3.2.41 \(\int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx\) [141]

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

[Out]

-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))*a^(1/2)/d+2*a*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)-2/3*a^2
*tan(d*x+c)^3/d/(a+a*sec(d*x+c))^(3/2)+2/5*a^3*tan(d*x+c)^5/d/(a+a*sec(d*x+c))^(5/2)+2*a^4*tan(d*x+c)^7/d/(a+a
*sec(d*x+c))^(7/2)+10/9*a^5*tan(d*x+c)^9/d/(a+a*sec(d*x+c))^(9/2)+2/11*a^6*tan(d*x+c)^11/d/(a+a*sec(d*x+c))^(1
1/2)

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Rubi [A]
time = 0.08, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3972, 472, 209} \begin {gather*} \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} \text {ArcTan}\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}} \end {gather*}

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 209

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

Rule 472

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 3972

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]

Rubi steps

\begin {align*} \int \sqrt {a+a \sec (c+d x)} \tan ^6(c+d x) \, dx &=-\frac {\left (2 a^4\right ) \text {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 ) \text {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) \text {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*}

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Mathematica [A]
time = 7.24, size = 134, normalized size = 0.60 \begin {gather*} -\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \sec ^5(c+d x) \sqrt {a (1+\sec (c+d x))} \left (3960 \sqrt {2} \text {ArcSin}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^{\frac {11}{2}}(c+d x)+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 )\right )}{3960 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3960*(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]]*Cos[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*S
in[(7*(c + d*x))/2] - 247*Sin[(11*(c + d*x))/2]))/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(565\) vs. \(2(196)=392\).
time = 0.26, size = 566, normalized size = 2.55

method	result	size
default	\(-\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (495 \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {11}{2}}+2475 \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {11}{2}}+4950 \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {11}{2}}+4950 \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {2}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {11}{2}}+2475 \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {2}\, \sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {11}{2}}+495 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {11}{2}} \sin \left (d x +c \right )+31616 \left (\cos ^{6}\left (d x +c \right )\right )-15808 \left (\cos ^{5}\left (d x +c \right )\right )-27712 \left (\cos ^{4}\left (d x +c \right )\right )+1984 \left (\cos ^{3}\left (d x +c \right )\right )+13120 \left (\cos ^{2}\left (d x +c \right )\right )-320 \cos \left (d x +c \right )-2880\right )}{15840 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}\)	\(566\)

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,method=_RETURNVERBOSE)

[Out]

-1/15840/d*(a*(1+cos(d*x+c))/cos(d*x+c))^(1/2)*(495*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c
)/cos(d*x+c)*2^(1/2))*2^(1/2)*sin(d*x+c)*cos(d*x+c)^5*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(11/2)+2475*arctanh(1/2*(
-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*sin(d*x+c)*cos(d*x+c)^4*(-2*cos(d*x
+c)/(1+cos(d*x+c)))^(11/2)+4950*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2)
)*2^(1/2)*sin(d*x+c)*cos(d*x+c)^3*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(11/2)+4950*arctanh(1/2*(-2*cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*sin(d*x+c)*cos(d*x+c)^2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^
(11/2)+2475*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*sin(d*x+c)
*cos(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(11/2)+495*2^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)
*sin(d*x+c)/cos(d*x+c)*2^(1/2))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(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)/sin(d*x+c)/cos(d*x+c)^5

