3.326 \(\int \frac{\tan ^5(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\)
Optimal. Leaf size=148 \[ \frac{2 \left (3 a^2-2 b^2\right ) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}-\frac{2 a \left (a^2-2 b^2\right ) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 (a+b \sec (c+d x))^{7/2}}{7 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
[Out]
(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (2*a*(a^2 - 2*b^2)*Sqrt[a + b*Sec[c + d*x]])/(b^4
*d) + (2*(3*a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(3/2))/(3*b^4*d) - (6*a*(a + b*Sec[c + d*x])^(5/2))/(5*b^4*d) +
(2*(a + b*Sec[c + d*x])^(7/2))/(7*b^4*d)
________________________________________________________________________________________
Rubi [A] time = 0.143342, antiderivative size = 148, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{3885, 898, 1153, 207} \[ \frac{2 \left (3 a^2-2 b^2\right ) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}-\frac{2 a \left (a^2-2 b^2\right ) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 (a+b \sec (c+d x))^{7/2}}{7 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{5/2}}{5 b^4 d}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
[In]
Int[Tan[c + d*x]^5/Sqrt[a + b*Sec[c + d*x]],x]
[Out]
(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (2*a*(a^2 - 2*b^2)*Sqrt[a + b*Sec[c + d*x]])/(b^4
*d) + (2*(3*a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(3/2))/(3*b^4*d) - (6*a*(a + b*Sec[c + d*x])^(5/2))/(5*b^4*d) +
(2*(a + b*Sec[c + d*x])^(7/2))/(7*b^4*d)
Rule 3885
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Rule 898
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 + a*e^2)/e^2 - (2*c
*d*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]
Rule 1153
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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 \frac{\tan ^5(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{x \sqrt{a+x}} \, dx,x,b \sec (c+d x)\right )}{b^4 d}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (-a^2+b^2+2 a x^2-x^4\right )^2}{-a+x^2} \, dx,x,\sqrt{a+b \sec (c+d x)}\right )}{b^4 d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-a^3+2 a b^2+\left (3 a^2-2 b^2\right ) x^2-3 a x^4+x^6+\frac{b^4}{-a+x^2}\right ) \, dx,x,\sqrt{a+b \sec (c+d x)}\right )}{b^4 d}\\ &=-\frac{2 a \left (a^2-2 b^2\right ) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 \left (3 a^2-2 b^2\right ) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{5/2}}{5 b^4 d}+\frac{2 (a+b \sec (c+d x))^{7/2}}{7 b^4 d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-a+x^2} \, dx,x,\sqrt{a+b \sec (c+d x)}\right )}{d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}-\frac{2 a \left (a^2-2 b^2\right ) \sqrt{a+b \sec (c+d x)}}{b^4 d}+\frac{2 \left (3 a^2-2 b^2\right ) (a+b \sec (c+d x))^{3/2}}{3 b^4 d}-\frac{6 a (a+b \sec (c+d x))^{5/2}}{5 b^4 d}+\frac{2 (a+b \sec (c+d x))^{7/2}}{7 b^4 d}\\ \end{align*}
Mathematica [A] time = 6.30109, size = 248, normalized size = 1.68 \[ \frac{\sec (c+d x) (a \cos (c+d x)+b) \left (-\frac{4 \left (35 b^2-12 a^2\right ) \sec (c+d x)}{105 b^3}+\frac{8 a \left (35 b^2-12 a^2\right )}{105 b^4}-\frac{12 a \sec ^2(c+d x)}{35 b^2}+\frac{2 \sec ^3(c+d x)}{7 b}\right )}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sin (c+d x) \tan (c+d x) \sqrt{a \cos (c+d x)} \sqrt{a \cos (c+d x)+b} \left (\log \left (\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}+1\right )-\log \left (1-\frac{\sqrt{a \cos (c+d x)+b}}{\sqrt{a \cos (c+d x)}}\right )\right )}{a d \left (1-\cos ^2(c+d x)\right ) \sqrt{a+b \sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Tan[c + d*x]^5/Sqrt[a + b*Sec[c + d*x]],x]
[Out]
((b + a*Cos[c + d*x])*Sec[c + d*x]*((8*a*(-12*a^2 + 35*b^2))/(105*b^4) - (4*(-12*a^2 + 35*b^2)*Sec[c + d*x])/(
105*b^3) - (12*a*Sec[c + d*x]^2)/(35*b^2) + (2*Sec[c + d*x]^3)/(7*b)))/(d*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[a*
Cos[c + d*x]]*Sqrt[b + a*Cos[c + d*x]]*(-Log[1 - Sqrt[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]] + Log[1 + Sqrt
[b + a*Cos[c + d*x]]/Sqrt[a*Cos[c + d*x]]])*Sin[c + d*x]*Tan[c + d*x])/(a*d*(1 - Cos[c + d*x]^2)*Sqrt[a + b*Se
c[c + d*x]])
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Maple [B] time = 0.705, size = 4997, normalized size = 33.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(d*x+c)^5/(a+b*sec(d*x+c))^(1/2),x)
[Out]
-1/420/d/b^4/a/(a-b)^(3/2)*4^(1/2)*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(cos(d*x+c)+1)*(-1+cos(d*x+c))^4*(-420*
cos(d*x+c)^5*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1
