3.121 \(\int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=262 \[ -\frac {a \left (a^2+b^2\right )^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac {\left (a^2+b^2\right )^{5/2} \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {\left (a^2+b^2\right )^2 \sec (c+d x)}{b^5 d}-\frac {a \left (a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac {a \left (a^2+b^2\right ) \tan (c+d x) \sec (c+d x)}{2 b^4 d}+\frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}-\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 b^2 d}-\frac {a \tan (c+d x) \sec ^3(c+d x)}{4 b^2 d}-\frac {3 a \tan (c+d x) \sec (c+d x)}{8 b^2 d}+\frac {\sec ^5(c+d x)}{5 b d} \]

[Out]

-3/8*a*arctanh(sin(d*x+c))/b^2/d-1/2*a*(a^2+b^2)*arctanh(sin(d*x+c))/b^4/d-a*(a^2+b^2)^2*arctanh(sin(d*x+c))/b
^6/d-(a^2+b^2)^(5/2)*arctanh((b*cos(d*x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/b^6/d+(a^2+b^2)^2*sec(d*x+c)/b^5/d+1
/3*(a^2+b^2)*sec(d*x+c)^3/b^3/d+1/5*sec(d*x+c)^5/b/d-3/8*a*sec(d*x+c)*tan(d*x+c)/b^2/d-1/2*a*(a^2+b^2)*sec(d*x
+c)*tan(d*x+c)/b^4/d-1/4*a*sec(d*x+c)^3*tan(d*x+c)/b^2/d

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Rubi [A]  time = 0.26, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3104, 3768, 3770, 3074, 206} \[ \frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}+\frac {\left (a^2+b^2\right )^2 \sec (c+d x)}{b^5 d}-\frac {a \left (a^2+b^2\right )^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac {a \left (a^2+b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac {a \left (a^2+b^2\right ) \tan (c+d x) \sec (c+d x)}{2 b^4 d}-\frac {\left (a^2+b^2\right )^{5/2} \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 b^2 d}-\frac {a \tan (c+d x) \sec ^3(c+d x)}{4 b^2 d}-\frac {3 a \tan (c+d x) \sec (c+d x)}{8 b^2 d}+\frac {\sec ^5(c+d x)}{5 b d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]

[Out]

(-3*a*ArcTanh[Sin[c + d*x]])/(8*b^2*d) - (a*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (a*(a^2 + b^2)^2*Ar
cTanh[Sin[c + d*x]])/(b^6*d) - ((a^2 + b^2)^(5/2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/
(b^6*d) + ((a^2 + b^2)^2*Sec[c + d*x])/(b^5*d) + ((a^2 + b^2)*Sec[c + d*x]^3)/(3*b^3*d) + Sec[c + d*x]^5/(5*b*
d) - (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*b^2*d) - (a*(a^2 + b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*b^4*d) - (a*Sec[
c + d*x]^3*Tan[c + d*x])/(4*b^2*d)

Rule 206

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

Rule 3074

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Dist[d^(-1), Subst[Int
[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2,
0]

Rule 3104

Int[cos[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[Cos[c + d*x]^(m + 1)/(b*d*(m + 1)), x] + (-Dist[a/b^2, Int[Cos[c + d*x]^(m + 1), x], x] + Dist[(a^2 + b
^2)/b^2, Int[Cos[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[
a^2 + b^2, 0] && LtQ[m, -1]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

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Mathematica [B]  time = 5.03, size = 661, normalized size = 2.52 \[ \frac {\sec (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (240 a^4 b+520 a^2 b^3+480 \left (a^2+b^2\right )^{5/2} \tanh ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )-b}{\sqrt {a^2+b^2}}\right )+\frac {2 b^3 \left (20 a^2+29 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {2 b^3 \left (20 a^2+29 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 b \left (120 a^4+260 a^2 b^2+149 b^4\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {2 b \left (120 a^4+260 a^2 b^2+149 b^4\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}+30 a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-30 a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {b^2 \left (-60 a^3+20 a^2 b-105 a b^2+29 b^3\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {b^2 \left (60 a^3+20 a^2 b+105 a b^2+29 b^3\right )}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {3 b^4 (2 b-5 a)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {3 b^4 (5 a+2 b)}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {12 b^5 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}-\frac {12 b^5 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5}+298 b^5\right )}{240 b^6 d (a+b \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]

[Out]

