\(\int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx\) [197]

Optimal result

Integrand size = 21, antiderivative size = 78 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {2}{3 a d (a+a \sec (c+d x))^{3/2}}+\frac {2}{a^2 d \sqrt {a+a \sec (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3965, 53, 65, 213} \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {2}{a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {2}{3 a d (a \sec (c+d x)+a)^{3/2}} \]

[In]

Int[Tan[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]

[Out]

(-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + 2/(3*a*d*(a + a*Sec[c + d*x])^(3/2)) + 2/(a^2*d*S
qrt[a + a*Sec[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 213

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

Rule 3965

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

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

Mathematica [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.51 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\sec (c+d x)\right )}{3 a d (a (1+\sec (c+d x)))^{3/2}} \]

[In]

Integrate[Tan[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +a \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {a +a \sec \left (d x +c \right )}}+\frac {2}{3 a \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}}{d}\)	\(62\)
default	\(\frac {-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +a \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{a^{2} \sqrt {a +a \sec \left (d x +c \right )}}+\frac {2}{3 a \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}}{d}\)	\(62\)

[In]

int(tan(d*x+c)/(a+a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (66) = 132\).

Time = 0.30 (sec) , antiderivative size = 321, normalized size of antiderivative = 4.12 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\left [\frac {3 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (4 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{6 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}, \frac {3 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) + 2 \, {\left (4 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{3 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}\right ] \]

[In]

integrate(tan(d*x+c)/(a+a*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x
+ c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(4*cos(d*x + c)^2 + 3*cos(d*
x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d), 1/3*(3
*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(
d*x + c)/(2*a*cos(d*x + c) + a)) + 2*(4*cos(d*x + c)^2 + 3*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c
)))/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d)]

Sympy [F]

\[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\tan {\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(tan(d*x+c)/(a+a*sec(d*x+c))**(5/2),x)

[Out]

Integral(tan(c + d*x)/(a*(sec(c + d*x) + 1))**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {3 \, \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (4 \, a + \frac {3 \, a}{\cos \left (d x + c\right )}\right )}}{{\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} a^{2}}}{3 \, d} \]

[In]

integrate(tan(d*x+c)/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

1/3*(3*log((sqrt(a + a/cos(d*x + c)) - sqrt(a))/(sqrt(a + a/cos(d*x + c)) + sqrt(a)))/a^(5/2) + 2*(4*a + 3*a/c
os(d*x + c))/((a + a/cos(d*x + c))^(3/2)*a^2))/d

Giac [A] (verification not implemented)

none

Time = 1.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.49 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {12 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} + \frac {\sqrt {2} {\left ({\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{8} + 6 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{9}\right )}}{a^{12} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{6 \, d} \]

[In]

integrate(tan(d*x+c)/(a+a*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/6*(12*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a^2*sgn(cos(d*x + c))) + sq
rt(2)*((-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2)*a^8 + 6*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^9)/(a^12*sgn(cos(d*
x + c))))/d

Mupad [B] (verification not implemented)

Time = 14.76 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \frac {\tan (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {2\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}{a^2}+\frac {2}{3\,a}}{d\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}}-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}}{\sqrt {a}}\right )}{a^{5/2}\,d} \]

[In]

int(tan(c + d*x)/(a + a/cos(c + d*x))^(5/2),x)

[Out]

((2*(a + a/cos(c + d*x)))/a^2 + 2/(3*a))/(d*(a + a/cos(c + d*x))^(3/2)) - (2*atanh((a + a/cos(c + d*x))^(1/2)/
a^(1/2)))/(a^(5/2)*d)