3.219 \(\int \frac{1}{(a+b \sec ^2(c+d x))^4} \, dx\)

Optimal. Leaf size=204 \[ -\frac{\sqrt{b} \left (70 a^2 b+35 a^3+56 a b^2+16 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b}}\right )}{16 a^4 d (a+b)^{7/2}}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tan (c+d x)}{16 a^3 d (a+b)^3 \left (a+b \tan ^2(c+d x)+b\right )}-\frac{b (11 a+6 b) \tan (c+d x)}{24 a^2 d (a+b)^2 \left (a+b \tan ^2(c+d x)+b\right )^2}+\frac{x}{a^4}-\frac{b \tan (c+d x)}{6 a d (a+b) \left (a+b \tan ^2(c+d x)+b\right )^3} \]

[Out]

x/a^4 - (Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b]])/(16*a^4*(
a + b)^(7/2)*d) - (b*Tan[c + d*x])/(6*a*(a + b)*d*(a + b + b*Tan[c + d*x]^2)^3) - (b*(11*a + 6*b)*Tan[c + d*x]
)/(24*a^2*(a + b)^2*d*(a + b + b*Tan[c + d*x]^2)^2) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tan[c + d*x])/(16*a^3*(a +
b)^3*d*(a + b + b*Tan[c + d*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.334845, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4128, 414, 527, 522, 203, 205} \[ -\frac{\sqrt{b} \left (70 a^2 b+35 a^3+56 a b^2+16 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b}}\right )}{16 a^4 d (a+b)^{7/2}}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tan (c+d x)}{16 a^3 d (a+b)^3 \left (a+b \tan ^2(c+d x)+b\right )}-\frac{b (11 a+6 b) \tan (c+d x)}{24 a^2 d (a+b)^2 \left (a+b \tan ^2(c+d x)+b\right )^2}+\frac{x}{a^4}-\frac{b \tan (c+d x)}{6 a d (a+b) \left (a+b \tan ^2(c+d x)+b\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[c + d*x]^2)^(-4),x]

[Out]

x/a^4 - (Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b]])/(16*a^4*(
a + b)^(7/2)*d) - (b*Tan[c + d*x])/(6*a*(a + b)*d*(a + b + b*Tan[c + d*x]^2)^3) - (b*(11*a + 6*b)*Tan[c + d*x]
)/(24*a^2*(a + b)^2*d*(a + b + b*Tan[c + d*x]^2)^2) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tan[c + d*x])/(16*a^3*(a +
b)^3*d*(a + b + b*Tan[c + d*x]^2))

