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

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

[Out]

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

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Rubi [A]  time = 0.32453, 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} \[ \frac{b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{16 a^3 d (a+b)^3 \left (a+b \cot ^2(c+d x)+b\right )}+\frac{\sqrt{b} \left (70 a^2 b+35 a^3+56 a b^2+16 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b}}\right )}{16 a^4 d (a+b)^{7/2}}+\frac{b (11 a+6 b) \cot (c+d x)}{24 a^2 d (a+b)^2 \left (a+b \cot ^2(c+d x)+b\right )^2}+\frac{x}{a^4}+\frac{b \cot (c+d x)}{6 a d (a+b) \left (a+b \cot ^2(c+d x)+b\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csc[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]*Cot[c + d*x])/Sqrt[a + b]])/(16*a^4*(
a + b)^(7/2)*d) + (b*Cot[c + d*x])/(6*a*(a + b)*d*(a + b + b*Cot[c + d*x]^2)^3) + (b*(11*a + 6*b)*Cot[c + d*x]
)/(24*a^2*(a + b)^2*d*(a + b + b*Cot[c + d*x]^2)^2) + (b*(19*a^2 + 22*a*b + 8*b^2)*Cot[c + d*x])/(16*a^3*(a +
b)^3*d*(a + b + b*Cot[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 \csc ^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,\cot (c+d x)\right )}{d}\\ &=\frac{b \cot (c+d x)}{6 a (a+b) d \left (a+b+b \cot ^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,\cot (c+d x)\right )}{6 a (a+b) d}\\ &=\frac{b \cot (c+d x)}{6 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^3}+\frac{b (11 a+6 b) \cot (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \cot ^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,\cot (c+d x)\right )}{24 a^2 (a+b)^2 d}\\ &=\frac{b \cot (c+d x)}{6 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^3}+\frac{b (11 a+6 b) \cot (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^2}+\frac{b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{16 a^3 (a+b)^3 d \left (a+b+b \cot ^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,\cot (c+d x)\right )}{48 a^3 (a+b)^3 d}\\ &=\frac{b \cot (c+d x)}{6 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^3}+\frac{b (11 a+6 b) \cot (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^2}+\frac{b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{16 a^3 (a+b)^3 d \left (a+b+b \cot ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (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,\cot (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} \cot (c+d x)}{\sqrt{a+b}}\right )}{16 a^4 (a+b)^{7/2} d}+\frac{b \cot (c+d x)}{6 a (a+b) d \left (a+b+b \cot ^2(c+d x)\right )^3}+\frac{b (11 a+6 b) \cot (c+d x)}{24 a^2 (a+b)^2 d \left (a+b+b \cot ^2(c+d x)\right )^2}+\frac{b \left (19 a^2+22 a b+8 b^2\right ) \cot (c+d x)}{16 a^3 (a+b)^3 d \left (a+b+b \cot ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 3.30773, size = 263, normalized size = 1.29 \[ \frac{\csc ^8(c+d x) (a \cos (2 (c+d x))-a-2 b) \left (-\frac{a b \left (87 a^2+116 a b+44 b^2\right ) \sin (2 (c+d x)) (a (-\cos (2 (c+d x)))+a+2 b)^2}{(a+b)^3}+\frac{3 \sqrt{b} \left (70 a^2 b+35 a^3+56 a b^2+16 b^3\right ) (a (-\cos (2 (c+d x)))+a+2 b)^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{b}}\right )}{(a+b)^{7/2}}-\frac{32 a b^3 \sin (2 (c+d x))}{a+b}-\frac{4 a b^2 (19 a+14 b) \sin (2 (c+d x)) (a \cos (2 (c+d x))-a-2 b)}{(a+b)^2}+48 (c+d x) (a \cos (2 (c+d x))-a-2 b)^3\right )}{768 a^4 d \left (a+b \csc ^2(c+d x)\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-a - 2*b + a*Cos[2*(c + d*x)])*Csc[c + d*x]^8*((3*Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTan[(Sq
rt[a + b]*Tan[c + d*x])/Sqrt[b]]*(a + 2*b - a*Cos[2*(c + d*x)])^3)/(a + b)^(7/2) + 48*(c + d*x)*(-a - 2*b + a*
Cos[2*(c + d*x)])^3 - (32*a*b^3*Sin[2*(c + d*x)])/(a + b) - (a*b*(87*a^2 + 116*a*b + 44*b^2)*(a + 2*b - a*Cos[
2*(c + d*x)])^2*Sin[2*(c + d*x)])/(a + b)^3 - (4*a*b^2*(19*a + 14*b)*(-a - 2*b + a*Cos[2*(c + d*x)])*Sin[2*(c
+ d*x)])/(a + b)^2))/(768*a^4*d*(a + b*Csc[c + d*x]^2)^4)

