3.6.85 \(\int \frac {\tan ^{\frac {3}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx\) [585]

Optimal. Leaf size=232 \[ \frac {(a-b) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 a^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \left (a^2+b^2\right ) d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \]

[Out]

-1/2*(a-b)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/2*(a-b)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2)
)/(a^2+b^2)/d*2^(1/2)+1/4*(a+b)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)-1/4*(a+b)*ln(1+2
^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)+2*a^(3/2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/(a^
2+b^2)/d/b^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3654, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} \frac {(a-b) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a-b) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a+b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(a+b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 a^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x]),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

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

Rule 211

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

Rule 631

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 642

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]

Rule 1176

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 1179

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 1182

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 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3654

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1
/(c^2 + d^2), Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e +
f*x]], x], x] + Dist[(b*c - a*d)^2/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan
[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2
, 0]

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.29, size = 227, normalized size = 0.98 \begin {gather*} \frac {3 a \left (2 \sqrt {2} \sqrt {b} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \sqrt {2} \sqrt {b} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+8 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+\sqrt {2} \sqrt {b} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\sqrt {2} \sqrt {b} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )+8 b^{3/2} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 \sqrt {b} \left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x]),x]

[Out]

(3*a*(2*Sqrt[2]*Sqrt[b]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*Sqrt[b]*ArcTan[1 + Sqrt[2]*Sqrt[Tan
[c + d*x]]] + 8*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + Sqrt[2]*Sqrt[b]*Log[1 - Sqrt[2]*Sqrt[Ta
n[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Sqrt[b]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + 8*b^(3/2)*
Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2))/(12*Sqrt[b]*(a^2 + b^2)*d)

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Maple [A]
time = 0.12, size = 225, normalized size = 0.97

method	result	size
derivativedivides	\(\frac {\frac {2 a^{2} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(225\)
default	\(\frac {\frac {2 a^{2} \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{a^{2}+b^{2}}}{d}\)	\(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 171, normalized size = 0.74 \begin {gather*} \frac {\frac {8 \, a^{2} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} + b^{2}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{2} + b^{2}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3491 vs. \(2 (194) = 388\).
time = 8.68, size = 7094, normalized size = 30.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

[-1/4*(4*sqrt(2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^5*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 +
a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(((a^8 + 2*a^6*
b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))
*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d^7*sqrt((
a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*
d^4)) + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 +
4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*
b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^6*b - a^4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4
))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3
+ a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*
(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2
*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((a^11 + 3*a^9*b^2 + 2*a^7*b^4 - 2*a^5*b^6 - 3*a^3*b^8 - a*b^10)*d^7*sqrt((a
^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d
^4)) + (a^8*b + 2*a^6*b^3 - 2*a^2*b^7 - b^9)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 +
4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b
^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3
/4))/(a^4 - 2*a^2*b^2 + b^4)) + 4*sqrt(2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^5*sqrt((a^4 + 2*a^2*b^2 + b^4
- 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4
- 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3
/4)*arctan(-((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^
4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a
^3*b^6 + a*b^8)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1
/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)
*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2
*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^6*b - a^4*b^3 - a^2*b^5 + b^7)*d^3*sqrt(1/((
a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 +
b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(s
in(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(
d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((a^11 + 3*a^9*b^2 + 2*a^7*b^4 - 2*a^5*b^6 - 3*a^3
*b^8 - a*b^10)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/
((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^8*b + 2*a^6*b^3 - 2*a^2*b^7 - b^9)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*
d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 +
2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4)) + sqrt(2)*(2*(a^3*b + a*b^3)*d^3*sqrt(1/((a^4 + 2*a^2*
b^2 + b^4)*d^4)) + (a^2 + b^2)*d)*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4
+ 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((a^6 - a^4*b^
2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^6*b - a^4*b^3 - a^2*b^
5 + b^7)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c))*sq
rt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*
a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b
^4)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*(2*(a^3*b + a*b^3)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^2
+ b^2)*d)*sqrt((a^4 + 2*a^2*b^2 + b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)
))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{\frac {3}{2}}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**(3/2)/(a+b*tan(d*x+c)),x)

[Out]

Integral(tan(c + d*x)**(3/2)/(a + b*tan(c + d*x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 5.26, size = 2500, normalized size = 10.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^(3/2)/(a + b*tan(c + d*x)),x)

[Out]