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 203

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sec ^2(c+d x)\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+b+b x^2\right )^4} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{b \tan (c+d x)}{6 a (a+b) d \left (a+b+b \tan ^2(c+d x)\right )^3}+\frac{\operatorname{Subst}\left (\int \frac{6 a+b-5 b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{6 a (a+b) d}\\ &=-\frac{b \tan (c+d x)}{6 a (a+b) d \left (a+b+b \tan ^2(c+d x)\right )^3}-\frac{b (11 a+6 b) \tan (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \tan ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (8 a^2+5 a b+2 b^2\right )-3 b (11 a+6 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{24 a^2 (a+b)^2 d}\\ &=-\frac{b \tan (c+d x)}{6 a (a+b) d \left (a+b+b \tan ^2(c+d x)\right )^3}-\frac{b (11 a+6 b) \tan (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \tan ^2(c+d x)\right )^2}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tan (c+d x)}{16 a^3 (a+b)^3 d \left (a+b+b \tan ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (16 a^3+29 a^2 b+26 a b^2+8 b^3\right )-3 b \left (19 a^2+22 a b+8 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (c+d x)\right )}{48 a^3 (a+b)^3 d}\\ &=-\frac{b \tan (c+d x)}{6 a (a+b) d \left (a+b+b \tan ^2(c+d x)\right )^3}-\frac{b (11 a+6 b) \tan (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \tan ^2(c+d x)\right )^2}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tan (c+d x)}{16 a^3 (a+b)^3 d \left (a+b+b \tan ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{a^4 d}-\frac{\left (b \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (c+d x)\right )}{16 a^4 (a+b)^3 d}\\ &=\frac{x}{a^4}-\frac{\sqrt{b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b}}\right )}{16 a^4 (a+b)^{7/2} d}-\frac{b \tan (c+d x)}{6 a (a+b) d \left (a+b+b \tan ^2(c+d x)\right )^3}-\frac{b (11 a+6 b) \tan (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \tan ^2(c+d x)\right )^2}-\frac{b \left (19 a^2+22 a b+8 b^2\right ) \tan (c+d x)}{16 a^3 (a+b)^3 d \left (a+b+b \tan ^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 6.85346, size = 1411, normalized size = 6.92 \[ \frac{\left (35 a^3+70 b a^2+56 b^2 a+16 b^3\right ) (\cos (2 c+2 d x) a+a+2 b)^4 \left (\frac{b \tan ^{-1}\left (\sec (d x) \left (\frac{\cos (2 c)}{2 \sqrt{a+b} \sqrt{b \cos (4 c)-i b \sin (4 c)}}-\frac{i \sin (2 c)}{2 \sqrt{a+b} \sqrt{b \cos (4 c)-i b \sin (4 c)}}\right ) (-a \sin (d x)-2 b \sin (d x)+a \sin (2 