3.177 \(\int \frac{\tan ^6(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx\)

Optimal. Leaf size=189 \[ \frac{2 a^4 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{6 a^3 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]

[Out]

(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[
c + d*x]]) - (2*a*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c
+ d*x])^(5/2)) + (6*a^3*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (2*a^4*Tan[c + d*x]^9)/(9*d*(a + a*
Sec[c + d*x])^(9/2))

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Rubi [A]  time = 0.100538, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3887, 461, 203} \[ \frac{2 a^4 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{6 a^3 \tan ^7(c+d x)}{7 d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]

[Out]

(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[
c + d*x]]) - (2*a*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c
+ d*x])^(5/2)) + (6*a^3*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (2*a^4*Tan[c + d*x]^9)/(9*d*(a + a*
Sec[c + d*x])^(9/2))

Rule 3887

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

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\tan ^6(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx &=-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{x^6 \left (2+a x^2\right )^2}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^3}-\frac{x^2}{a^2}+\frac{x^4}{a}+3 x^6+a x^8-\frac{1}{a^3 \left (1+a x^2\right )}\right ) \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{2 \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{2 a \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{2 a^2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{6 a^3 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac{2 a^4 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{\sqrt{a} d}+\frac{2 \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{2 a \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{2 a^2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{6 a^3 \tan ^7(c+d x)}{7 d (a+a \sec (c+d x))^{7/2}}+\frac{2 a^4 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}\\ \end{align*}

Mathematica [C]  time = 19.3146, size = 469, normalized size = 2.48 \[ \frac{16 \left (-3-2 \sqrt{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\left (10-7 \sqrt{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )-5 \sqrt{2}+7}{\cos \left (\frac{1}{2} (c+d x)\right )+1}} \sqrt{\frac{-\left (\sqrt{2}-2\right ) \cos \left (\frac{1}{2} (c+d x)\right )+\sqrt{2}-1}{\cos \left (\frac{1}{2} (c+d x)\right )+1}} \left (\left (\sqrt{2}-2\right ) \cos \left (\frac{1}{2} (c+d x)\right )-\sqrt{2}+1\right ) \cos ^4\left (\frac{1}{4} (c+d x)\right ) \sqrt{-\tan ^2\left (\frac{1}{4} (c+d x)\right )-2 \sqrt{2}+3} \sec ^2(c+d x) \sqrt{\left (\left (2+\sqrt{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right )-\sqrt{2}-1\right ) \sec ^2\left (\frac{1}{4} (c+d x)\right )} \left (\text{EllipticF}\left (\sin ^{-1}\left (\frac{\tan \left (\frac{1}{4} (c+d x)\right )}{\sqrt{3-2 \sqrt{2}}}\right ),17-12 \sqrt{2}\right )+2 \Pi \left (-3+2 \sqrt{2};-\sin ^{-1}\left (\frac{\tan \left (\frac{1}{4} (c+d x)\right )}{\sqrt{3-2 \sqrt{2}}}\right )|17-12 \sqrt{2}\right )\right )}{d \sqrt{a (\sec (c+d x)+1)}}+\frac{\cos \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (\frac{1532}{315} \sin \left (\frac{1}{2} (c+d x)\right )+\frac{4}{9} \sin \left (\frac{1}{2} (c+d x)\right ) \sec ^4(c+d x)-\frac{4}{63} \sin \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x)-\frac{176}{105} \sin \left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x)+\frac{136}{315} \sin \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x)\right )}{d \sqrt{a (\sec (c+d x)+1)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tan[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]],x]

[Out]

(Cos[(c + d*x)/2]*Sec[c + d*x]*((1532*Sin[(c + d*x)/2])/315 + (136*Sec[c + d*x]*Sin[(c + d*x)/2])/315 - (176*S
ec[c + d*x]^2*Sin[(c + d*x)/2])/105 - (4*Sec[c + d*x]^3*Sin[(c + d*x)/2])/63 + (4*Sec[c + d*x]^4*Sin[(c + d*x)
/2])/9))/(d*Sqrt[a*(1 + Sec[c + d*x])]) + (16*(-3 - 2*Sqrt[2])*Cos[(c + d*x)/4]^4*Cos[(c + d*x)/2]*Sqrt[(7 - 5
*Sqrt[2] + (10 - 7*Sqrt[2])*Cos[(c + d*x)/2])/(1 + Cos[(c + d*x)/2])]*Sqrt[(-1 + Sqrt[2] - (-2 + Sqrt[2])*Cos[
(c + d*x)/2])/(1 + Cos[(c + d*x)/2])]*(1 - Sqrt[2] + (-2 + Sqrt[2])*Cos[(c + d*x)/2])*(EllipticF[ArcSin[Tan[(c
+ d*x)/4]/Sqrt[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]] + 2*EllipticPi[-3 + 2*Sqrt[2], -ArcSin[Tan[(c + d*x)/4]/Sqrt
[3 - 2*Sqrt[2]]], 17 - 12*Sqrt[2]])*Sqrt[(-1 - Sqrt[2] + (2 + Sqrt[2])*Cos[(c + d*x)/2])*Sec[(c + d*x)/4]^2]*S
ec[c + d*x]^2*Sqrt[3 - 2*Sqrt[2] - Tan[(c + d*x)/4]^2])/(d*Sqrt[a*(1 + Sec[c + d*x])])

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Maple [B]  time = 0.261, size = 480, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x)

[Out]

1/5040/d/a*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(315*2^(1/2)*sin(d*x+c)*cos(d*x+c)^4*arctanh(1/2*2^(1/2)*(-2*co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)+1260*2^(1/2)*sin(d*
x+c)*cos(d*x+c)^3*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+
c)/(cos(d*x+c)+1))^(9/2)+1890*2^(1/2)*sin(d*x+c)*cos(d*x+c)^2*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*arctanh(1/2
*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+1260*2^(1/2)*sin(d*x+c)*cos(d*x+c)*(-2*co
s(d*x+c)/(cos(d*x+c)+1))^(9/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))
+315*2^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(c
os(d*x+c)+1))^(9/2)*sin(d*x+c)-12256*cos(d*x+c)^5+11168*cos(d*x+c)^4+5312*cos(d*x+c)^3-4064*cos(d*x+c)^2-1280*
cos(d*x+c)+1120)/sin(d*x+c)/cos(d*x+c)^4

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Maxima [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(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.81951, size = 957, normalized size = 5.06 \begin{align*} \left [-\frac{315 \,{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4}\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (d x + c\right )^{2} - 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 ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) - 2 \,{\left (383 \, \cos \left (d x + c\right )^{4} + 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}}, \frac{2 \,{\left (315 \,{\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4}\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right ) +{\left (383 \, \cos \left (d x + c\right )^{4} + 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{315 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[-1/315*(315*(cos(d*x + c)^5 + cos(d*x + c)^4)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x +
c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(383*cos(d*x +
c)^4 + 34*cos(d*x + c)^3 - 132*cos(d*x + c)^2 - 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*s
in(d*x + c))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4), 2/315*(315*(cos(d*x + c)^5 + cos(d*x + c)^4)*sqrt(a)*a
rctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (383*cos(d*x + c)^4 + 34*
cos(d*x + c)^3 - 132*cos(d*x + c)^2 - 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c
))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{6}{\left (c + d x \right )}}{\sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**6/(a+a*sec(d*x+c))**(1/2),x)

[Out]

Integral(tan(c + d*x)**6/sqrt(a*(sec(c + d*x) + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError