\(\int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\) [111]

Optimal result

Integrand size = 23, antiderivative size = 103 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\frac {a \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}} \]

[Out]

a*arctan((e*sin(d*x+c))^(1/2)/e^(1/2))/d/e^(1/2)+a*arctanh((e*sin(d*x+c))^(1/2)/e^(1/2))/d/e^(1/2)-2*a*(sin(1/
2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)
^(1/2)/d/(e*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3957, 2917, 2644, 335, 218, 212, 209, 2721, 2720} \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\frac {a \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{d \sqrt {e \sin (c+d x)}} \]

[In]

Int[(a + a*Sec[c + d*x])/Sqrt[e*Sin[c + d*x]],x]

[Out]

(a*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (a*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) +
(2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]])

Rule 209

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

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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {(-a-a \cos (c+d x)) \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx \\ & = a \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx+a \int \frac {\sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac {\left (a \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{\sqrt {e \sin (c+d x)}} \\ & = \frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e} \\ & = \frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}+\frac {a \text {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d} \\ & = \frac {a \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {a \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 0.95 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.19 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\frac {9 a \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\cot ^2\left (\frac {1}{2} (c+d x)\right ),\cot ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x)) \sin ^3(c+d x)}{5 d \left (4 \left (-2 \operatorname {AppellF1}\left (\frac {9}{4},\frac {1}{2},2,\frac {13}{4},-\cot ^2\left (\frac {1}{2} (c+d x)\right ),\cot ^2\left (\frac {1}{2} (c+d x)\right )\right )+\operatorname {AppellF1}\left (\frac {9}{4},\frac {3}{2},1,\frac {13}{4},-\cot ^2\left (\frac {1}{2} (c+d x)\right ),\cot ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right )+9 \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\cot ^2\left (\frac {1}{2} (c+d x)\right ),\cot ^2\left (\frac {1}{2} (c+d x)\right )\right ) (-1+\cos (c+d x))\right ) \sqrt {e \sin (c+d x)}} \]

[In]

Integrate[(a + a*Sec[c + d*x])/Sqrt[e*Sin[c + d*x]],x]

[Out]

(9*a*AppellF1[5/4, 1/2, 1, 9/4, -Cot[(c + d*x)/2]^2, Cot[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])
*Sin[c + d*x]^3)/(5*d*(4*(-2*AppellF1[9/4, 1/2, 2, 13/4, -Cot[(c + d*x)/2]^2, Cot[(c + d*x)/2]^2] + AppellF1[9
/4, 3/2, 1, 13/4, -Cot[(c + d*x)/2]^2, Cot[(c + d*x)/2]^2])*Cos[(c + d*x)/2]^2 + 9*AppellF1[5/4, 1/2, 1, 9/4,
-Cot[(c + d*x)/2]^2, Cot[(c + d*x)/2]^2]*(-1 + Cos[c + d*x]))*Sqrt[e*Sin[c + d*x]])

Maple [A] (verified)

Time = 7.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.12

method	result	size
parts	\(-\frac {a \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {a \left (\arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{\sqrt {e}\, d}\)	\(115\)
default	\(\frac {\frac {a \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{\sqrt {e}}+\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{\sqrt {e}}-\frac {a \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\)	\(117\)

[In]

int((a+a*sec(d*x+c))/(e*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-a*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))/
cos(d*x+c)/(e*sin(d*x+c))^(1/2)/d+a*(arctan((e*sin(d*x+c))^(1/2)/e^(1/2))+arctanh((e*sin(d*x+c))^(1/2)/e^(1/2)
))/e^(1/2)/d

Fricas [C] (verification not implemented)

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

Time = 0.19 (sec) , antiderivative size = 550, normalized size of antiderivative = 5.34 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\left [\frac {8 \, \sqrt {2} a \sqrt {-i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 8 \, \sqrt {2} a \sqrt {i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, a \sqrt {-e} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {-e}}{4 \, {\left (e \cos \left (d x + c\right )^{2} - e \sin \left (d x + c\right ) - e\right )}}\right ) - a \sqrt {-e} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} - 8 \, {\left (7 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 8\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {-e} + 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right )}{8 \, d e}, \frac {8 \, \sqrt {2} a \sqrt {-i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 8 \, \sqrt {2} a \sqrt {i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, a \sqrt {e} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {e}}{4 \, {\left (e \cos \left (d x + c\right )^{2} + e \sin \left (d x + c\right ) - e\right )}}\right ) + a \sqrt {e} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} - 8 \, {\left (7 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 8\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {e} - 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right )}{8 \, d e}\right ] \]

[In]

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

[Out]

[1/8*(8*sqrt(2)*a*sqrt(-I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + 8*sqrt(2)*a*sqrt(I*e)*
weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*a*sqrt(-e)*arctan(1/4*(cos(d*x + c)^2 - 6*sin(d*x
+ c) - 2)*sqrt(e*sin(d*x + c))*sqrt(-e)/(e*cos(d*x + c)^2 - e*sin(d*x + c) - e)) - a*sqrt(-e)*log((e*cos(d*x
+ c)^4 - 72*e*cos(d*x + c)^2 - 8*(7*cos(d*x + c)^2 - (cos(d*x + c)^2 - 8)*sin(d*x + c) - 8)*sqrt(e*sin(d*x + c
))*sqrt(-e) + 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x
+ c)^2 - 2)*sin(d*x + c) + 8)))/(d*e), 1/8*(8*sqrt(2)*a*sqrt(-I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I
*sin(d*x + c)) + 8*sqrt(2)*a*sqrt(I*e)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*a*sqrt(e)*
arctan(1/4*(cos(d*x + c)^2 + 6*sin(d*x + c) - 2)*sqrt(e*sin(d*x + c))*sqrt(e)/(e*cos(d*x + c)^2 + e*sin(d*x +
c) - e)) + a*sqrt(e)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 - 8*(7*cos(d*x + c)^2 + (cos(d*x + c)^2 - 8)*
sin(d*x + c) - 8)*sqrt(e*sin(d*x + c))*sqrt(e) - 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c
)^4 - 8*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)))/(d*e)]

Sympy [F]

\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=a \left (\int \frac {1}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx\right ) \]

[In]

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

[Out]

a*(Integral(1/sqrt(e*sin(c + d*x)), x) + Integral(sec(c + d*x)/sqrt(e*sin(c + d*x)), x))

Maxima [F]

\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\sqrt {e \sin \left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

integrate((a*sec(d*x + c) + a)/sqrt(e*sin(d*x + c)), x)

Giac [F]

\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\sqrt {e \sin \left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

integrate((a*sec(d*x + c) + a)/sqrt(e*sin(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{\sqrt {e\,\sin \left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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