\(\int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx\) [139]

Optimal result

Integrand size = 28, antiderivative size = 383 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {4 a^3 \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {3 a \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {6 a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {8 a^2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^4 d}-\frac {2 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {4 a^2 \sec (c+d x)}{b^5 d}+\frac {2 \left (a^2+b^2\right ) \sec (c+d x)}{b^5 d}+\frac {\sec ^3(c+d x)}{3 b^3 d}-\frac {\left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac {4 a \left (a^2+b^2\right )}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac {3 a \sec (c+d x) \tan (c+d x)}{2 b^4 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3185, 3155, 3153, 212, 3183, 3855, 3173, 3853} \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {4 a^3 \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {6 a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {8 a^2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {2 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^4 d}+\frac {4 a^2 \sec (c+d x)}{b^5 d}+\frac {2 \left (a^2+b^2\right ) \sec (c+d x)}{b^5 d}+\frac {4 a \left (a^2+b^2\right )}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac {\left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {3 a \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {3 a \tan (c+d x) \sec (c+d x)}{2 b^4 d}+\frac {\sec ^3(c+d x)}{3 b^3 d} \]

[In]

Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]

[Out]

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

Rule 212

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

Rule 3153

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 3155

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

Rule 3173

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

Rule 3183

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 3185

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

Rule 3853

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 3855

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.66 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.80 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (\frac {6 b^2 \left (a^2+b^2\right )^2 \sin (c+d x)}{a}+\frac {6 (a-i b) (a+i b) b \left (8 a^2-b^2\right ) (a \cos (c+d x)+b \sin (c+d x))}{a}+2 b \left (36 a^2+13 b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2+60 \sqrt {a^2+b^2} \left (4 a^2+b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (c+d x)+b \sin (c+d x))^2+30 a \left (4 a^2+3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2-30 a \left (4 a^2+3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2+\frac {b^2 (-9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 b^3 \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 b \left (36 a^2+13 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {2 b^3 \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {b^2 (9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {2 b \left (36 a^2+13 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{12 b^6 d (a+b \tan (c+d x))^3} \]

[In]

Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]

[Out]

(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*((6*b^2*(a^2 + b^2)^2*Sin[c + d*x])/a + (6*(a - I*b)*(a + I*
b)*b*(8*a^2 - b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x]))/a + 2*b*(36*a^2 + 13*b^2)*(a*Cos[c + d*x] + b*Sin[c + d*
x])^2 + 60*Sqrt[a^2 + b^2]*(4*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] +
b*Sin[c + d*x])^2 + 30*a*(4*a^2 + 3*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c +
d*x])^2 - 30*a*(4*a^2 + 3*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 +
(b^2*(-9*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*b^3*Sin[(c +
d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b*(36*a^2 + 13*b^2)
*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (2*b^3*Sin[(c +
d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (b^2*(9*a + b)*(a*Cos[
c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (2*b*(36*a^2 + 13*b^2)*Sin[(c + d*x)/2
]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(12*b^6*d*(a + b*Tan[c + d*x])^
3)

Maple [A] (verified)

Time = 3.25 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {-\frac {1}{3 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3 a +b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {12 a^{2}+3 a b +5 b^{2}}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{6}}+\frac {1}{3 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-3 a +b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-12 a^{2}+3 a b -5 b^{2}}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{6}}-\frac {2 \left (\frac {\frac {b^{2} \left (7 a^{4}+5 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {b \left (8 a^{6}-9 a^{4} b^{2}-15 a^{2} b^{4}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}-\frac {b^{2} \left (25 a^{4}+23 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-4 a^{4} b -\frac {7 a^{2} b^{3}}{2}+\frac {b^{5}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {5 \left (4 a^{4}+5 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6}}}{d}\)	\(444\)
default	\(\frac {-\frac {1}{3 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3 a +b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {12 a^{2}+3 a b +5 b^{2}}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{6}}+\frac {1}{3 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-3 a +b}{2 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-12 a^{2}+3 a b -5 b^{2}}{2 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {5 a \left (4 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{6}}-\frac {2 \left (\frac {\frac {b^{2} \left (7 a^{4}+5 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {b \left (8 a^{6}-9 a^{4} b^{2}-15 a^{2} b^{4}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}-\frac {b^{2} \left (25 a^{4}+23 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-4 a^{4} b -\frac {7 a^{2} b^{3}}{2}+\frac {b^{5}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {5 \left (4 a^{4}+5 a^{2} b^{2}+b^{4}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6}}}{d}\)	\(444\)
risch	\(\frac {60 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-180 i a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}-60 i a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+180 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+240 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+60 a^{4} {\mathrm e}^{i \left (d x +c \right )}-20 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-15 b^{4} {\mathrm e}^{i \left (d x +c \right )}+60 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}-15 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}+240 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}-20 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}+360 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}+22 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+15 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+140 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+250 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+90 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+100 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-90 i a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-100 i a \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+140 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+15 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )} a +i b +a \right )^{2} b^{5} d}+\frac {10 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right ) a^{2}}{d \,b^{6}}+\frac {5 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 d \,b^{4}}-\frac {10 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right ) a^{2}}{d \,b^{6}}-\frac {5 \sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a -b}{\sqrt {a^{2}+b^{2}}}\right )}{2 d \,b^{4}}-\frac {10 a^{3} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{b^{6} d}-\frac {15 a \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 b^{4} d}+\frac {10 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{b^{6} d}+\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 b^{4} d}\)	\(693\)

