3.148 \(\int (a+a \sec (c+d x))^{3/2} \tan ^3(c+d x) \, dx\)

Optimal. Leaf size=121 \[ \frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}+\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sqrt{a \sec (c+d x)+a}}{d} \]

[Out]

(2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*a*Sqrt[a + a*Sec[c + d*x]])/d - (2*(a + a*Sec[c +
d*x])^(3/2))/(3*d) - (2*(a + a*Sec[c + d*x])^(5/2))/(5*a*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a^2*d)

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Rubi [A]  time = 0.10358, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3880, 80, 50, 63, 207} \[ \frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}+\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sqrt{a \sec (c+d x)+a}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^3,x]

[Out]

(2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*a*Sqrt[a + a*Sec[c + d*x]])/d - (2*(a + a*Sec[c +
d*x])^(3/2))/(3*d) - (2*(a + a*Sec[c + d*x])^(5/2))/(5*a*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a^2*d)

Rule 3880

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]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

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

Rule 63

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 207

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

Rubi steps

\begin{align*} \int (a+a \sec (c+d x))^{3/2} \tan ^3(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-a+a x) (a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 a \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 a \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{d}\\ &=\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{d}-\frac{2 a \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.247848, size = 92, normalized size = 0.76 \[ \frac{2 (a (\sec (c+d x)+1))^{3/2} \left (\sqrt{\sec (c+d x)+1} \left (15 \sec ^3(c+d x)+24 \sec ^2(c+d x)-32 \sec (c+d x)-146\right )+105 \tanh ^{-1}\left (\sqrt{\sec (c+d x)+1}\right )\right )}{105 d (\sec (c+d x)+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^3,x]

[Out]

(2*(a*(1 + Sec[c + d*x]))^(3/2)*(105*ArcTanh[Sqrt[1 + Sec[c + d*x]]] + Sqrt[1 + Sec[c + d*x]]*(-146 - 32*Sec[c
+ d*x] + 24*Sec[c + d*x]^2 + 15*Sec[c + d*x]^3)))/(105*d*(1 + Sec[c + d*x])^(3/2))

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Maple [B]  time = 0.204, size = 291, normalized size = 2.4 \begin{align*}{\frac{a}{840\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{7/2}+315\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{7/2}+315\,\cos \left ( dx+c \right ) \sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{7/2}+105\,\sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{7/2}-2336\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-512\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+384\,\cos \left ( dx+c \right ) +240 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^(3/2)*tan(d*x+c)^3,x)

[Out]

1/840/d*a*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(105*cos(d*x+c)^3*2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos
(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)+315*cos(d*x+c)^2*2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*
x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)+315*cos(d*x+c)*2^(1/2)*arctan(1/2*2^(1/2)*(-2
*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)+105*2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)-2336*cos(d*x+c)^3-512*cos(d*x+c)^2+384*cos
(d*x+c)+240)/cos(d*x+c)^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.01883, size = 782, normalized size = 6.46 \begin{align*} \left [\frac{105 \, a^{\frac{3}{2}} \cos \left (d x + c\right )^{3} \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 (146 \, a \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} - 24 \, a \cos \left (d x + c\right ) - 15 \, a\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{210 \, d \cos \left (d x + c\right )^{3}}, -\frac{105 \, \sqrt{-a} 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 ) \cos \left (d x + c\right )^{3} + 2 \,{\left (146 \, a \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} - 24 \, a \cos \left (d x + c\right ) - 15 \, a\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(105*a^(3/2)*cos(d*x + c)^3*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*(146*a*cos(d*x + c)^3 + 32*a*cos(d*x + c)^2 - 2
4*a*cos(d*x + c) - 15*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^3), -1/105*(105*sqrt(-a)*a*a
rctan(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))*cos(d*x + c)^3 +
2*(146*a*cos(d*x + c)^3 + 32*a*cos(d*x + c)^2 - 24*a*cos(d*x + c) - 15*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x +
c)))/(d*cos(d*x + c)^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 4.98603, size = 238, normalized size = 1.97 \begin{align*} -\frac{\sqrt{2}{\left (\frac{105 \, \sqrt{2} a^{3} \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}} + \frac{2 \,{\left (105 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} a^{3} - 70 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} a^{4} + 84 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a^{5} + 120 \, a^{6}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}\right )} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )}{105 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*sqrt(2)*(105*sqrt(2)*a^3*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/sqrt(-a) + 2*
(105*(a*tan(1/2*d*x + 1/2*c)^2 - a)^3*a^3 - 70*(a*tan(1/2*d*x + 1/2*c)^2 - a)^2*a^4 + 84*(a*tan(1/2*d*x + 1/2*
c)^2 - a)*a^5 + 120*a^6)/((a*tan(1/2*d*x + 1/2*c)^2 - a)^3*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)))*sgn(cos(d*x +
c))/(a*d)