3.385 \(\int \frac{1}{a+b \tan ^4(c+d x)} \, dx\)

Optimal. Leaf size=302 \[ \frac{\sqrt [4]{b} \left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt [4]{b} \left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{x}{a+b} \]

[Out]

x/(a + b) + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4
)*(a + b)*d) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(
3/4)*(a + b)*d) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Ta
n[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d) + ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d)

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Rubi [A]  time = 0.321428, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {3661, 1171, 203, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\sqrt [4]{b} \left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt [4]{b} \left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{x}{a+b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x]^4)^(-1),x]

[Out]

x/(a + b) + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4
)*(a + b)*d) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(
3/4)*(a + b)*d) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Ta
n[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d) + ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d)

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

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 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{a+b \tan ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+b x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{(a+b) \left (1+x^2\right )}+\frac{b-b x^2}{(a+b) \left (a+b x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{(a+b) d}+\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{(a+b) d}\\ &=\frac{x}{a+b}-\frac{\left (1-\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{2 (a+b) d}+\frac{\left (1+\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{2 (a+b) d}\\ &=\frac{x}{a+b}-\frac{\left (1-\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\tan (c+d x)\right )}{4 (a+b) d}-\frac{\left (1-\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\tan (c+d x)\right )}{4 (a+b) d}-\frac{\left (\left (\sqrt{a}+\sqrt{b}\right ) \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\tan (c+d x)\right )}{4 \sqrt{2} a^{3/4} (a+b) d}-\frac{\left (\left (\sqrt{a}+\sqrt{b}\right ) \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\tan (c+d x)\right )}{4 \sqrt{2} a^{3/4} (a+b) d}\\ &=\frac{x}{a+b}-\frac{\left (\sqrt{a}+\sqrt{b}\right ) \sqrt [4]{b} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{b} \tan ^2(c+d x)\right )}{4 \sqrt{2} a^{3/4} (a+b) d}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) \sqrt [4]{b} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{b} \tan ^2(c+d x)\right )}{4 \sqrt{2} a^{3/4} (a+b) d}-\frac{\left (\left (\sqrt{a}-\sqrt{b}\right ) \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} (a+b) d}+\frac{\left (\left (\sqrt{a}-\sqrt{b}\right ) \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} (a+b) d}\\ &=\frac{x}{a+b}+\frac{\left (\sqrt{a}-\sqrt{b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} (a+b) d}-\frac{\left (\sqrt{a}-\sqrt{b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} (a+b) d}-\frac{\left (\sqrt{a}+\sqrt{b}\right ) \sqrt [4]{b} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{b} \tan ^2(c+d x)\right )}{4 \sqrt{2} a^{3/4} (a+b) d}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) \sqrt [4]{b} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{b} \tan ^2(c+d x)\right )}{4 \sqrt{2} a^{3/4} (a+b) d}\\ \end{align*}

Mathematica [A]  time = 0.540501, size = 228, normalized size = 0.75 \[ \frac{8 a^{3/4} \tan ^{-1}(\tan (c+d x))+\sqrt{2} \sqrt [4]{b} \left (2 \left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )-2 \left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )-\left (\sqrt{a}+\sqrt{b}\right ) \left (\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )-\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right )\right )\right )}{8 a^{3/4} d (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x]^4)^(-1),x]

[Out]

(8*a^(3/4)*ArcTan[Tan[c + d*x]] + Sqrt[2]*b^(1/4)*(2*(Sqrt[a] - Sqrt[b])*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d
*x])/a^(1/4)] - 2*(Sqrt[a] - Sqrt[b])*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)] - (Sqrt[a] + Sqrt[b])
*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)
*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])))/(8*a^(3/4)*(a + b)*d)