c+d x))\right ) \cos (2 c)}{256 a^4 \sqrt{a+b} d \sqrt{b \cos (4 c)-i b \sin (4 c)}}-\frac{i b \tan ^{-1}\left (\sec (d x) \left (\frac{\cos (2 c)}{2 \sqrt{a+b} \sqrt{b \cos (4 c)-i b \sin (4 c)}}-\frac{i \sin (2 c)}{2 \sqrt{a+b} \sqrt{b \cos (4 c)-i b \sin (4 c)}}\right ) (-a \sin (d x)-2 b \sin (d x)+a \sin (2 c+d x))\right ) \sin (2 c)}{256 a^4 \sqrt{a+b} d \sqrt{b \cos (4 c)-i b \sin (4 c)}}\right ) \sec ^8(c+d x)}{(a+b)^3 \left (b \sec ^2(c+d x)+a\right )^4}+\frac{(\cos (2 c+2 d x) a+a+2 b) \sec (2 c) \left (480 d x \cos (2 c) a^6+360 d x \cos (2 d x) a^6+360 d x \cos (4 c+2 d x) a^6+144 d x \cos (2 c+4 d x) a^6+144 d x \cos (6 c+4 d x) a^6+24 d x \cos (4 c+6 d x) a^6+24 d x \cos (8 c+6 d x) a^6+3168 b d x \cos (2 c) a^5+2232 b d x \cos (2 d x) a^5+2232 b d x \cos (4 c+2 d x) a^5+720 b d x \cos (2 c+4 d x) a^5+720 b d x \cos (6 c+4 d x) a^5+72 b d x \cos (4 c+6 d x) a^5+72 b d x \cos (8 c+6 d x) a^5+870 b \sin (2 c) a^5-870 b \sin (2 d x) a^5+435 b \sin (4 c+2 d x) a^5-435 b \sin (2 c+4 d x) a^5+87 b \sin (6 c+4 d x) a^5-87 b \sin (4 c+6 d x) a^5+8928 b^2 d x \cos (2 c) a^4+5688 b^2 d x \cos (2 d x) a^4+5688 b^2 d x \cos (4 c+2 d x) a^4+1296 b^2 d x \cos (2 c+4 d x) a^4+1296 b^2 d x \cos (6 c+4 d x) a^4+72 b^2 d x \cos (4 c+6 d x) a^4+72 b^2 d x \cos (8 c+6 d x) a^4+4292 b^2 \sin (2 c) a^4-3792 b^2 \sin (2 d x) a^4+2124 b^2 \sin (4 c+2 d x) a^4-1374 b^2 \sin (2 c+4 d x) a^4+366 b^2 \sin (6 c+4 d x) a^4-116 b^2 \sin (4 c+6 d x) a^4+14112 b^3 d x \cos (2 c) a^3+7272 b^3 d x \cos (2 d x) a^3+7272 b^3 d x \cos (4 c+2 d x) a^3+1008 b^3 d x \cos (2 c+4 d x) a^3+1008 b^3 d x \cos (6 c+4 d x) a^3+24 b^3 d x \cos (4 c+6 d x) a^3+24 b^3 d x \cos (8 c+6 d x) a^3+8792 b^3 \sin (2 c) a^3-6432 b^3 \sin (2 d x) a^3+3972 b^3 \sin (4 c+2 d x) a^3-1248 b^3 \sin (2 c+4 d x) a^3+408 b^3 \sin (6 c+4 d x) a^3-44 b^3 \sin (4 c+6 d x) a^3+13248 b^4 d x \cos (2 c) a^2+4608 b^4 d x \cos (2 d x) a^2+4608 b^4 d x \cos (4 c+2 d x) a^2+288 b^4 d x \cos (2 c+4 d x) a^2+288 b^4 d x \cos (6 c+4 d x) a^2+9936 b^4 \sin (2 c) a^2-4608 b^4 \sin (2 d x) a^2+3072 b^4 \sin (4 c+2 d x) a^2-384 b^4 \sin (2 c+4 d x) a^2+144 b^4 \sin (6 c+4 d x) a^2+6912 b^5 d x \cos (2 c) a+1152 b^5 d x \cos (2 d x) a+1152 b^5 d x \cos (4 c+2 d x) a+5824 b^5 \sin (2 c) a-1248 b^5 \sin (2 d x) a+864 b^5 \sin (4 c+2 d x) a+1536 b^6 d x \cos (2 c)+1408 b^6 \sin (2 c)\right ) \sec ^8(c+d x)}{3072 a^4 (a+b)^3 d \left (b \sec ^2(c+d x)+a\right )^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sec[c + d*x]^2)^(-4),x]