/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)
-b)/sin(d*x+c)^2)*a^3*b^4+315*cos(d*x+c)^5*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*c
os(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(
d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^2*b^5-105*cos(d*x+c)^5*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*
cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+
a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a*b^6-60*((b+a*cos(d*x+c))*cos(d
*x+c)/(cos(d*x+c)+1)^2)^(3/2)*(a-b)^(3/2)*a*b^3-210*cos(d*x+c)^6*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+
c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*
x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^6*b-105*cos(d*x+c)^6*ln(-2/(a-b)^(1/2)
*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)
+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^5*b^2+420*
cos(d*x+c)^6*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1
/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)
-b)/sin(d*x+c)^2)*a^4*b^3-315*cos(d*x+c)^6*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*c
os(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(
d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^3*b^4+105*cos(d*x+c)^6*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*
cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+
a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^2*b^5+210*cos(d*x+c)^6*ln(-1/(
a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*
cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^
6*b+105*cos(d*x+c)^6*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+
1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-
b)^(1/2)-b)/sin(d*x+c)^2)*a^5*b^2+105*cos(d*x+c)^6*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d
*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+
c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^7-105*cos(d*x+c)^6*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))
*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*
((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^7+420*cos(d*x+c)^5*ln(-2/(
a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*
cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^
3*b^4-315*cos(d*x+c)^5*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c
)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(
a-b)^(1/2)-b)/sin(d*x+c)^2)*a^2*b^5+105*cos(d*x+c)^5*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos
(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*
x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a*b^6-105*cos(d*x+c)^5*ln(-1/(a-b)^(1/2)*(-1+cos(d*x
+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c
)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^6*b+210*cos(d*x+c)^5*l
n(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2
)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)
^2)*a^5*b^2+192*cos(d*x+c)^6*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^5+192*cos(d*x+
c)^5*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^5+105*cos(d*x+c)^5*ln(-1/(a-b)^(1/2)*(
-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b
*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^4*b^3-420*co
s(d*x+c)^6*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2
)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b
)/sin(d*x+c)^2)*a^4*b^3+315*cos(d*x+c)^6*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos
(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*
x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^3*b^4-105*cos(d*x+c)^6*ln(-1/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*co
s(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*
cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^2*b^5+105*cos(d*x+c)^5*ln(-2/(a-
b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*co
s(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^6*
b-210*cos(d*x+c)^5*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)
^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)
^(1/2)-b)/sin(d*x+c)^2)*a^5*b^2-105*cos(d*x+c)^5*ln(-2/(a-b)^(1/2)*(-1+cos(d*x+c))*(2*cos(d*x+c)*((b+a*cos(d*x
+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-2*a*cos(d*x+c)+b*cos(d*x+c)+2*((b+a*cos(d*x+c))*cos(d*x+c)
/(cos(d*x+c)+1)^2)^(1/2)*(a-b)^(1/2)-b)/sin(d*x+c)^2)*a^4*b^3+192*cos(d*x+c)^5*(a-b)^(3/2)*((b+a*cos(d*x+c))*c
os(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^4*b-524*cos(d*x+c)^5*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+