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Maple [A]  time = 0.031, size = 374, normalized size = 1.2 \begin{align*}{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) }}+{\frac{b\sqrt{2}}{8\,d \left ( a+b \right ) a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+\sqrt [4]{{\frac{a}{b}}}\tan \left ( dx+c \right ) \sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}-\sqrt [4]{{\frac{a}{b}}}\tan \left ( dx+c \right ) \sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{2}}{4\,d \left ( a+b \right ) a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\tan \left ( dx+c \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{b\sqrt{2}}{4\,d \left ( a+b \right ) a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ( -{\tan \left ( dx+c \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{\sqrt{2}}{8\,d \left ( a+b \right ) }\ln \left ({ \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}-\sqrt [4]{{\frac{a}{b}}}\tan \left ( dx+c \right ) \sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+\sqrt [4]{{\frac{a}{b}}}\tan \left ( dx+c \right ) \sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{4\,d \left ( a+b \right ) }\arctan \left ({\tan \left ( dx+c \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}}{4\,d \left ( a+b \right ) }\arctan \left ( -{\tan \left ( dx+c \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/(a+b)*arctan(tan(d*x+c))+1/8/d*b/(a+b)*(a/b)^(1/4)/a*2^(1/2)*ln((tan(d*x+c)^2+(a/b)^(1/4)*tan(d*x+c)*2^(1/
2)+(a/b)^(1/2))/(tan(d*x+c)^2-(a/b)^(1/4)*tan(d*x+c)*2^(1/2)+(a/b)^(1/2)))+1/4/d*b/(a+b)*(a/b)^(1/4)/a*2^(1/2)
*arctan(2^(1/2)/(a/b)^(1/4)*tan(d*x+c)+1)-1/4/d*b/(a+b)*(a/b)^(1/4)/a*2^(1/2)*arctan(-2^(1/2)/(a/b)^(1/4)*tan(
d*x+c)+1)-1/8/d/(a+b)/(a/b)^(1/4)*2^(1/2)*ln((tan(d*x+c)^2-(a/b)^(1/4)*tan(d*x+c)*2^(1/2)+(a/b)^(1/2))/(tan(d*
x+c)^2+(a/b)^(1/4)*tan(d*x+c)*2^(1/2)+(a/b)^(1/2)))-1/4/d/(a+b)/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)
*tan(d*x+c)+1)+1/4/d/(a+b)/(a/b)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(a/b)^(1/4)*tan(d*x+c)+1)

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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+tan(d*x+c)^4*b),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.99136, size = 3252, normalized size = 10.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a + b)*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a
^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log((2*(a^3 - a*b^2)*d*sqrt(((a^3 + 2*a^2*b + a*
b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 +
2*a^2*b + a*b^2)*d^2))*tan(d*x + c) + (a*b - b^2)*tan(d*x + c)^2 + a^2 - a*b + ((a^4 + 2*a^3*b + a^2*b^2)*d^2
*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4
*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) - (a + b)*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*
a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*lo
g(-(2*(a^3 - a*b^2)*d*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*
b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) - (a*b - b^2)*tan(d*x + c)
^2 - a^2 + a*b - ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b -
2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) + (a + b)*sqrt
(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4
)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log(-(2*(a^3 - a*b^2)*d*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(
-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b
^2)*d^2))*tan(d*x + c) + (a*b - b^2)*tan(d*x + c)^2 + a^2 - a*b - ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^
2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3
*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) - (a + b)*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/
((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log((2*(a^3 - a
*b^2)*d*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b
^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) - (a*b - b^2)*tan(d*x + c)^2 - a^2 + a*
b + ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^
3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) + 8*x)/(a + b)