[In]

int(sec(d*x+c)^4/(cos(d*x+c)*a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/3/b^3/(tan(1/2*d*x+1/2*c)-1)^3-1/2*(3*a+b)/b^4/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(12*a^2+3*a*b+5*b^2)/b^5/(
tan(1/2*d*x+1/2*c)-1)+5/2*a*(4*a^2+3*b^2)/b^6*ln(tan(1/2*d*x+1/2*c)-1)+1/3/b^3/(tan(1/2*d*x+1/2*c)+1)^3-1/2*(-
3*a+b)/b^4/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(-12*a^2+3*a*b-5*b^2)/b^5/(tan(1/2*d*x+1/2*c)+1)-5/2*a*(4*a^2+3*b^2)/b
^6*ln(tan(1/2*d*x+1/2*c)+1)-2/b^6*((1/2*b^2*(7*a^4+5*a^2*b^2-2*b^4)/a*tan(1/2*d*x+1/2*c)^3+1/2*b*(8*a^6-9*a^4*
b^2-15*a^2*b^4+2*b^6)/a^2*tan(1/2*d*x+1/2*c)^2-1/2*b^2*(25*a^4+23*a^2*b^2-2*b^4)/a*tan(1/2*d*x+1/2*c)-4*a^4*b-
7/2*a^2*b^3+1/2*b^5)/(tan(1/2*d*x+1/2*c)^2*a-2*b*tan(1/2*d*x+1/2*c)-a)^2-5/2*(4*a^4+5*a^2*b^2+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))))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.47 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {4 \, b^{5} + 30 \, {\left (4 \, a^{4} b + a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{4} + 20 \, {\left (2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (4 \, a^{4} - 3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a^{2} + b^{2}} \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 ({\left (4 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left ({\left (4 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \, {\left (a b^{4} \cos \left (d x + c\right ) - 6 \, {\left (3 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (2 \, a b^{7} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + b^{8} d \cos \left (d x + c\right )^{3} + {\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{5}\right )}} \]

[In]

integrate(sec(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/12*(4*b^5 + 30*(4*a^4*b + a^2*b^3 - b^5)*cos(d*x + c)^4 + 20*(2*a^2*b^3 + b^5)*cos(d*x + c)^2 + 15*((4*a^4 -
3*a^2*b^2 - b^4)*cos(d*x + c)^5 + 2*(4*a^3*b + a*b^3)*cos(d*x + c)^4*sin(d*x + c) + (4*a^2*b^2 + b^4)*cos(d*x
+ c)^3)*sqrt(a^2 + b^2)*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*((4*a^5 - a^3*b^2 - 3*a*b^4)*cos(d*x + c)^5 + 2*(4*a^4*b + 3*a^2*b^3)*cos(d*x + c)^4*sin(d*x +
c) + (4*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) + 15*((4*a^5 - a^3*b^2 - 3*a*b^4)*cos(d*x +
c)^5 + 2*(4*a^4*b + 3*a^2*b^3)*cos(d*x + c)^4*sin(d*x + c) + (4*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^3)*log(-sin(d*
x + c) + 1) - 10*(a*b^4*cos(d*x + c) - 6*(3*a^3*b^2 + 2*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(2*a*b^7*d*cos(d*
x + c)^4*sin(d*x + c) + b^8*d*cos(d*x + c)^3 + (a^2*b^6 - b^8)*d*cos(d*x + c)^5)

Sympy [F]

\[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

integrate(sec(d*x+c)**4/(a*cos(d*x+c)+b*sin(d*x+c))**3,x)

[Out]

Integral(sec(c + d*x)**4/(a*cos(c + d*x) + b*sin(c + d*x))**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (361) = 722\).