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Maple [B]  time = 0.09, size = 737, normalized size = 3.6 \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*csc(d*x+c)^2)^4,x)

[Out]

1/d/a^4*arctan(tan(d*x+c))+29/16/d*b/a/(a*tan(d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a+b)*tan(d*x+c)^5+13/8/d*b^2/a^2/(
a*tan(d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a+b)*tan(d*x+c)^5+1/2/d*b^3/a^3/(a*tan(d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a+b)*
tan(d*x+c)^5+17/6/d*b^2/a/(a*tan(d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a^2+2*a*b+b^2)*tan(d*x+c)^3+3/d*b^3/a^2/(a*tan(
d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a^2+2*a*b+b^2)*tan(d*x+c)^3+1/d*b^4/a^3/(a*tan(d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a^2
+2*a*b+b^2)*tan(d*x+c)^3+19/16/d*b^3/a/(a*tan(d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c
)+11/8/d*b^4/a^2/(a*tan(d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(d*x+c)+1/2/d*b^5/a^3/(a*tan
(d*x+c)^2+b*tan(d*x+c)^2+b)^3/(a^3+3*a^2*b+3*a*b^2+b^3)*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((a+b)*tan(d*x+c)/((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)*a
rctan((a+b)*tan(d*x+c)/((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((a+b)*t
an(d*x+c)/((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((a+b)*tan(d*x+c)/((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*csc(d*x+c)^2)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.805293, size = 3672, normalized size = 18. \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*csc(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^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*
b^3 + a^2*b^4)*d*x*cos(d*x + c)^4 + 576*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*d*x*cos(
d*x + c)^2 - 192*(a^6 + 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*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^6 - 175*a^5*b - 371*a^4*b^2 - 429*a^3*b^3 - 286*a^2*b^4
- 104*a*b^5 - 16*b^6 - 3*(35*a^6 + 105*a^5*b + 126*a^4*b^2 + 72*a^3*b^3 + 16*a^2*b^4)*cos(d*x + c)^4 + 3*(35*
a^6 + 140*a^5*b + 231*a^4*b^2 + 198*a^3*b^3 + 88*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*(a^2 + 5*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^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-b/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(a^2*cos(d*x + c
)^4 - 2*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*((87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3)*cos(d*x +
c)^5 - 2*(87*a^5*b + 184*a^4*b^2 + 127*a^3*b^3 + 30*a^2*b^4)*cos(d*x + c)^3 + 3*(29*a^5*b + 84*a^4*b^2 + 89*a^
3*b^3 + 42*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^10 + 4*a^9*b + 6*a^8*b^2 + 4*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2
+ 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 - (a^10 + 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4
+ 6*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^6 + 4*a^
5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cos(d*x + c)^4 + 288*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a
^2*b^4 + a*b^5)*d*x*cos(d*x + c)^2 - 96*(a^6 + 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*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^6 - 175*a^5*b - 371*a^4*b^2 - 42
9*a^3*b^3 - 286*a^2*b^4 - 104*a*b^5 - 16*b^6 - 3*(35*a^6 + 105*a^5*b + 126*a^4*b^2 + 72*a^3*b^3 + 16*a^2*b^4)*
cos(d*x + c)^4 + 3*(35*a^6 + 140*a^5*b + 231*a^4*b^2 + 198*a^3*b^3 + 88*a^2*b^4 + 16*a*b^5)*cos(d*x + c)^2)*sq
rt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(d*x + c)^2 - a - 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*(87*a^5*b + 184*a^4*b^2 + 127*a^3*b^3 + 30*a^2*b^4)
*cos(d*x + c)^3 + 3*(29*a^5*b + 84*a^4*b^2 + 89*a^3*b^3 + 42*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^10 + 4*a^9*b + 6*a^8*b^2 + 4*a^7*b^3 + a^6*b^4)*
d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 - (a^10
+ 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4 + 6*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*csc(d*x+c)**2)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.48535, size = 478, normalized size = 2.34 \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 (2 \, a + 2 \, b\right ) + \arctan \left (\frac{a \tan \left (d x + c\right ) + 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{87 \, a^{4} b \tan \left (d x + c\right )^{5} + 252 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 267 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 126 \, 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} + 57 \, a^{2} b^{3} \tan \left (d x + c\right ) + 66 \, 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 (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + 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*csc(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(2*a + 2*b) + arctan((a*
tan(d*x + c) + 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)) - (87
*a^4*b*tan(d*x + c)^5 + 252*a^3*b^2*tan(d*x + c)^5 + 267*a^2*b^3*tan(d*x + c)^5 + 126*a*b^4*tan(d*x + c)^5 + 2
4*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 + 57*a^2*b^3*tan(d*x + c) + 66*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)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + b)^3) - 48*(d*x + c)/a^4)/d