[Out]

((35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*(a + 2*b + a*Cos[2*c + 2*d*x])^4*Sec[c + d*x]^8*((b*ArcTan[Sec[d*x]*(
Cos[2*c]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]]) - ((I/2)*Sin[2*c])/(Sqrt[a + b]*Sqrt[b*Cos[4*c] - I*b
*Sin[4*c]]))*(-(a*Sin[d*x]) - 2*b*Sin[d*x] + a*Sin[2*c + d*x])]*Cos[2*c])/(256*a^4*Sqrt[a + b]*d*Sqrt[b*Cos[4*
c] - I*b*Sin[4*c]]) - ((I/256)*b*ArcTan[Sec[d*x]*(Cos[2*c]/(2*Sqrt[a + b]*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]]) - (
(I/2)*Sin[2*c])/(Sqrt[a + b]*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]]))*(-(a*Sin[d*x]) - 2*b*Sin[d*x] + a*Sin[2*c + d*x
])]*Sin[2*c])/(a^4*Sqrt[a + b]*d*Sqrt[b*Cos[4*c] - I*b*Sin[4*c]])))/((a + b)^3*(a + b*Sec[c + d*x]^2)^4) + ((a
+ 2*b + a*Cos[2*c + 2*d*x])*Sec[2*c]*Sec[c + d*x]^8*(480*a^6*d*x*Cos[2*c] + 3168*a^5*b*d*x*Cos[2*c] + 8928*a^
4*b^2*d*x*Cos[2*c] + 14112*a^3*b^3*d*x*Cos[2*c] + 13248*a^2*b^4*d*x*Cos[2*c] + 6912*a*b^5*d*x*Cos[2*c] + 1536*
b^6*d*x*Cos[2*c] + 360*a^6*d*x*Cos[2*d*x] + 2232*a^5*b*d*x*Cos[2*d*x] + 5688*a^4*b^2*d*x*Cos[2*d*x] + 7272*a^3
*b^3*d*x*Cos[2*d*x] + 4608*a^2*b^4*d*x*Cos[2*d*x] + 1152*a*b^5*d*x*Cos[2*d*x] + 360*a^6*d*x*Cos[4*c + 2*d*x] +
2232*a^5*b*d*x*Cos[4*c + 2*d*x] + 5688*a^4*b^2*d*x*Cos[4*c + 2*d*x] + 7272*a^3*b^3*d*x*Cos[4*c + 2*d*x] + 460
8*a^2*b^4*d*x*Cos[4*c + 2*d*x] + 1152*a*b^5*d*x*Cos[4*c + 2*d*x] + 144*a^6*d*x*Cos[2*c + 4*d*x] + 720*a^5*b*d*
x*Cos[2*c + 4*d*x] + 1296*a^4*b^2*d*x*Cos[2*c + 4*d*x] + 1008*a^3*b^3*d*x*Cos[2*c + 4*d*x] + 288*a^2*b^4*d*x*C
os[2*c + 4*d*x] + 144*a^6*d*x*Cos[6*c + 4*d*x] + 720*a^5*b*d*x*Cos[6*c + 4*d*x] + 1296*a^4*b^2*d*x*Cos[6*c + 4
*d*x] + 1008*a^3*b^3*d*x*Cos[6*c + 4*d*x] + 288*a^2*b^4*d*x*Cos[6*c + 4*d*x] + 24*a^6*d*x*Cos[4*c + 6*d*x] + 7
2*a^5*b*d*x*Cos[4*c + 6*d*x] + 72*a^4*b^2*d*x*Cos[4*c + 6*d*x] + 24*a^3*b^3*d*x*Cos[4*c + 6*d*x] + 24*a^6*d*x*
Cos[8*c + 6*d*x] + 72*a^5*b*d*x*Cos[8*c + 6*d*x] + 72*a^4*b^2*d*x*Cos[8*c + 6*d*x] + 24*a^3*b^3*d*x*Cos[8*c +
6*d*x] + 870*a^5*b*Sin[2*c] + 4292*a^4*b^2*Sin[2*c] + 8792*a^3*b^3*Sin[2*c] + 9936*a^2*b^4*Sin[2*c] + 5824*a*b
^5*Sin[2*c] + 1408*b^6*Sin[2*c] - 870*a^5*b*Sin[2*d*x] - 3792*a^4*b^2*Sin[2*d*x] - 6432*a^3*b^3*Sin[2*d*x] - 4
608*a^2*b^4*Sin[2*d*x] - 1248*a*b^5*Sin[2*d*x] + 435*a^5*b*Sin[4*c + 2*d*x] + 2124*a^4*b^2*Sin[4*c + 2*d*x] +
3972*a^3*b^3*Sin[4*c + 2*d*x] + 3072*a^2*b^4*Sin[4*c + 2*d*x] + 864*a*b^5*Sin[4*c + 2*d*x] - 435*a^5*b*Sin[2*c
+ 4*d*x] - 1374*a^4*b^2*Sin[2*c + 4*d*x] - 1248*a^3*b^3*Sin[2*c + 4*d*x] - 384*a^2*b^4*Sin[2*c + 4*d*x] + 87*
a^5*b*Sin[6*c + 4*d*x] + 366*a^4*b^2*Sin[6*c + 4*d*x] + 408*a^3*b^3*Sin[6*c + 4*d*x] + 144*a^2*b^4*Sin[6*c + 4
*d*x] - 87*a^5*b*Sin[4*c + 6*d*x] - 116*a^4*b^2*Sin[4*c + 6*d*x] - 44*a^3*b^3*Sin[4*c + 6*d*x]))/(3072*a^4*(a
+ b)^3*d*(a + b*Sec[c + d*x]^2)^4)

