\(\int \frac {\tan ^{\frac {3}{2}}(a+b \log (c x^n))}{x} \, dx\) [181]

Optimal result

Integrand size = 19, antiderivative size = 199 \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3554, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n}+\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[In]

Int[Tan[a + b*Log[c*x^n]]^(3/2)/x,x]

[Out]

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

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2}}+2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {2 \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\)	\(139\)
default	\(\frac {2 \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\)	\(139\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.63 \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}\right ) + i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}\right ) - i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (-i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}\right ) - b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (-b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}\right ) - 4 \, \sqrt {\frac {\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}}}{2 \, b n} \]

[In]

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

[Out]

-1/2*(b*n*(-1/(b^4*n^4))^(1/4)*log(b*n*(-1/(b^4*n^4))^(1/4) + sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(cos(2
*b*n*log(x) + 2*b*log(c) + 2*a) + 1))) + I*b*n*(-1/(b^4*n^4))^(1/4)*log(I*b*n*(-1/(b^4*n^4))^(1/4) + sqrt(sin(
2*b*n*log(x) + 2*b*log(c) + 2*a)/(cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1))) - I*b*n*(-1/(b^4*n^4))^(1/4)*log
(-I*b*n*(-1/(b^4*n^4))^(1/4) + sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(cos(2*b*n*log(x) + 2*b*log(c) + 2*a)
+ 1))) - b*n*(-1/(b^4*n^4))^(1/4)*log(-b*n*(-1/(b^4*n^4))^(1/4) + sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(
cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1))) - 4*sqrt(sin(2*b*n*log(x) + 2*b*log(c) + 2*a)/(cos(2*b*n*log(x) +
2*b*log(c) + 2*a) + 1)))/(b*n)

Sympy [F]

\[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\tan ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\tan \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x} \,d x } \]

[In]

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

[Out]

integrate(tan(b*log(c*x^n) + a)^(3/2)/x, x)

Giac [F(-1)]

Timed out. \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 29.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.39 \[ \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{b\,n}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n} \]

[In]

int(tan(a + b*log(c*x^n))^(3/2)/x,x)

[Out]

(2*tan(a + b*log(c*x^n))^(1/2))/(b*n) + ((-1)^(1/4)*atan((-1)^(1/4)*tan(a + b*log(c*x^n))^(1/2))*1i)/(b*n) + (
(-1)^(1/4)*atanh((-1)^(1/4)*tan(a + b*log(c*x^n))^(1/2))*1i)/(b*n)