3.2.91 \(\int \frac {a+b \sec (e+f x)}{(c+d \sec (e+f x))^3} \, dx\) [191]

Optimal. Leaf size=204 \[ \frac {a x}{c^3}+\frac {\left (b c^3 \left (2 c^2+d^2\right )-a d \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^3 (c-d)^{5/2} (c+d)^{5/2} f}-\frac {d (b c-a d) \tan (e+f x)}{2 c \left (c^2-d^2\right ) f (c+d \sec (e+f x))^2}-\frac {d \left (3 b c^3-5 a c^2 d+2 a d^3\right ) \tan (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (c+d \sec (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.37, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4008, 4145, 4004, 3916, 2738, 214} \begin {gather*} -\frac {d (b c-a d) \tan (e+f x)}{2 c f \left (c^2-d^2\right ) (c+d \sec (e+f x))^2}-\frac {d \left (-5 a c^2 d+2 a d^3+3 b c^3\right ) \tan (e+f x)}{2 c^2 f \left (c^2-d^2\right )^2 (c+d \sec (e+f x))}+\frac {\left (b c^3 \left (2 c^2+d^2\right )-a d \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac {a x}{c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^3,x]

[Out]

(a*x)/c^3 + ((b*c^3*(2*c^2 + d^2) - a*d*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sq
rt[c + d]])/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) - (d*(b*c - a*d)*Tan[e + f*x])/(2*c*(c^2 - d^2)*f*(c + d*Sec[e
+ f*x])^2) - (d*(3*b*c^3 - 5*a*c^2*d + 2*a*d^3)*Tan[e + f*x])/(2*c^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))

Rule 214

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

Rule 2738

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

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4008

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[b*(b
*c - a*d)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 -
b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))*Csc[e + f*x] + b
*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m,
-1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rule 4145

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)
*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 1.32, size = 267, normalized size = 1.31 \begin {gather*} \frac {(d+c \cos (e+f x)) \sec ^2(e+f x) (a+b \sec (e+f x)) \left (2 a (e+f x) (d+c \cos (e+f x))^2-\frac {2 \left (b c^3 \left (2 c^2+d^2\right )+a d \left (-6 c^4+5 c^2 d^2-2 d^4\right )\right ) \tanh ^{-1}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) (d+c \cos (e+f x))^2}{\left (c^2-d^2\right )^{5/2}}+\frac {c d^2 (b c-a d) \sin (e+f x)}{(c-d) (c+d)}-\frac {c d \left (4 b c^3-6 a c^2 d-b c d^2+3 a d^3\right ) (d+c \cos (e+f x)) \sin (e+f x)}{(c-d)^2 (c+d)^2}\right )}{2 c^3 f (b+a \cos (e+f x)) (c+d \sec (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^3,x]

[Out]