- atan(((((1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(((32*(4*a^2*b^6*d^4 + 8*a^4*b^4*d^4 + 4*a^6*b^2*d^4
))/d^5 - (32*tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 -
16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*tan(c + d*x)^(1/
2)*(14*a*b^6*d^2 - 4*a^3*b^4*d^2 + 14*a^5*b^2*d^2))/d^4) + (32*(a*b^5*d^2 + 4*a^5*b*d^2 - 15*a^3*b^3*d^2))/d^5
)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*tan(c + d*x)^(1/2)*(2*a^4*b + b^5))/d^4)*(1i/(4*(b^2*d
^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*1i - (((1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(((32*(4*a^2*b^6*d^4
+ 8*a^4*b^4*d^4 + 4*a^6*b^2*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2
)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i
)))^(1/2) - (32*tan(c + d*x)^(1/2)*(14*a*b^6*d^2 - 4*a^3*b^4*d^2 + 14*a^5*b^2*d^2))/d^4) + (32*(a*b^5*d^2 + 4*
a^5*b*d^2 - 15*a^3*b^3*d^2))/d^5)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*tan(c + d*x)^(1/2)*(2*
a^4*b + b^5))/d^4)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*1i)/((((1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*
2i)))^(1/2)*(((32*(4*a^2*b^6*d^4 + 8*a^4*b^4*d^4 + 4*a^6*b^2*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(1i/(4*(b^2*d^
2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(1i/(4
*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*tan(c + d*x)^(1/2)*(14*a*b^6*d^2 - 4*a^3*b^4*d^2 + 14*a^5*b^2*
d^2))/d^4) + (32*(a*b^5*d^2 + 4*a^5*b*d^2 - 15*a^3*b^3*d^2))/d^5)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1
/2) - (32*tan(c + d*x)^(1/2)*(2*a^4*b + b^5))/d^4)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + (((1i/(4*
(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(((32*(4*a^2*b^6*d^4 + 8*a^4*b^4*d^4 + 4*a^6*b^2*d^4))/d^5 + (32*tan(
c + d*x)^(1/2)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 -
16*a^6*b^3*d^4))/d^4)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - (32*tan(c + d*x)^(1/2)*(14*a*b^6*d^2
- 4*a^3*b^4*d^2 + 14*a^5*b^2*d^2))/d^4) + (32*(a*b^5*d^2 + 4*a^5*b*d^2 - 15*a^3*b^3*d^2))/d^5)*(1i/(4*(b^2*d^2
- a^2*d^2 + a*b*d^2*2i)))^(1/2) + (32*tan(c + d*x)^(1/2)*(2*a^4*b + b^5))/d^4)*(1i/(4*(b^2*d^2 - a^2*d^2 + a*
b*d^2*2i)))^(1/2) + (64*a^2*b^2)/d^5))*(1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*2i - atan((((1/(b^2*d^2
*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(((((((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((32*(4*a^2*b^6*d^4
+ 8*a^4*b^4*d^4 + 4*a^6*b^2*d^4))/d^5 - (16*tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2
)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4))/2 + (32*tan(c + d*x)^(1/2)*(14*a*b^6*
d^2 - 4*a^3*b^4*d^2 + 14*a^5*b^2*d^2))/d^4)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 + (32*(a*b^5*d^
2 + 4*a^5*b*d^2 - 15*a^3*b^3*d^2))/d^5)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - (32*tan(c + d*x)^
(1/2)*(2*a^4*b + b^5))/d^4)*1i)/2 - ((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(((((((1/(b^2*d^2*1i - a^
2*d^2*1i + 2*a*b*d^2))^(1/2)*((32*(4*a^2*b^6*d^4 + 8*a^4*b^4*d^4 + 4*a^6*b^2*d^4))/d^5 + (16*tan(c + d*x)^(1/2
)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d
^4))/d^4))/2 - (32*tan(c + d*x)^(1/2)*(14*a*b^6*d^2 - 4*a^3*b^4*d^2 + 14*a^5*b^2*d^2))/d^4)*(1/(b^2*d^2*1i - a
^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 + (32*(a*b^5*d^2 + 4*a^5*b*d^2 - 15*a^3*b^3*d^2))/d^5)*(1/(b^2*d^2*1i - a^2*d
^2*1i + 2*a*b*d^2))^(1/2))/2 + (32*tan(c + d*x)^(1/2)*(2*a^4*b + b^5))/d^4)*1i)/2)/(((1/(b^2*d^2*1i - a^2*d^2*
1i + 2*a*b*d^2))^(1/2)*(((((((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((32*(4*a^2*b^6*d^4 + 8*a^4*b^4*d
^4 + 4*a^6*b^2*d^4))/d^5 - (16*tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^9*d^4
+ 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4))/2 + (32*tan(c + d*x)^(1/2)*(14*a*b^6*d^2 - 4*a^3*b^
4*d^2 + 14*a^5*b^2*d^2))/d^4)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 + (32*(a*b^5*d^2 + 4*a^5*b*d^
2 - 15*a^3*b^3*d^2))/d^5)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - (32*tan(c + d*x)^(1/2)*(2*a^4*b
+ b^5))/d^4))/2 + ((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(((((((1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*
d^2))^(1/2)*((32*(4*a^2*b^6*d^4 + 8*a^4*b^4*d^4 + 4*a^6*b^2*d^4))/d^5 + (16*tan(c + d*x)^(1/2)*(1/(b^2*d^2*1i
- a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4))/2 - (3
2*tan(c + d*x)^(1/2)*(14*a*b^6*d^2 - 4*a^3*b^4*d^2 + 14*a^5*b^2*d^2))/d^4)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b
*d^2))^(1/2))/2 + (32*(a*b^5*d^2 + 4*a^5*b*d^2 - 15*a^3*b^3*d^2))/d^5)*(1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2
))^(1/2))/2 + (32*tan(c + d*x)^(1/2)*(2*a^4*b + b^5))/d^4))/2 + (64*a^2*b^2)/d^5))*(1/(b^2*d^2*1i - a^2*d^2*1i
+ 2*a*b*d^2))^(1/2)*1i - (atan((((-a^3*b)^(1/2)*((((32*(a*b^5*d^2 + 4*a^5*b*d^2 - 15*a^3*b^3*d^2))/d^5 + ((-a
^3*b)^(1/2)*((((32*(4*a^2*b^6*d^4 + 8*a^4*b^4*d...

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