sympy [F]

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

[In]

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))**(3/2),x)

[Out]

Integral((a + b*sin(c + d*x))/(e*cos(c + d*x))**(3/2), x)

Maxima [F]

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

[In]

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((a+b*sin(d*x+c))/(e*cos(d*x+c))^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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