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

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Maple [A]
time = 0.34, size = 287, normalized size = 1.41 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(f*x+e))/(c+d*sec(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(2*a/c^3*arctan(tan(1/2*f*x+1/2*e))+2/c^3*((-1/2*(6*a*c^2*d+a*c*d^2-2*a*d^3-4*b*c^3-b*c^2*d)*c*d/(c-d)/(c^
2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3+1/2*d*c*(6*a*c^2*d-a*c*d^2-2*a*d^3-4*b*c^3+b*c^2*d)/(c+d)/(c-d)^2*tan(1/2*f*
x+1/2*e))/(c*tan(1/2*f*x+1/2*e)^2-d*tan(1/2*f*x+1/2*e)^2-c-d)^2-1/2*(6*a*c^4*d-5*a*c^2*d^3+2*a*d^5-2*b*c^5-b*c
^3*d^2)/(c^4-2*c^2*d^2+d^4)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1/2*f*x+1/2*e)/((c+d)*(c-d))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))/(c+d*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*c^2-4*d^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (196) = 392\).
time = 3.04, size = 1176, normalized size = 5.76 \begin {gather*} \left [\frac {4 \, {\left (a c^{8} - 3 \, a c^{6} d^{2} + 3 \, a c^{4} d^{4} - a c^{2} d^{6}\right )} f x \cos \left (f x + e\right )^{2} + 8 \, {\left (a c^{7} d - 3 \, a c^{5} d^{3} + 3 \, a c^{3} d^{5} - a c d^{7}\right )} f x \cos \left (f x + e\right ) + 4 \, {\left (a c^{6} d^{2} - 3 \, a c^{4} d^{4} + 3 \, a c^{2} d^{6} - a d^{8}\right )} f x - {\left (2 \, b c^{5} d^{2} - 6 \, a c^{4} d^{3} + b c^{3} d^{4} + 5 \, a c^{2} d^{5} - 2 \, a d^{7} + {\left (2 \, b c^{7} - 6 \, a c^{6} d + b c^{5} d^{2} + 5 \, a c^{4} d^{3} - 2 \, a c^{2} d^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, b c^{6} d - 6 \, a c^{5} d^{2} + b c^{4} d^{3} + 5 \, a c^{3} d^{4} - 2 \, a c d^{6}\right )} \cos \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) - 2 \, {\left (3 \, b c^{6} d^{2} - 5 \, a c^{5} d^{3} - 3 \, b c^{4} d^{4} + 7 \, a c^{3} d^{5} - 2 \, a c d^{7} + {\left (4 \, b c^{7} d - 6 \, a c^{6} d^{2} - 5 \, b c^{5} d^{3} + 9 \, a c^{4} d^{4} + b c^{3} d^{5} - 3 \, a c^{2} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (c^{11} - 3 \, c^{9} d^{2} + 3 \, c^{7} d^{4} - c^{5} d^{6}\right )} f \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{10} d - 3 \, c^{8} d^{3} + 3 \, c^{6} d^{5} - c^{4} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{9} d^{2} - 3 \, c^{7} d^{4} + 3 \, c^{5} d^{6} - c^{3} d^{8}\right )} f\right )}}, \frac {2 \, {\left (a c^{8} - 3 \, a c^{6} d^{2} + 3 \, a c^{4} d^{4} - a c^{2} d^{6}\right )} f x \cos \left (f x + e\right )^{2} + 4 \, {\left (a c^{7} d - 3 \, a c^{5} d^{3} + 3 \, a c^{3} d^{5} - a c d^{7}\right )} f x \cos \left (f x + e\right ) + 2 \, {\left (a c^{6} d^{2} - 3 \, a c^{4} d^{4} + 3 \, a c^{2} d^{6} - a d^{8}\right )} f x + {\left (2 \, b c^{5} d^{2} - 6 \, a c^{4} d^{3} + b c^{3} d^{4} + 5 \, a c^{2} d^{5} - 2 \, a d^{7} + {\left (2 \, b c^{7} - 6 \, a c^{6} d + b c^{5} d^{2} + 5 \, a c^{4} d^{3} - 2 \, a