________________________________________________________________________________________

Maple [B]  time = 0.092, size = 649, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sec(d*x+c)^2)^4,x)

[Out]

1/d/a^4*arctan(tan(d*x+c))-19/16/d*b^3/a/(a+b+b*tan(d*x+c)^2)^3/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c)^5-11/8/d*
b^4/a^2/(a+b+b*tan(d*x+c)^2)^3/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c)^5-1/2/d*b^5/a^3/(a+b+b*tan(d*x+c)^2)^3/(a^
3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c)^5-17/6/d*b^2/a/(a+b+b*tan(d*x+c)^2)^3/(a^2+2*a*b+b^2)*tan(d*x+c)^3-3/d*b^3/a
^2/(a+b+b*tan(d*x+c)^2)^3/(a^2+2*a*b+b^2)*tan(d*x+c)^3-1/d*b^4/a^3/(a+b+b*tan(d*x+c)^2)^3/(a^2+2*a*b+b^2)*tan(
d*x+c)^3-29/16*b*tan(d*x+c)/a/(a+b)/d/(a+b+b*tan(d*x+c)^2)^3-13/8/d*b^2/a^2/(a+b+b*tan(d*x+c)^2)^3/(a+b)*tan(d
*x+c)-1/2/d*b^3/a^3/(a+b+b*tan(d*x+c)^2)^3/(a+b)*tan(d*x+c)-35/16/d*b/a/(a^3+3*a^2*b+3*a*b^2+b^3)/((a+b)*b)^(1
/2)*arctan(tan(d*x+c)*b/((a+b)*b)^(1/2))-35/8/d*b^2/a^2/(a^3+3*a^2*b+3*a*b^2+b^3)/((a+b)*b)^(1/2)*arctan(tan(d
*x+c)*b/((a+b)*b)^(1/2))-7/2/d*b^3/a^3/(a^3+3*a^2*b+3*a*b^2+b^3)/((a+b)*b)^(1/2)*arctan(tan(d*x+c)*b/((a+b)*b)
^(1/2))-1/d*b^4/a^4/(a^3+3*a^2*b+3*a*b^2+b^3)/((a+b)*b)^(1/2)*arctan(tan(d*x+c)*b/((a+b)*b)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(d*x+c)^2)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.883143, size = 2967, normalized size = 14.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(d*x+c)^2)^4,x, algorithm="fricas")

[Out]