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/990*(495*((cos(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x + 2*c)^2 + 2*cos(2*
d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*arctan2((cos(2*d*x + 2*c)^
2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)
), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c) + 1)) + 1) - (cos(2*d*x + 2*c)^4 + sin(2*d*x + 2*c)^4 + 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x +
2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*arctan2(
(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos
(2*d*x + 2*c) + 1)), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(
sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - 1) - 2*(d*cos(2*d*x + 2*c)^4 + d*sin(2*d*x + 2*c)^4 + 4*d*cos(2*d*x
+ 2*c)^3 + 6*d*cos(2*d*x + 2*c)^2 + 2*(d*cos(2*d*x + 2*c)^2 + 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c)^2 +
4*d*cos(2*d*x + 2*c) + d)*integrate((((cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*cos(12*d*x + 12*c)*cos(2*d*x +
2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 15*cos(6*d*x + 6*c)*cos
(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14*d*x + 14*c)*sin(2*d*x + 2*c)
+ 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c) + 20*sin(8*d*x + 8*c)*sin(2*
d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*
cos(13/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + (cos(2*d*x + 2*c)*sin(14*d*x + 14*c) + 6*cos(2*d*x + 2
*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 15*co
s(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(14*d*x + 14*c)*sin(2*d*x + 2*c) -
6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x
+ 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(13/2*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*s
in(14*d*x + 14*c) + 6*cos(2*d*x + 2*c)*sin(12*d*x + 12*c) + 15*cos(2*d*x + 2*c)*sin(10*d*x + 10*c) + 20*cos(2*
d*x + 2*c)*sin(8*d*x + 8*c) + 15*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 6*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos
(14*d*x + 14*c)*sin(2*d*x + 2*c) - 6*cos(12*d*x + 12*c)*sin(2*d*x + 2*c) - 15*cos(10*d*x + 10*c)*sin(2*d*x + 2
*c) - 20*cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 15*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 6*cos(4*d*x + 4*c)*sin(2*d
*x + 2*c))*cos(13/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(14*d*x + 14*c)*cos(2*d*x + 2*c) + 6*co
s(12*d*x + 12*c)*cos(2*d*x + 2*c) + 15*cos(10*d*x + 10*c)*cos(2*d*x + 2*c) + 20*cos(8*d*x + 8*c)*cos(2*d*x + 2
*c) + 15*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 6*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(14
*d*x + 14*c)*sin(2*d*x + 2*c) + 6*sin(12*d*x + 12*c)*sin(2*d*x + 2*c) + 15*sin(10*d*x + 10*c)*sin(2*d*x + 2*c)
+ 20*sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 15*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 6*sin(4*d*x + 4*c)*sin(2*d*x
+ 2*c) + sin(2*d*x + 2*c)^2)*sin(13/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*sin(1/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c) + 1)))/(((2*(6*cos(12*d*x + 12*c) + 15*cos(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*
cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(14*d*x + 14*c) + cos(14*d*x + 14*c)^2 + 12*(15*c
os(10*d*x + 10*c) + 20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos(12*
d*x + 12*c) + 36*cos(12*d*x + 12*c)^2 + 30*(20*cos(8*d*x + 8*c) + 15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + c
os(2*d*x + 2*c))*cos(10*d*x + 10*c) + 225*cos(10*d*x + 10*c)^2 + 40*(15*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c)
+ cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + 400*cos(8*d*x + 8*c)^2 + 30*(6*cos(4*d*x + 4*c) + cos(2*d*x + 2*c))*cos
(6*d*x + 6*c) + 225*cos(6*d*x + 6*c)^2 + 36*cos(4*d*x + 4*c)^2 + 12*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2*
d*x + 2*c)^2 + 2*(6*sin(12*d*x + 12*c) + 15*sin(10*d*x + 10*c) + 20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6
*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(14*d*x + 14*c) + sin(14*d*x + 14*c)^2 + 12*(15*sin(10*d*x + 10*c) +
20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 36*sin
(12*d*x + 12*c)^2 + 30*(20*sin(8*d*x + 8*c) + 15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin
(10*d*x + 10*c) + 225*sin(10*d*x + 10*c)^2 + 40*(15*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*
sin(8*d*x + 8*c) + 400*sin(8*d*x + 8*c)^2 + 30*(6*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 225*
sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 12*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(1/
2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + ...

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Fricas [A]
time = 3.70, size = 371, normalized size = 1.67 \begin {gather*} \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 {gather*}

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)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \tan ^{6}{\left (c + d x \right )}\, dx \end {gather*}

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)

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Giac [A]
time = 2.67, size = 284, normalized size = 1.28 \begin {gather*} \frac {\sqrt {2} {\left (\frac {495 \, \sqrt {2} \sqrt {-a} a \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{{\left | a \right |}} - \frac {4 \, {\left (495 \, a^{6} - {\left (2805 \, a^{6} - {\left (6666 \, a^{6} - {\left (4158 \, a^{6} + {\left (221 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1463 \, a^{6}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{5} \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 )}{990 \, d} \end {gather*}

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]

1/990*sqrt(2)*(495*sqrt(2)*sqrt(-a)*a*log(abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^
2 + a))^2 - 4*sqrt(2)*abs(a) - 6*a)/abs(2*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)
)^2 + 4*sqrt(2)*abs(a) - 6*a))/abs(a) - 4*(495*a^6 - (2805*a^6 - (6666*a^6 - (4158*a^6 + (221*a^6*tan(1/2*d*x
+ 1/2*c)^2 - 1463*a^6)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1
/2*c)^2)*tan(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 - a)^5*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)))*sgn(cos(
d*x + c))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {tan}\left (c+d\,x\right )}^6\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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