c^{2} d^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, b c^{6} d - 6 \, a c^{5} d^{2} + b c^{4} d^{3} + 5 \, a c^{3} d^{4} - 2 \, a c d^{6}\right )} \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) - {\left (3 \, b c^{6} d^{2} - 5 \, a c^{5} d^{3} - 3 \, b c^{4} d^{4} + 7 \, a c^{3} d^{5} - 2 \, a c d^{7} + {\left (4 \, b c^{7} d - 6 \, a c^{6} d^{2} - 5 \, b c^{5} d^{3} + 9 \, a c^{4} d^{4} + b c^{3} d^{5} - 3 \, a c^{2} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{11} - 3 \, c^{9} d^{2} + 3 \, c^{7} d^{4} - c^{5} d^{6}\right )} f \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{10} d - 3 \, c^{8} d^{3} + 3 \, c^{6} d^{5} - c^{4} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{9} d^{2} - 3 \, c^{7} d^{4} + 3 \, c^{5} d^{6} - c^{3} d^{8}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))/(c+d*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/4*(4*(a*c^8 - 3*a*c^6*d^2 + 3*a*c^4*d^4 - a*c^2*d^6)*f*x*cos(f*x + e)^2 + 8*(a*c^7*d - 3*a*c^5*d^3 + 3*a*c^
3*d^5 - a*c*d^7)*f*x*cos(f*x + e) + 4*(a*c^6*d^2 - 3*a*c^4*d^4 + 3*a*c^2*d^6 - a*d^8)*f*x - (2*b*c^5*d^2 - 6*a
*c^4*d^3 + b*c^3*d^4 + 5*a*c^2*d^5 - 2*a*d^7 + (2*b*c^7 - 6*a*c^6*d + b*c^5*d^2 + 5*a*c^4*d^3 - 2*a*c^2*d^5)*c
os(f*x + e)^2 + 2*(2*b*c^6*d - 6*a*c^5*d^2 + b*c^4*d^3 + 5*a*c^3*d^4 - 2*a*c*d^6)*cos(f*x + e))*sqrt(c^2 - d^2
)*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e)
+ 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) - 2*(3*b*c^6*d^2 - 5*a*c^5*d^3 - 3*b*c^4*d^4
+ 7*a*c^3*d^5 - 2*a*c*d^7 + (4*b*c^7*d - 6*a*c^6*d^2 - 5*b*c^5*d^3 + 9*a*c^4*d^4 + b*c^3*d^5 - 3*a*c^2*d^6)*co
s(f*x + e))*sin(f*x + e))/((c^11 - 3*c^9*d^2 + 3*c^7*d^4 - c^5*d^6)*f*cos(f*x + e)^2 + 2*(c^10*d - 3*c^8*d^3 +
3*c^6*d^5 - c^4*d^7)*f*cos(f*x + e) + (c^9*d^2 - 3*c^7*d^4 + 3*c^5*d^6 - c^3*d^8)*f), 1/2*(2*(a*c^8 - 3*a*c^6
*d^2 + 3*a*c^4*d^4 - a*c^2*d^6)*f*x*cos(f*x + e)^2 + 4*(a*c^7*d - 3*a*c^5*d^3 + 3*a*c^3*d^5 - a*c*d^7)*f*x*cos
(f*x + e) + 2*(a*c^6*d^2 - 3*a*c^4*d^4 + 3*a*c^2*d^6 - a*d^8)*f*x + (2*b*c^5*d^2 - 6*a*c^4*d^3 + b*c^3*d^4 + 5
*a*c^2*d^5 - 2*a*d^7 + (2*b*c^7 - 6*a*c^6*d + b*c^5*d^2 + 5*a*c^4*d^3 - 2*a*c^2*d^5)*cos(f*x + e)^2 + 2*(2*b*c
^6*d - 6*a*c^5*d^2 + b*c^4*d^3 + 5*a*c^3*d^4 - 2*a*c*d^6)*cos(f*x + e))*sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d
^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) - (3*b*c^6*d^2 - 5*a*c^5*d^3 - 3*b*c^4*d^4 + 7*a*c^3*d^5
- 2*a*c*d^7 + (4*b*c^7*d - 6*a*c^6*d^2 - 5*b*c^5*d^3 + 9*a*c^4*d^4 + b*c^3*d^5 - 3*a*c^2*d^6)*cos(f*x + e))*si
n(f*x + e))/((c^11 - 3*c^9*d^2 + 3*c^7*d^4 - c^5*d^6)*f*cos(f*x + e)^2 + 2*(c^10*d - 3*c^8*d^3 + 3*c^6*d^5 - c
^4*d^7)*f*cos(f*x + e) + (c^9*d^2 - 3*c^7*d^4 + 3*c^5*d^6 - c^3*d^8)*f)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \sec {\left (e + f x \right )}}{\left (c + d \sec {\left (e + f x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))/(c+d*sec(f*x+e))**3,x)