(Sec[c + d*x]*(240*a^4*b + 520*a^2*b^3 + 298*b^5 + 480*(a^2 + b^2)^(5/2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqr
t[a^2 + b^2]] + 30*a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 30*a*(8*a^4 + 20
*a^2*b^2 + 15*b^4)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*b^4*(-5*a + 2*b))/(Cos[(c + d*x)/2] - Sin[(c
+ d*x)/2])^4 + (b^2*(-60*a^3 + 20*a^2*b - 105*a*b^2 + 29*b^3))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*b
^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (2*b^3*(20*a^2 + 29*b^2)*Sin[(c + d*x)/2])/(Cos
[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b*(120*a^4 + 260*a^2*b^2 + 149*b^4)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/
2] - Sin[(c + d*x)/2]) - (12*b^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + (3*b^4*(5*a + 2*b
))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - (2*b^3*(20*a^2 + 29*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Si
n[(c + d*x)/2])^3 + (b^2*(60*a^3 + 20*a^2*b + 105*a*b^2 + 29*b^3))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (
2*b*(120*a^4 + 260*a^2*b^2 + 149*b^4)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))*(a*Cos[c + d*x]
+ b*Sin[c + d*x]))/(240*b^6*d*(a + b*Tan[c + d*x]))

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fricas [A]  time = 0.90, size = 346, normalized size = 1.32 \[ \frac {120 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}} \cos \left (d x + c\right )^{5} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \, b^{5} + 240 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 80 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} - 30 \, {\left (2 \, a b^{4} \cos \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{240 \, b^{6} d \cos \left (d x + c\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/240*(120*(a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)*cos(d*x + c)^5*log(-(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2
- b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/(2*a*b*cos(d*x + c
)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) - 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5*log(si
n(d*x + c) + 1) + 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 48*b^5 + 240*(a^4
*b + 2*a^2*b^3 + b^5)*cos(d*x + c)^4 + 80*(a^2*b^3 + b^5)*cos(d*x + c)^2 - 30*(2*a*b^4*cos(d*x + c) + (4*a^3*b
^2 + 7*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(b^6*d*cos(d*x + c)^5)

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giac [B]  time = 0.36, size = 554, normalized size = 2.11 \[ -\frac {\frac {15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{6}} - \frac {15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{6}} + \frac {120 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} + \frac {2 \, {\left (60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 360 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 480 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1200 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 720 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 720 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1120 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1040 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, a^{4} + 280 \, a^{2} b^{2} + 184 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} b^{5}}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/120*(15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^6 - 15*(8*a^5 + 20*a^3*b^2 + 1
5*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^6 + 120*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(abs(2*a*tan(1/2*
d*x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b
^2)*b^6) + 2*(60*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 135*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*a^4*tan(1/2*d*x + 1/2*c
)^8 + 360*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 + 360*b^4*tan(1/2*d*x + 1/2*c)^8 - 120*a^3*b*tan(1/2*d*x + 1/2*c)^7 -
150*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 480*a^4*tan(1/2*d*x + 1/2*c)^6 - 1200*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 - 720
*b^4*tan(1/2*d*x + 1/2*c)^6 + 720*a^4*tan(1/2*d*x + 1/2*c)^4 + 1600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 1120*b^4*
tan(1/2*d*x + 1/2*c)^4 + 120*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 150*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 480*a^4*tan(1/2
*d*x + 1/2*c)^2 - 1040*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 560*b^4*tan(1/2*d*x + 1/2*c)^2 - 60*a^3*b*tan(1/2*d*x
+ 1/2*c) - 135*a*b^3*tan(1/2*d*x + 1/2*c) + 120*a^4 + 280*a^2*b^2 + 184*b^4)/((tan(1/2*d*x + 1/2*c)^2 - 1)^5*b
^5))/d

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maple [B]  time = 0.26, size = 994, normalized size = 3.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x)

[Out]