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Sympy [A]  time = 24.9238, size = 2759, normalized size = 9.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x/tan(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((x + 1/(d*tan(c + d*x)) - 1/(3*d*tan(c + d*x)**3
))/b, Eq(a, 0)), (-4*d*x*tan(c + d*x)**2/(8*b*d*tan(c + d*x)**2 + 8*b*d) - 4*d*x/(8*b*d*tan(c + d*x)**2 + 8*b*
d) + log(tan(c + d*x) - 1)*tan(c + d*x)**2/(8*b*d*tan(c + d*x)**2 + 8*b*d) + log(tan(c + d*x) - 1)/(8*b*d*tan(
c + d*x)**2 + 8*b*d) - log(tan(c + d*x) + 1)*tan(c + d*x)**2/(8*b*d*tan(c + d*x)**2 + 8*b*d) - log(tan(c + d*x
) + 1)/(8*b*d*tan(c + d*x)**2 + 8*b*d) - 2*tan(c + d*x)/(8*b*d*tan(c + d*x)**2 + 8*b*d), Eq(a, -b)), (x/(a + b
*tan(c)**4), Eq(d, 0)), (x/a, Eq(b, 0)), (-2*(-1)**(1/4)*a**(33/4)*b**4*(atan((-1)**(3/4)*tan(c + d*x)/(a**(1/
4)*(1/b)**(1/4))) + pi*floor((c + d*x - pi/2)/pi)*sign((-1)**(3/4)/(a**(1/4)*(1/b)**(1/4))))*(1/b)**(21/4)/(-4
*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12
*a**9*d/b - 8*a**8*d + 4*a**7*b*d) - (-1)**(1/4)*a**(33/4)*b*(1/b)**(9/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4
) + tan(c + d*x))/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**
4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) + (-1)**(1/4)*a**(33/4)*(1/b)**(5/4)*log(-(-1)**(1/4)*
a**(1/4)*(1/b)**(1/4) + tan(c + d*x))/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2)
+ 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) + 8*(-1)**(3/4)*a**(31/4)*b**5*(at
an((-1)**(3/4)*tan(c + d*x)/(a**(1/4)*(1/b)**(1/4))) + pi*floor((c + d*x - pi/2)/pi)*sign((-1)**(3/4)/(a**(1/4
)*(1/b)**(1/4))))*(1/b)**(23/4)/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I
*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) + 2*(-1)**(3/4)*a**(31/4)*b**2*(1/b)**(1
1/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + tan(c + d*x))/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)
*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) - 2*(-1)**(3/
4)*a**(31/4)*b*(1/b)**(7/4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + tan(c + d*x))/(-4*I*a**(19/2)*b**2*d*(1/b
)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4
*a**7*b*d) + 12*(-1)**(1/4)*a**(29/4)*b**5*(atan((-1)**(3/4)*tan(c + d*x)/(a**(1/4)*(1/b)**(1/4))) + pi*floor(
(c + d*x - pi/2)/pi)*sign((-1)**(3/4)/(a**(1/4)*(1/b)**(1/4))))*(1/b)**(21/4)/(-4*I*a**(19/2)*b**2*d*(1/b)**(7
/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7
*b*d) - 2*(-1)**(3/4)*a**(27/4)*b**7*(atan((-1)**(3/4)*tan(c + d*x)/(a**(1/4)*(1/b)**(1/4))) + pi*floor((c + d
*x - pi/2)/pi)*sign((-1)**(3/4)/(a**(1/4)*(1/b)**(1/4))))*(1/b)**(27/4)/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) +
8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d)
- 6*(-1)**(3/4)*a**(27/4)*b**6*(atan((-1)**(3/4)*tan(c + d*x)/(a**(1/4)*(1/b)**(1/4))) + pi*floor((c + d*x - p
i/2)/pi)*sign((-1)**(3/4)/(a**(1/4)*(1/b)**(1/4))))*(1/b)**(23/4)/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a*
*(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) + 2*(-
1)**(3/4)*a**(27/4)*b**3*(1/b)**(11/4)*log((-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + tan(c + d*x))/(-4*I*a**(19/2)*b
**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*
a**8*d + 4*a**7*b*d) - 2*(-1)**(3/4)*a**(27/4)*b**2*(1/b)**(7/4)*log(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + tan(
c + d*x))/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b
)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) - 2*(-1)**(1/4)*a**(25/4)*b**6*(atan((-1)**(3/4)*tan(c + d*x)/
(a**(1/4)*(1/b)**(1/4))) + pi*floor((c + d*x - pi/2)/pi)*sign((-1)**(3/4)/(a**(1/4)*(1/b)**(1/4))))*(1/b)**(21
/4)/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/
2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) + (-1)**(1/4)*a**(25/4)*b**3*(1/b)**(9/4)*log((-1)**(1/4)*a**(1/4)*(
1/b)**(1/4) + tan(c + d*x))/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**
(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) - (-1)**(1/4)*a**(25/4)*b**2*(1/b)**(5/4)*lo
g(-(-1)**(1/4)*a**(1/4)*(1/b)**(1/4) + tan(c + d*x))/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*
d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) - 4*I*a**(17/2)*b**
2*d*x*(1/b)**(7/2)/(-4*I*a**(19/2)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b*
*4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a**8*d + 4*a**7*b*d) + 12*I*a**(15/2)*b**3*d*x*(1/b)**(7/2)/(-4*I*a**(19/2
)*b**2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b -
8*a**8*d + 4*a**7*b*d) - 12*a**8*d*x/(-4*I*a**(19/2)*b**3*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**4*d*(1/b)**(7/2)
+ 12*I*a**(15/2)*b**5*d*(1/b)**(7/2) - 12*a**9*d - 8*a**8*b*d + 4*a**7*b**2*d) + 4*a**7*d*x/(-4*I*a**(19/2)*b*
*2*d*(1/b)**(7/2) + 8*I*a**(17/2)*b**3*d*(1/b)**(7/2) + 12*I*a**(15/2)*b**4*d*(1/b)**(7/2) - 12*a**9*d/b - 8*a
**8*d + 4*a**7*b*d), True))