[Out]

Integral((a + b*sec(e + f*x))/(c + d*sec(e + f*x))**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (191) = 382\).
time = 0.58, size = 457, normalized size = 2.24 \begin {gather*} \frac {\frac {{\left (2 \, b c^{5} - 6 \, a c^{4} d + b c^{3} d^{2} + 5 \, a c^{2} d^{3} - 2 \, a d^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c^{7} - 2 \, c^{5} d^{2} + c^{3} d^{4}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {{\left (f x + e\right )} a}{c^{3}} + \frac {4 \, b c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, a c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, a c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - b c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, a c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{6} - 2 \, c^{4} d^{2} + c^{2} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{2}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))/(c+d*sec(f*x+e))^3,x, algorithm="giac")

[Out]

((2*b*c^5 - 6*a*c^4*d + b*c^3*d^2 + 5*a*c^2*d^3 - 2*a*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) +
arctan(-(c*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2)))/((c^7 - 2*c^5*d^2 + c^3*d^4)*sqr
t(-c^2 + d^2)) + (f*x + e)*a/c^3 + (4*b*c^4*d*tan(1/2*f*x + 1/2*e)^3 - 6*a*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 - 3*
b*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 + 5*a*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 - b*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 + 3*a
*c*d^4*tan(1/2*f*x + 1/2*e)^3 - 2*a*d^5*tan(1/2*f*x + 1/2*e)^3 - 4*b*c^4*d*tan(1/2*f*x + 1/2*e) + 6*a*c^3*d^2*
tan(1/2*f*x + 1/2*e) - 3*b*c^3*d^2*tan(1/2*f*x + 1/2*e) + 5*a*c^2*d^3*tan(1/2*f*x + 1/2*e) + b*c^2*d^3*tan(1/2
*f*x + 1/2*e) - 3*a*c*d^4*tan(1/2*f*x + 1/2*e) - 2*a*d^5*tan(1/2*f*x + 1/2*e))/((c^6 - 2*c^4*d^2 + c^2*d^4)*(c
*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)^2))/f

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Mupad [B]
time = 11.35, size = 2500, normalized size = 12.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(e + f*x))/(c + d/cos(e + f*x))^3,x)

[Out]