Time = 0.33 (sec) , antiderivative size = 902, normalized size of antiderivative = 2.36 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/6*(2*(60*a^6 + 35*a^4*b^2 - 3*a^2*b^4 + (210*a^5*b + 125*a^3*b^3 - 6*a*b^5)*sin(d*x + c)/(cos(d*x + c) + 1)
- 2*(120*a^6 - 10*a^4*b^2 - 55*a^2*b^4 + 3*b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*(330*a^5*b + 205*a^3*b
^3 - 12*a*b^5)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2*(180*a^6 - 95*a^4*b^2 - 120*a^2*b^4 + 9*b^6)*sin(d*x +
c)^4/(cos(d*x + c) + 1)^4 + 12*(60*a^5*b + 35*a^3*b^3 - 3*a*b^5)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 6*(40*a
^6 - 30*a^4*b^2 - 35*a^2*b^4 + 3*b^6)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 6*(50*a^5*b + 25*a^3*b^3 - 4*a*b^5
)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 3*(20*a^6 - 15*a^4*b^2 - 15*a^2*b^4 + 2*b^6)*sin(d*x + c)^8/(cos(d*x +
c) + 1)^8 + 3*(10*a^5*b + 5*a^3*b^3 - 2*a*b^5)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/(a^4*b^5 + 4*a^3*b^6*sin(
d*x + c)/(cos(d*x + c) + 1) - 16*a^3*b^6*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 24*a^3*b^6*sin(d*x + c)^5/(cos(
d*x + c) + 1)^5 - 16*a^3*b^6*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 4*a^3*b^6*sin(d*x + c)^9/(cos(d*x + c) + 1)
^9 - a^4*b^5*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - (5*a^4*b^5 - 4*a^2*b^7)*sin(d*x + c)^2/(cos(d*x + c) + 1)
^2 + 2*(5*a^4*b^5 - 6*a^2*b^7)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 2*(5*a^4*b^5 - 6*a^2*b^7)*sin(d*x + c)^6/
(cos(d*x + c) + 1)^6 + (5*a^4*b^5 - 4*a^2*b^7)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) - 15*(4*a^3 + 3*a*b^2)*log
(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/b^6 + 15*(4*a^3 + 3*a*b^2)*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/b^6
- 15*(4*a^4 + 5*a^2*b^2 + b^4)*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) - sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^6))/d

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.33 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {\frac {15 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{6}} - \frac {15 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{6}} + \frac {15 \, {\left (4 \, a^{4} + 5 \, a^{2} b^{2} + b^{4}\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 (9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 72 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} + 14 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} b^{5}} + \frac {6 \, {\left (7 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 25 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 23 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{6} - 7 \, a^{4} b^{2} + a^{2} b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}^{2} a^{2} b^{5}}}{6 \, d} \]

[In]

integrate(sec(d*x+c)^4/(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/6*(15*(4*a^3 + 3*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^6 - 15*(4*a^3 + 3*a*b^2)*log(abs(tan(1/2*d*x +
1/2*c) - 1))/b^6 + 15*(4*a^4 + 5*a^2*b^2 + b^4)*log(abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2))/a
bs(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*(9*a*b*tan(1/2*d*x + 1/2*c)^
5 + 36*a^2*tan(1/2*d*x + 1/2*c)^4 + 18*b^2*tan(1/2*d*x + 1/2*c)^4 - 72*a^2*tan(1/2*d*x + 1/2*c)^2 - 24*b^2*tan
(1/2*d*x + 1/2*c)^2 - 9*a*b*tan(1/2*d*x + 1/2*c) + 36*a^2 + 14*b^2)/((tan(1/2*d*x + 1/2*c)^2 - 1)^3*b^5) + 6*(
7*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 5*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 2*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 8*a^6*tan
(1/2*d*x + 1/2*c)^2 - 9*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 + 2*b^6*tan(1/2*d*x
+ 1/2*c)^2 - 25*a^5*b*tan(1/2*d*x + 1/2*c) - 23*a^3*b^3*tan(1/2*d*x + 1/2*c) + 2*a*b^5*tan(1/2*d*x + 1/2*c) -
8*a^6 - 7*a^4*b^2 + a^2*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c) - a)^2*a^2*b^5))/d

Mupad [B] (verification not implemented)

Time = 26.43 (sec) , antiderivative size = 1203, normalized size of antiderivative = 3.14 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^4*(a*cos(c + d*x) + b*sin(c + d*x))^3),x)

[Out]

