\(\int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\) [165]

Optimal result

Integrand size = 23, antiderivative size = 116 \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {5 \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} \sqrt {a} d}-\frac {5 a}{12 d (a+a \sin (c+d x))^{3/2}}-\frac {5}{8 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a+a \sin (c+d x)}} \]

[Out]

-5/12*a/d/(a+a*sin(d*x+c))^(3/2)+5/16*arctanh(1/2*(a+a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/d*2^(1/2)/a^(1/2)-5/
8/d/(a+a*sin(d*x+c))^(1/2)+1/2*sec(d*x+c)^2/d/(a+a*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2766, 2746, 53, 65, 212} \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {5 \text {arctanh}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} \sqrt {a} d}-\frac {5}{8 d \sqrt {a \sin (c+d x)+a}}-\frac {5 a}{12 d (a \sin (c+d x)+a)^{3/2}}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a \sin (c+d x)+a}} \]

[In]

Int[Sec[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(5*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*Sqrt[a]*d) - (5*a)/(12*d*(a + a*Sin[c + d*x
])^(3/2)) - 5/(8*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^2/(2*d*Sqrt[a + a*Sin[c + d*x]])

Rule 53

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 2766

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[(-b)*((
g*Cos[e + f*x])^(p + 1)/(a*f*g*(p + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[a*((2*p + 1)/(2*g^2*(p + 1))), In
t[(g*Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0
] && LtQ[p, -1] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^2(c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{4} (5 a) \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = \frac {\sec ^2(c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{4 d} \\ & = -\frac {5 a}{12 d (a+a \sin (c+d x))^{3/2}}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {(5 a) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{8 d} \\ & = -\frac {5 a}{12 d (a+a \sin (c+d x))^{3/2}}-\frac {5}{8 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {5 \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{16 d} \\ & = -\frac {5 a}{12 d (a+a \sin (c+d x))^{3/2}}-\frac {5}{8 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a+a \sin (c+d x)}}+\frac {5 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{8 d} \\ & = \frac {5 \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} \sqrt {a} d}-\frac {5 a}{12 d (a+a \sin (c+d x))^{3/2}}-\frac {5}{8 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^2(c+d x)}{2 d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.36 \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},2,-\frac {1}{2},\frac {1}{2} (1+\sin (c+d x))\right )}{6 d (a+a \sin (c+d x))^{3/2}} \]

[In]

Integrate[Sec[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

-1/6*(a*Hypergeometric2F1[-3/2, 2, -1/2, (1 + Sin[c + d*x])/2])/(d*(a + a*Sin[c + d*x])^(3/2))

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {2 a^{3} \left (-\frac {1}{4 a^{3} \sqrt {a +a \sin \left (d x +c \right )}}-\frac {1}{12 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}}{4 a \sin \left (d x +c \right )-4 a}-\frac {5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 \sqrt {a}}}{4 a^{3}}\right )}{d}\)	\(107\)

[In]

int(sec(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*a^3*(-1/4/a^3/(a+a*sin(d*x+c))^(1/2)-1/12/a^2/(a+a*sin(d*x+c))^(3/2)-1/4/a^3*(1/4*(a+a*sin(d*x+c))^(1/2)/(a*
sin(d*x+c)-a)-5/8*2^(1/2)/a^(1/2)*arctanh(1/2*(a+a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))))/d

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.25 \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {15 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (15 \, \cos \left (d x + c\right )^{2} - 10 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (a d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{2}\right )}} \]

[In]

integrate(sec(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/96*(15*sqrt(2)*(cos(d*x + c)^2*sin(d*x + c) + cos(d*x + c)^2)*sqrt(a)*log(-(a*sin(d*x + c) + 2*sqrt(2)*sqrt(
a*sin(d*x + c) + a)*sqrt(a) + 3*a)/(sin(d*x + c) - 1)) - 4*(15*cos(d*x + c)^2 - 10*sin(d*x + c) - 2)*sqrt(a*si
n(d*x + c) + a))/(a*d*cos(d*x + c)^2*sin(d*x + c) + a*d*cos(d*x + c)^2)

Sympy [F]

\[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]

[In]

integrate(sec(d*x+c)**3/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Integral(sec(c + d*x)**3/sqrt(a*(sin(c + d*x) + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {15 \, \sqrt {2} \sqrt {a} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (15 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a - 20 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{2} - 8 \, a^{3}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 2 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a}}{96 \, a d} \]

[In]

integrate(sec(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-1/96*(15*sqrt(2)*sqrt(a)*log(-(sqrt(2)*sqrt(a) - sqrt(a*sin(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(a*sin(d*x
+ c) + a))) + 4*(15*(a*sin(d*x + c) + a)^2*a - 20*(a*sin(d*x + c) + a)*a^2 - 8*a^3)/((a*sin(d*x + c) + a)^(5/2
) - 2*(a*sin(d*x + c) + a)^(3/2)*a))/(a*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (93) = 186\).

Time = 0.37 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\sqrt {a} {\left (\frac {15 \, \sqrt {2} \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {15 \, \sqrt {2} \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {6 \, \sqrt {2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {4 \, \sqrt {2} {\left (6 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{96 \, d} \]

[In]

integrate(sec(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/96*sqrt(a)*(15*sqrt(2)*log(cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 15*
sqrt(2)*log(-cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 6*sqrt(2)*cos(-1/4*
pi + 1/2*d*x + 1/2*c)/((cos(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 4*sqrt(
2)*(6*cos(-1/4*pi + 1/2*d*x + 1/2*c)^2 + 1)/(a*cos(-1/4*pi + 1/2*d*x + 1/2*c)^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/
2*c))))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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