________________________________________________________________________________________

Giac [A]  time = 2.27099, size = 478, normalized size = 1.58 \begin{align*} \frac{\frac{2 \,{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} - \left (a b^{3}\right )^{\frac{3}{4}}\right )}{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \tan \left (d x + c\right )\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )\right )}}{\sqrt{2} a^{2} b^{2} + \sqrt{2} a b^{3}} + \frac{2 \,{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} - \left (a b^{3}\right )^{\frac{3}{4}}\right )}{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \tan \left (d x + c\right )\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )\right )}}{\sqrt{2} a^{2} b^{2} + \sqrt{2} a b^{3}} + \frac{{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} + \left (a b^{3}\right )^{\frac{3}{4}}\right )} \log \left (\tan \left (d x + c\right )^{2} + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} \tan \left (d x + c\right ) + \sqrt{\frac{a}{b}}\right )}{\sqrt{2} a^{2} b^{2} + \sqrt{2} a b^{3}} - \frac{{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} + \left (a b^{3}\right )^{\frac{3}{4}}\right )} \log \left (\tan \left (d x + c\right )^{2} - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} \tan \left (d x + c\right ) + \sqrt{\frac{a}{b}}\right )}{\sqrt{2} a^{2} b^{2} + \sqrt{2} a b^{3}} + \frac{4 \,{\left (d x + c\right )}}{a + b}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*((a*b^3)^(1/4)*b^2 - (a*b^3)^(3/4))*(pi*floor((d*x + c)/pi + 1/2) + arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(
1/4) + 2*tan(d*x + c))/(a/b)^(1/4)))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + 2*((a*b^3)^(1/4)*b^2 - (a*b^3)^(3/4))
*(pi*floor((d*x + c)/pi + 1/2) + arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*tan(d*x + c))/(a/b)^(1/4)))/(sqr
t(2)*a^2*b^2 + sqrt(2)*a*b^3) + ((a*b^3)^(1/4)*b^2 + (a*b^3)^(3/4))*log(tan(d*x + c)^2 + sqrt(2)*(a/b)^(1/4)*t
an(d*x + c) + sqrt(a/b))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) - ((a*b^3)^(1/4)*b^2 + (a*b^3)^(3/4))*log(tan(d*x +
c)^2 - sqrt(2)*(a/b)^(1/4)*tan(d*x + c) + sqrt(a/b))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + 4*(d*x + c)/(a + b))
/d