((60*a^4 - 3*b^4 + 35*a^2*b^2)/(3*b^5) + (tan(c/2 + (d*x)/2)*(210*a^4 - 6*b^4 + 125*a^2*b^2))/(3*a*b^4) + (tan
(c/2 + (d*x)/2)^8*(20*a^6 + 2*b^6 - 15*a^2*b^4 - 15*a^4*b^2))/(a^2*b^5) - (2*tan(c/2 + (d*x)/2)^6*(40*a^6 + 3*
b^6 - 35*a^2*b^4 - 30*a^4*b^2))/(a^2*b^5) - (2*tan(c/2 + (d*x)/2)^2*(120*a^6 + 3*b^6 - 55*a^2*b^4 - 10*a^4*b^2
))/(3*a^2*b^5) + (2*tan(c/2 + (d*x)/2)^4*(180*a^6 + 9*b^6 - 120*a^2*b^4 - 95*a^4*b^2))/(3*a^2*b^5) + (tan(c/2
+ (d*x)/2)^9*(10*a^4 - 2*b^4 + 5*a^2*b^2))/(a*b^4) - (2*tan(c/2 + (d*x)/2)^7*(50*a^4 - 4*b^4 + 25*a^2*b^2))/(a
*b^4) + (4*tan(c/2 + (d*x)/2)^5*(60*a^4 - 3*b^4 + 35*a^2*b^2))/(a*b^4) - (2*tan(c/2 + (d*x)/2)^3*(330*a^4 - 12
*b^4 + 205*a^2*b^2))/(3*a*b^4))/(d*(tan(c/2 + (d*x)/2)^8*(5*a^2 - 4*b^2) - tan(c/2 + (d*x)/2)^2*(5*a^2 - 4*b^2
) + tan(c/2 + (d*x)/2)^4*(10*a^2 - 12*b^2) - tan(c/2 + (d*x)/2)^6*(10*a^2 - 12*b^2) - a^2*tan(c/2 + (d*x)/2)^1
0 + a^2 - 16*a*b*tan(c/2 + (d*x)/2)^3 + 24*a*b*tan(c/2 + (d*x)/2)^5 - 16*a*b*tan(c/2 + (d*x)/2)^7 + 4*a*b*tan(
c/2 + (d*x)/2)^9 + 4*a*b*tan(c/2 + (d*x)/2))) - (atanh((3000*a^2*tan(c/2 + (d*x)/2))/(3000*a^2 + (7000*a^4)/b^
2 + (4000*a^6)/b^4) + (7000*a^4*tan(c/2 + (d*x)/2))/(7000*a^4 + 3000*a^2*b^2 + (4000*a^6)/b^2) + (4000*a^6*tan
(c/2 + (d*x)/2))/(4000*a^6 + 3000*a^2*b^4 + 7000*a^4*b^2))*(15*a*b^2 + 20*a^3))/(b^6*d) + (5*atanh((1000*a^2*(
a^2 + b^2)^(1/2))/(1000*a^2*b + (5000*a^4)/b + (4000*a^6)/b^3 + 10000*a^3*tan(c/2 + (d*x)/2) + 2000*a*b^2*tan(
c/2 + (d*x)/2) + (8000*a^5*tan(c/2 + (d*x)/2))/b^2) + (4000*a^4*(a^2 + b^2)^(1/2))/(5000*a^4*b + 1000*a^2*b^3
+ (4000*a^6)/b + 8000*a^5*tan(c/2 + (d*x)/2) + 2000*a*b^4*tan(c/2 + (d*x)/2) + 10000*a^3*b^2*tan(c/2 + (d*x)/2
)) + (9000*a^3*tan(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2))/(5000*a^4 + 1000*a^2*b^2 + (4000*a^6)/b^2 + 2000*a*b^3*ta
n(c/2 + (d*x)/2) + 10000*a^3*b*tan(c/2 + (d*x)/2) + (8000*a^5*tan(c/2 + (d*x)/2))/b) + (4000*a^5*tan(c/2 + (d*
x)/2)*(a^2 + b^2)^(1/2))/(4000*a^6 + 1000*a^2*b^4 + 5000*a^4*b^2 + 2000*a*b^5*tan(c/2 + (d*x)/2) + 8000*a^5*b*
tan(c/2 + (d*x)/2) + 10000*a^3*b^3*tan(c/2 + (d*x)/2)) + (2000*a*tan(c/2 + (d*x)/2)*(a^2 + b^2)^(1/2))/(1000*a
^2 + (5000*a^4)/b^2 + (4000*a^6)/b^4 + (10000*a^3*tan(c/2 + (d*x)/2))/b + (8000*a^5*tan(c/2 + (d*x)/2))/b^3 +
2000*a*b*tan(c/2 + (d*x)/2)))*(4*a^2 + b^2)*(a^2 + b^2)^(1/2))/(b^6*d)