1)^2)^(1/2)*a^3*b^2-524*cos(d*x+c)^5*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^2*b^3-
524*cos(d*x+c)^6*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^3*b^2+210*cos(d*x+c)^6*(a-
b)^(3/2)*a^(3/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4
*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*b^4+210*cos(d*x+c)^5*(a-b)^(3/2)*a^(1/2)*ln
(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(
d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*b^5-108*cos(d*x+c)^5*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/
(cos(d*x+c)+1)^2)^(3/2)*a^2*b^2-288*cos(d*x+c)^4*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3
/2)*a^3*b+840*cos(d*x+c)^4*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*a*b^3+180*cos(d*x+
c)^3*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*a^2*b^2-96*cos(d*x+c)^2*(a-b)^(3/2)*((b+
a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*a^3*b+100*cos(d*x+c)^2*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+
c)/(cos(d*x+c)+1)^2)^(3/2)*a*b^3+72*cos(d*x+c)*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2
)*a^2*b^2+192*cos(d*x+c)^4*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^4*b-180*cos(d*x+
c)*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*a*b^3-524*cos(d*x+c)^4*(a-b)^(3/2)*((b+a*c
os(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^2*b^3-36*cos(d*x+c)^6*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)
/(cos(d*x+c)+1)^2)^(3/2)*a^2*b^2-96*cos(d*x+c)^5*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3
/2)*a^3*b+280*cos(d*x+c)^5*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*a*b^3-36*cos(d*x+c
)^4*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*a^2*b^2-288*cos(d*x+c)^3*(a-b)^(3/2)*((b+
a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)*a^3*b+780*cos(d*x+c)^3*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+
c)/(cos(d*x+c)+1)^2)^(3/2)*a*b^3+216*cos(d*x+c)^2*(a-b)^(3/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(
3/2)*a^2*b^2)/cos(d*x+c)^3/sin(d*x+c)^8/((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(3/2)
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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(tan(d*x+c)^5/(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 4.08129, size = 945, normalized size = 6.39 \begin{align*} \left [\frac{105 \, \sqrt{a} b^{4} \cos \left (d x + c\right )^{3} \log \left (-8 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a b \cos \left (d x + c\right ) - b^{2} + 4 \,{\left (2 \, a \cos \left (d x + c\right )^{2} + b \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}\right ) - 4 \,{\left (18 \, a^{2} b^{2} \cos \left (d x + c\right ) - 15 \, a b^{3} + 4 \,{\left (12 \, a^{4} - 35 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} - 2 \,{\left (12 \, a^{3} b - 35 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}}{210 \, a b^{4} d \cos \left (d x + c\right )^{3}}, \frac{105 \, \sqrt{-a} b^{4} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + b}\right ) \cos \left (d x + c\right )^{3} - 2 \,{\left (18 \, a^{2} b^{2} \cos \left (d x + c\right ) - 15 \, a b^{3} + 4 \,{\left (12 \, a^{4} - 35 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} - 2 \,{\left (12 \, a^{3} b - 35 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}}{105 \, a b^{4} d \cos \left (d x + c\right )^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^5/(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
[1/210*(105*sqrt(a)*b^4*cos(d*x + c)^3*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x +
c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - 4*(18*a^2*b^2*cos(d*x + c) - 15*a*b
^3 + 4*(12*a^4 - 35*a^2*b^2)*cos(d*x + c)^3 - 2*(12*a^3*b - 35*a*b^3)*cos(d*x + c)^2)*sqrt((a*cos(d*x + c) + b
)/cos(d*x + c)))/(a*b^4*d*cos(d*x + c)^3), 1/105*(105*sqrt(-a)*b^4*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)
/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b))*cos(d*x + c)^3 - 2*(18*a^2*b^2*cos(d*x + c) - 15*a*b^3 + 4
*(12*a^4 - 35*a^2*b^2)*cos(d*x + c)^3 - 2*(12*a^3*b - 35*a*b^3)*cos(d*x + c)^2)*sqrt((a*cos(d*x + c) + b)/cos(
d*x + c)))/(a*b^4*d*cos(d*x + c)^3)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{5}{\left (c + d x \right )}}{\sqrt{a + b \sec{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)**5/(a+b*sec(d*x+c))**(1/2),x)
[Out]
Integral(tan(c + d*x)**5/sqrt(a + b*sec(c + d*x)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{5}}{\sqrt{b \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^5/(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate(tan(d*x + c)^5/sqrt(b*sec(d*x + c) + a), x)