[1/192*(192*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*x*cos(d*x + c)^6 + 576*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^
2*b^4)*d*x*cos(d*x + c)^4 + 576*(a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*d*x*cos(d*x + c)^2 + 192*(a^3*b^3 +
3*a^2*b^4 + 3*a*b^5 + b^6)*d*x + 3*((35*a^6 + 70*a^5*b + 56*a^4*b^2 + 16*a^3*b^3)*cos(d*x + c)^6 + 35*a^3*b^3
+ 70*a^2*b^4 + 56*a*b^5 + 16*b^6 + 3*(35*a^5*b + 70*a^4*b^2 + 56*a^3*b^3 + 16*a^2*b^4)*cos(d*x + c)^4 + 3*(35*
a^4*b^2 + 70*a^3*b^3 + 56*a^2*b^4 + 16*a*b^5)*cos(d*x + c)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(
d*x + c)^4 - 2*(3*a*b + 4*b^2)*cos(d*x + c)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x
+ c))*sqrt(-b/(a + b))*sin(d*x + c) + b^2)/(a^2*cos(d*x + c)^4 + 2*a*b*cos(d*x + c)^2 + b^2)) - 4*((87*a^5*b +
116*a^4*b^2 + 44*a^3*b^3)*cos(d*x + c)^5 + 2*(68*a^4*b^2 + 83*a^3*b^3 + 30*a^2*b^4)*cos(d*x + c)^3 + 3*(19*a^
3*b^3 + 22*a^2*b^4 + 8*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x +
c)^6 + 3*(a^9*b + 3*a^8*b^2 + 3*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 + a^5
*b^5)*d*cos(d*x + c)^2 + (a^7*b^3 + 3*a^6*b^4 + 3*a^5*b^5 + a^4*b^6)*d), 1/96*(96*(a^6 + 3*a^5*b + 3*a^4*b^2 +
a^3*b^3)*d*x*cos(d*x + c)^6 + 288*(a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*d*x*cos(d*x + c)^4 + 288*(a^4*b^2
+ 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*d*x*cos(d*x + c)^2 + 96*(a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6)*d*x + 3*((35*a
^6 + 70*a^5*b + 56*a^4*b^2 + 16*a^3*b^3)*cos(d*x + c)^6 + 35*a^3*b^3 + 70*a^2*b^4 + 56*a*b^5 + 16*b^6 + 3*(35*
a^5*b + 70*a^4*b^2 + 56*a^3*b^3 + 16*a^2*b^4)*cos(d*x + c)^4 + 3*(35*a^4*b^2 + 70*a^3*b^3 + 56*a^2*b^4 + 16*a*
b^5)*cos(d*x + c)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(d*x + c)^2 - b)*sqrt(b/(a + b))/(b*cos(d*x + c)
*sin(d*x + c))) - 2*((87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3)*cos(d*x + c)^5 + 2*(68*a^4*b^2 + 83*a^3*b^3 + 30*a^
2*b^4)*cos(d*x + c)^3 + 3*(19*a^3*b^3 + 22*a^2*b^4 + 8*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^10 + 3*a^9*b + 3
*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 + 3*(a^9*b + 3*a^8*b^2 + 3*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^8*b
^2 + 3*a^7*b^3 + 3*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 + (a^7*b^3 + 3*a^6*b^4 + 3*a^5*b^5 + a^4*b^6)*d)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(d*x+c)**2)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.32772, size = 437, normalized size = 2.14 \begin{align*} -\frac{\frac{3 \,{\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, a b^{3} + 16 \, b^{4}\right )}{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (d x + c\right )}{\sqrt{a b + b^{2}}}\right )\right )}}{{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt{a b + b^{2}}} + \frac{57 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 66 \, a b^{4} \tan \left (d x + c\right )^{5} + 24 \, b^{5} \tan \left (d x + c\right )^{5} + 136 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 280 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} + 192 \, a b^{4} \tan \left (d x + c\right )^{3} + 48 \, b^{5} \tan \left (d x + c\right )^{3} + 87 \, a^{4} b \tan \left (d x + c\right ) + 252 \, a^{3} b^{2} \tan \left (d x + c\right ) + 267 \, a^{2} b^{3} \tan \left (d x + c\right ) + 126 \, a b^{4} \tan \left (d x + c\right ) + 24 \, b^{5} \tan \left (d x + c\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )}{\left (b \tan \left (d x + c\right )^{2} + a + b\right )}^{3}} - \frac{48 \,{\left (d x + c\right )}}{a^{4}}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sec(d*x+c)^2)^4,x, algorithm="giac")

[Out]

-1/48*(3*(35*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*(pi*floor((d*x + c)/pi + 1/2)*sgn(b) + arctan(b*tan(d*x +
c)/sqrt(a*b + b^2)))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt(a*b + b^2)) + (57*a^2*b^3*tan(d*x + c)^5 + 6
6*a*b^4*tan(d*x + c)^5 + 24*b^5*tan(d*x + c)^5 + 136*a^3*b^2*tan(d*x + c)^3 + 280*a^2*b^3*tan(d*x + c)^3 + 192
*a*b^4*tan(d*x + c)^3 + 48*b^5*tan(d*x + c)^3 + 87*a^4*b*tan(d*x + c) + 252*a^3*b^2*tan(d*x + c) + 267*a^2*b^3
*tan(d*x + c) + 126*a*b^4*tan(d*x + c) + 24*b^5*tan(d*x + c))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*(b*tan(d*
x + c)^2 + a + b)^3) - 48*(d*x + c)/a^4)/d