(2*a*atan(((a*((8*tan(e/2 + (f*x)/2)*(4*a^2*c^10 + 8*a^2*d^10 + 4*b^2*c^10 - 8*a^2*c*d^9 - 8*a^2*c^9*d - 32*a^
2*c^2*d^8 + 32*a^2*c^3*d^7 + 57*a^2*c^4*d^6 - 48*a^2*c^5*d^5 - 52*a^2*c^6*d^4 + 32*a^2*c^7*d^3 + 24*a^2*c^8*d^
2 + b^2*c^6*d^4 + 4*b^2*c^8*d^2 - 24*a*b*c^9*d - 4*a*b*c^3*d^7 + 2*a*b*c^5*d^5 + 8*a*b*c^7*d^3))/(c^10*d + c^1
1 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 - 3*c^9*d^2) + (a*((8*(4*a*c^15 + 4*b*c^15 - 4*a*c^6
*d^9 + 2*a*c^7*d^8 + 18*a*c^8*d^7 - 4*a*c^9*d^6 - 36*a*c^10*d^5 + 6*a*c^11*d^4 + 34*a*c^12*d^3 - 8*a*c^13*d^2
- 2*b*c^8*d^7 + 2*b*c^9*d^6 + 6*b*c^12*d^3 - 6*b*c^13*d^2 - 12*a*c^14*d - 4*b*c^14*d))/(c^12*d + c^13 - c^6*d^
7 - c^7*d^6 + 3*c^8*d^5 + 3*c^9*d^4 - 3*c^10*d^3 - 3*c^11*d^2) - (a*tan(e/2 + (f*x)/2)*(8*c^15*d - 8*c^6*d^10
+ 8*c^7*d^9 + 32*c^8*d^8 - 32*c^9*d^7 - 48*c^10*d^6 + 48*c^11*d^5 + 32*c^12*d^4 - 32*c^13*d^3 - 8*c^14*d^2)*8i
)/(c^3*(c^10*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 - 3*c^9*d^2)))*1i)/c^3))/c^3 + (
a*((8*tan(e/2 + (f*x)/2)*(4*a^2*c^10 + 8*a^2*d^10 + 4*b^2*c^10 - 8*a^2*c*d^9 - 8*a^2*c^9*d - 32*a^2*c^2*d^8 +
32*a^2*c^3*d^7 + 57*a^2*c^4*d^6 - 48*a^2*c^5*d^5 - 52*a^2*c^6*d^4 + 32*a^2*c^7*d^3 + 24*a^2*c^8*d^2 + b^2*c^6*
d^4 + 4*b^2*c^8*d^2 - 24*a*b*c^9*d - 4*a*b*c^3*d^7 + 2*a*b*c^5*d^5 + 8*a*b*c^7*d^3))/(c^10*d + c^11 - c^4*d^7
- c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 - 3*c^9*d^2) - (a*((8*(4*a*c^15 + 4*b*c^15 - 4*a*c^6*d^9 + 2*a*c
^7*d^8 + 18*a*c^8*d^7 - 4*a*c^9*d^6 - 36*a*c^10*d^5 + 6*a*c^11*d^4 + 34*a*c^12*d^3 - 8*a*c^13*d^2 - 2*b*c^8*d^
7 + 2*b*c^9*d^6 + 6*b*c^12*d^3 - 6*b*c^13*d^2 - 12*a*c^14*d - 4*b*c^14*d))/(c^12*d + c^13 - c^6*d^7 - c^7*d^6
+ 3*c^8*d^5 + 3*c^9*d^4 - 3*c^10*d^3 - 3*c^11*d^2) + (a*tan(e/2 + (f*x)/2)*(8*c^15*d - 8*c^6*d^10 + 8*c^7*d^9
+ 32*c^8*d^8 - 32*c^9*d^7 - 48*c^10*d^6 + 48*c^11*d^5 + 32*c^12*d^4 - 32*c^13*d^3 - 8*c^14*d^2)*8i)/(c^3*(c^10
*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 - 3*c^9*d^2)))*1i)/c^3))/c^3)/((16*(4*a^3*d^
9 + 4*a*b^2*c^9 - 4*a^2*b*c^9 - 2*a^3*c*d^8 + 12*a^3*c^8*d - 18*a^3*c^2*d^7 + 13*a^3*c^3*d^6 + 36*a^3*c^4*d^5
- 26*a^3*c^5*d^4 - 34*a^3*c^6*d^3 + 24*a^3*c^7*d^2 + a*b^2*c^5*d^4 + 4*a*b^2*c^7*d^2 - 2*a^2*b*c^2*d^7 - 2*a^2
*b*c^3*d^6 + 2*a^2*b*c^4*d^5 + 2*a^2*b*c^6*d^3 + 6*a^2*b*c^7*d^2 - 20*a^2*b*c^8*d))/(c^12*d + c^13 - c^6*d^7 -
c^7*d^6 + 3*c^8*d^5 + 3*c^9*d^4 - 3*c^10*d^3 - 3*c^11*d^2) - (a*((8*tan(e/2 + (f*x)/2)*(4*a^2*c^10 + 8*a^2*d^
10 + 4*b^2*c^10 - 8*a^2*c*d^9 - 8*a^2*c^9*d - 32*a^2*c^2*d^8 + 32*a^2*c^3*d^7 + 57*a^2*c^4*d^6 - 48*a^2*c^5*d^
