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

Optimal. Leaf size=222 \[ \frac{2 a^6 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{10 a^5 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{2 a^4 \tan ^7(c+d x)}{d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]

[Out]

(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[
c + d*x]]) - (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[
c + d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(d*(a + a*Sec[c + d*x])^(7/2)) + (10*a^5*Tan[c + d*x]^9)/(9*d*(a + a
*Sec[c + d*x])^(9/2)) + (2*a^6*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.106012, antiderivative size = 222, 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} \[ \frac{2 a^6 \tan ^{11}(c+d x)}{11 d (a \sec (c+d x)+a)^{11/2}}+\frac{10 a^5 \tan ^9(c+d x)}{9 d (a \sec (c+d x)+a)^{9/2}}+\frac{2 a^4 \tan ^7(c+d x)}{d (a \sec (c+d x)+a)^{7/2}}+\frac{2 a^3 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 a^2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[
c + d*x]]) - (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[
c + d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(d*(a + a*Sec[c + d*x])^(7/2)) + (10*a^5*Tan[c + d*x]^9)/(9*d*(a + a
*Sec[c + d*x])^(9/2)) + (2*a^6*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2))

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 \sqrt{a+a \sec (c+d x)} \tan ^6(c+d x) \, dx &=-\frac{\left (2 a^4\right ) \operatorname{Subst}\left (\int \frac{x^6 \left (2+a x^2\right )^3}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{\left (2 a^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^3}-\frac{x^2}{a^2}+\frac{x^4}{a}+7 x^6+5 a x^8+a^2 x^{10}-\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 a \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{2 a^2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{2 a^3 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{2 a^4 \tan ^7(c+d x)}{d (a+a \sec (c+d x))^{7/2}}+\frac{10 a^5 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac{2 a^6 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}+\frac{(2 a) \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 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{2 a \tan (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{2 a^2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}+\frac{2 a^3 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{2 a^4 \tan ^7(c+d x)}{d (a+a \sec (c+d x))^{7/2}}+\frac{10 a^5 \tan ^9(c+d x)}{9 d (a+a \sec (c+d x))^{9/2}}+\frac{2 a^6 \tan ^{11}(c+d x)}{11 d (a+a \sec (c+d x))^{11/2}}\\ \end{align*}

Mathematica [A]  time = 7.28604, size = 134, normalized size = 0.6 \[ -\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (792 \sin \left (\frac{1}{2} (c+d x)\right )-1386 \sin \left (\frac{3}{2} (c+d x)\right )+495 \sin \left (\frac{5}{2} (c+d x)\right )-616 \sin \left (\frac{7}{2} (c+d x)\right )-247 \sin \left (\frac{11}{2} (c+d x)\right )+3960 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \cos ^{\frac{11}{2}}(c+d x)\right )}{3960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sec[(c + d*x)/2]*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*(3960*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Co
s[c + d*x]^(11/2) + 792*Sin[(c + d*x)/2] - 1386*Sin[(3*(c + d*x))/2] + 495*Sin[(5*(c + d*x))/2] - 616*Sin[(7*(
c + d*x))/2] - 247*Sin[(11*(c + d*x))/2]))/(3960*d)

________________________________________________________________________________________

Maple [B]  time = 0.287, size = 566, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15840/d*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(495*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*s
in(d*x+c)/cos(d*x+c))*cos(d*x+c)^5*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)+2475*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))*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)*(-2*cos(d*x
+c)/(cos(d*x+c)+1))^(11/2)+4950*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)
)*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)+4950*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))*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^
(11/2)+2475*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))*cos(d*x+c)*sin(d*x
+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(11/2)+495*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)/(cos(d*x+c)+1))^(11/2)*sin(d*x+c)+31616*cos(d*x+c)^6-15808*cos(
d*x+c)^5-27712*cos(d*x+c)^4+1984*cos(d*x+c)^3+13120*cos(d*x+c)^2-320*cos(d*x+c)-2880)/cos(d*x+c)^5/sin(d*x+c)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 1.89183, size = 1007, normalized size = 4.54 \begin{align*} \left [\frac{495 \,{\left (\cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5}\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 (494 \, \cos \left (d x + c\right )^{5} + 247 \, \cos \left (d x + c\right )^{4} - 186 \, \cos \left (d x + c\right )^{3} - 155 \, \cos \left (d x + c\right )^{2} + 50 \, \cos \left (d x + c\right ) + 45\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{495 \,{\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}, \frac{2 \,{\left (495 \,{\left (\cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5}\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 (494 \, \cos \left (d x + c\right )^{5} + 247 \, \cos \left (d x + c\right )^{4} - 186 \, \cos \left (d x + c\right )^{3} - 155 \, \cos \left (d x + c\right )^{2} + 50 \, \cos \left (d x + c\right ) + 45\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{495 \,{\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/495*(495*(cos(d*x + c)^6 + cos(d*x + c)^5)*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*(494*cos(d*x + c
)^5 + 247*cos(d*x + c)^4 - 186*cos(d*x + c)^3 - 155*cos(d*x + c)^2 + 50*cos(d*x + c) + 45)*sqrt((a*cos(d*x + c
) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5), 2/495*(495*(cos(d*x + c)^6 + cos(d*x
+ c)^5)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (494*co
s(d*x + c)^5 + 247*cos(d*x + c)^4 - 186*cos(d*x + c)^3 - 155*cos(d*x + c)^2 + 50*cos(d*x + c) + 45)*sqrt((a*co
s(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: NotImplementedError