2/d/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2))-15/8/d/b/(tan(1/2*d*x+1/2*c)-1)+
15/8/d/b/(tan(1/2*d*x+1/2*c)+1)+6/d/b^4/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/
2))*a^4+6/d/b^2/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2))*a^2+2/d/b^6/(a^2+b^2
)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2))*a^6-13/12/d/b/(tan(1/2*d*x+1/2*c)-1)^3-9/8/d
/b/(tan(1/2*d*x+1/2*c)-1)^2+13/12/d/b/(tan(1/2*d*x+1/2*c)+1)^3-9/8/d/b/(tan(1/2*d*x+1/2*c)+1)^2-1/4/d/b^2/(tan
(1/2*d*x+1/2*c)-1)^4*a-1/3/d/b^3/(tan(1/2*d*x+1/2*c)-1)^3*a^2-1/2/d/b^2/(tan(1/2*d*x+1/2*c)-1)^3*a-1/2/d/b^4/(
tan(1/2*d*x+1/2*c)-1)^2*a^3-1/2/d/b^3/(tan(1/2*d*x+1/2*c)-1)^2*a^2-1/d/b^5/(tan(1/2*d*x+1/2*c)-1)*a^4-1/2/d/b^
4/(tan(1/2*d*x+1/2*c)-1)*a^3+1/d*a^5/b^6*ln(tan(1/2*d*x+1/2*c)-1)+1/4/d/b^2/(tan(1/2*d*x+1/2*c)+1)^4*a+1/3/d/b
^3/(tan(1/2*d*x+1/2*c)+1)^3*a^2-1/2/d/b^2/(tan(1/2*d*x+1/2*c)+1)^3*a+1/2/d/b^4/(tan(1/2*d*x+1/2*c)+1)^2*a^3-1/
2/d/b^3/(tan(1/2*d*x+1/2*c)+1)^2*a^2+1/d/b^5/(tan(1/2*d*x+1/2*c)+1)*a^4-1/2/d/b^4/(tan(1/2*d*x+1/2*c)+1)*a^3-5
/2/d/b^3/(tan(1/2*d*x+1/2*c)-1)*a^2-9/8/d/b^2/(tan(1/2*d*x+1/2*c)-1)*a+5/2/d*a^3/b^4*ln(tan(1/2*d*x+1/2*c)-1)+
11/8/d/b^2/(tan(1/2*d*x+1/2*c)+1)^2*a+5/2/d/b^3/(tan(1/2*d*x+1/2*c)+1)*a^2-9/8/d/b^2/(tan(1/2*d*x+1/2*c)+1)*a-
5/2/d*a^3/b^4*ln(tan(1/2*d*x+1/2*c)+1)-11/8/d/b^2/(tan(1/2*d*x+1/2*c)-1)^2*a-1/d*a^5/b^6*ln(tan(1/2*d*x+1/2*c)
+1)-1/5/d/b/(tan(1/2*d*x+1/2*c)-1)^5-1/2/d/b/(tan(1/2*d*x+1/2*c)-1)^4+1/5/d/b/(tan(1/2*d*x+1/2*c)+1)^5-1/2/d/b
/(tan(1/2*d*x+1/2*c)+1)^4+15/8/d*a/b^2*ln(tan(1/2*d*x+1/2*c)-1)-15/8/d*a/b^2*ln(tan(1/2*d*x+1/2*c)+1)

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maxima [B]  time = 0.44, size = 625, normalized size = 2.39 \[ \frac {\frac {2 \, {\left (120 \, a^{4} + 280 \, a^{2} b^{2} + 184 \, b^{4} - \frac {15 \, {\left (4 \, a^{3} b + 9 \, a b^{3}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {80 \, {\left (6 \, a^{4} + 13 \, a^{2} b^{2} + 7 \, b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {30 \, {\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {80 \, {\left (9 \, a^{4} + 20 \, a^{2} b^{2} + 14 \, b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {240 \, {\left (2 \, a^{4} + 5 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {30 \, {\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {120 \, {\left (a^{4} + 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {15 \, {\left (4 \, a^{3} b + 9 \, a b^{3}\right )} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )}}{b^{5} - \frac {5 \, b^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, b^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {10 \, b^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, b^{5} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {b^{5} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{6}} + \frac {15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{6}} - \frac {120 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/120*(2*(120*a^4 + 280*a^2*b^2 + 184*b^4 - 15*(4*a^3*b + 9*a*b^3)*sin(d*x + c)/(cos(d*x + c) + 1) - 80*(6*a^4
+ 13*a^2*b^2 + 7*b^4)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 30*(4*a^3*b + 5*a*b^3)*sin(d*x + c)^3/(cos(d*x +
c) + 1)^3 + 80*(9*a^4 + 20*a^2*b^2 + 14*b^4)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 240*(2*a^4 + 5*a^2*b^2 + 3*
b^4)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 30*(4*a^3*b + 5*a*b^3)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 120*(a
^4 + 3*a^2*b^2 + 3*b^4)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 15*(4*a^3*b + 9*a*b^3)*sin(d*x + c)^9/(cos(d*x +
c) + 1)^9)/(b^5 - 5*b^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*b^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 10
*b^5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*b^5*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - b^5*sin(d*x + c)^10/(co
s(d*x + c) + 1)^10) - 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/b^6 + 15*(8*
a^5 + 20*a^3*b^2 + 15*a*b^4)*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/b^6 - 120*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*log((b - a*sin(d*x + c)/(cos(d*x + c) + 1) + sqrt(a^2 + b^2))/(b - a*sin(d*x + c)/(cos(d*x + c) + 1) - s
qrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^6))/d

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mupad [B]  time = 2.84, size = 2979, normalized size = 11.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^6*(a*cos(c + d*x) + b*sin(c + d*x))),x)

[Out]