5 - 52*a^2*c^6*d^4 + 32*a^2*c^7*d^3 + 24*a^2*c^8*d^2 + b^2*c^6*d^4 + 4*b^2*c^8*d^2 - 24*a*b*c^9*d - 4*a*b*c^3*
d^7 + 2*a*b*c^5*d^5 + 8*a*b*c^7*d^3))/(c^10*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 -
3*c^9*d^2) + (a*((8*(4*a*c^15 + 4*b*c^15 - 4*a*c^6*d^9 + 2*a*c^7*d^8 + 18*a*c^8*d^7 - 4*a*c^9*d^6 - 36*a*c^10
*d^5 + 6*a*c^11*d^4 + 34*a*c^12*d^3 - 8*a*c^13*d^2 - 2*b*c^8*d^7 + 2*b*c^9*d^6 + 6*b*c^12*d^3 - 6*b*c^13*d^2 -
12*a*c^14*d - 4*b*c^14*d))/(c^12*d + c^13 - c^6*d^7 - c^7*d^6 + 3*c^8*d^5 + 3*c^9*d^4 - 3*c^10*d^3 - 3*c^11*d
^2) - (a*tan(e/2 + (f*x)/2)*(8*c^15*d - 8*c^6*d^10 + 8*c^7*d^9 + 32*c^8*d^8 - 32*c^9*d^7 - 48*c^10*d^6 + 48*c^
11*d^5 + 32*c^12*d^4 - 32*c^13*d^3 - 8*c^14*d^2)*8i)/(c^3*(c^10*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c
^7*d^4 - 3*c^8*d^3 - 3*c^9*d^2)))*1i)/c^3)*1i)/c^3 + (a*((8*tan(e/2 + (f*x)/2)*(4*a^2*c^10 + 8*a^2*d^10 + 4*b^
2*c^10 - 8*a^2*c*d^9 - 8*a^2*c^9*d - 32*a^2*c^2*d^8 + 32*a^2*c^3*d^7 + 57*a^2*c^4*d^6 - 48*a^2*c^5*d^5 - 52*a^
2*c^6*d^4 + 32*a^2*c^7*d^3 + 24*a^2*c^8*d^2 + b^2*c^6*d^4 + 4*b^2*c^8*d^2 - 24*a*b*c^9*d - 4*a*b*c^3*d^7 + 2*a
*b*c^5*d^5 + 8*a*b*c^7*d^3))/(c^10*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 - 3*c^9*d^
2) - (a*((8*(4*a*c^15 + 4*b*c^15 - 4*a*c^6*d^9 + 2*a*c^7*d^8 + 18*a*c^8*d^7 - 4*a*c^9*d^6 - 36*a*c^10*d^5 + 6*
a*c^11*d^4 + 34*a*c^12*d^3 - 8*a*c^13*d^2 - 2*b*c^8*d^7 + 2*b*c^9*d^6 + 6*b*c^12*d^3 - 6*b*c^13*d^2 - 12*a*c^1
4*d - 4*b*c^14*d))/(c^12*d + c^13 - c^6*d^7 - c^7*d^6 + 3*c^8*d^5 + 3*c^9*d^4 - 3*c^10*d^3 - 3*c^11*d^2) + (a*
tan(e/2 + (f*x)/2)*(8*c^15*d - 8*c^6*d^10 + 8*c^7*d^9 + 32*c^8*d^8 - 32*c^9*d^7 - 48*c^10*d^6 + 48*c^11*d^5 +
32*c^12*d^4 - 32*c^13*d^3 - 8*c^14*d^2)*8i)/(c^3*(c^10*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 -
3*c^8*d^3 - 3*c^9*d^2)))*1i)/c^3)*1i)/c^3)))/(c^3*f) - ((tan(e/2 + (f*x)/2)^3*(2*a*d^4 - 6*a*c^2*d^2 + b*c^2*d
^2 - a*c*d^3 + 4*b*c^3*d))/((c^2*d - c^3)*(c + d)^2) + (tan(e/2 + (f*x)/2)*(2*a*d^4 - 6*a*c^2*d^2 - b*c^2*d^2
+ a*c*d^3 + 4*b*c^3*d))/((c + d)*(c^4 - 2*c^3*d + c^2*d^2)))/(f*(2*c*d - tan(e/2 + (f*x)/2)^2*(2*c^2 - 2*d^2)
+ tan(e/2 + (f*x)/2)^4*(c^2 - 2*c*d + d^2) + c^2 + d^2)) + (atan(((((8*tan(e/2 + (f*x)/2)*(4*a^2*c^10 + 8*a^2*
d^10 + 4*b^2*c^10 - 8*a^2*c*d^9 - 8*a^2*c^9*d - 32*a^2*c^2*d^8 + 32*a^2*c^3*d^7 + 57*a^2*c^4*d^6 - 48*a^2*c^5*
d^5 - 52*a^2*c^6*d^4 + 32*a^2*c^7*d^3 + 24*a^2*c^8*d^2 + b^2*c^6*d^4 + 4*b^2*c^8*d^2 - 24*a*b*c^9*d - 4*a*b*c^
3*d^7 + 2*a*b*c^5*d^5 + 8*a*b*c^7*d^3))/(c^10*d...

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