(atan(((((a^2 + b^2)^5)^(1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^
14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b
^7 + 128*a^13*b^5))/(2*b^15) - (((a^2 + b^2)^5)^(1/2)*((28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c
/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) + (((a^2 + b^2)^5)^(1/2)*(32
*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15)))/b^6))/b^6)*1i)/b^6 + (((a^2 + b^2)^5)^(
1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*
(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^
15) - (((a^2 + b^2)^5)^(1/2)*((tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(
2*b^15) - (28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (((a^2 + b^2)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*
x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15)))/b^6))/b^6)*1i)/b^6)/((32*a^16 + 120*a^2*b^14 + 655*a^4*b^12 + 15
49*a^6*b^10 + 2069*a^8*b^8 + 1695*a^10*b^6 + 856*a^12*b^4 + 248*a^14*b^2)/b^14 + (((a^2 + b^2)^5)^(1/2)*(((225
*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17
+ 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((a^
2 + b^2)^5)^(1/2)*((28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*
b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) + (((a^2 + b^2)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*
a*b^19 + 128*a^3*b^17))/(2*b^15)))/b^6))/b^6))/b^6 - (((a^2 + b^2)^5)^(1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11
+ 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^
13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((a^2 + b^2)^5)^(1/2)*((tan(c/2
+ (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16 + 44*a^4*b^14 +
16*a^6*b^12)/b^14 + (((a^2 + b^2)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b
^15)))/b^6))/b^6))/b^6 - (tan(c/2 + (d*x)/2)*(128*a^17 + 450*a^3*b^14 + 2550*a^5*b^12 + 6230*a^7*b^10 + 8530*a
^9*b^8 + 7088*a^11*b^6 + 3584*a^13*b^4 + 1024*a^15*b^2))/b^15))*((a^2 + b^2)^5)^(1/2)*2i)/(b^6*d) - ((2*(15*a^
4 + 23*b^4 + 35*a^2*b^2))/(15*b^5) + (tan(c/2 + (d*x)/2)^3*(5*a*b^2 + 4*a^3))/(2*b^4) - (tan(c/2 + (d*x)/2)^7*
(5*a*b^2 + 4*a^3))/(2*b^4) + (tan(c/2 + (d*x)/2)^9*(9*a*b^2 + 4*a^3))/(4*b^4) + (2*tan(c/2 + (d*x)/2)^8*(a^4 +
3*b^4 + 3*a^2*b^2))/b^5 - (4*tan(c/2 + (d*x)/2)^6*(2*a^4 + 3*b^4 + 5*a^2*b^2))/b^5 - (4*tan(c/2 + (d*x)/2)^2*
(6*a^4 + 7*b^4 + 13*a^2*b^2))/(3*b^5) + (4*tan(c/2 + (d*x)/2)^4*(9*a^4 + 14*b^4 + 20*a^2*b^2))/(3*b^5) - (tan(
c/2 + (d*x)/2)*(9*a*b^2 + 4*a^3))/(4*b^4))/(d*(5*tan(c/2 + (d*x)/2)^2 - 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 +
(d*x)/2)^6 - 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 - 1)) + (atan(((((15*a*b^4)/8 + a^5 + (5*a^3*b^2)
/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(
64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^1
5) - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*((28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/
2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(1
92*a*b^19 + 128*a^3*b^17))/(2*b^15))*((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2))/b^6))/b^6)*1i)/b^6 + (((15*a*b^4)/8
+ a^5 + (5*a^3*b^2)/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (
tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 1
28*a^13*b^5))/(2*b^15) - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*((tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16
+ 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 + ((32*a^2*b^3 + (t
an(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15))*((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2))/b^6))/b^6)*1i)/
b^6)/((32*a^16 + 120*a^2*b^14 + 655*a^4*b^12 + 1549*a^6*b^10 + 2069*a^8*b^8 + 1695*a^10*b^6 + 856*a^12*b^4 + 2
48*a^14*b^2)/b^14 + (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 16
0*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11
+ 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*((28*a^2*b^16
+ 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b
^12))/(2*b^15) + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15))*((15*a*b^4)/8 + a^5
+ (5*a^3*b^2)/2))/b^6))/b^6))/b^6 - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 +
320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^1
3 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)
/2)*((tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16 +
44*a^4*b^14 + 16*a^6*b^12)/b^14 + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15))*((
15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2))/b^6))/b^6))/b^6 - (tan(c/2 + (d*x)/2)*(128*a^17 + 450*a^3*b^14 + 2550*a^5*
b^12 + 6230*a^7*b^10 + 8530*a^9*b^8 + 7088*a^11*b^6 + 3584*a^13*b^4 + 1024*a^15*b^2))/b^15))*((15*a*b^4)/8 + a
^5 + (5*a^3*b^2)/2)*2i)/(b^6*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{6}{\left (c + d x \right )}}{a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**6/(a*cos(d*x+c)+b*sin(d*x+c)),x)

[Out]

Integral(sec(c + d*x)**6/(a*cos(c + d*x) + b*sin(c + d*x)), x)

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