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

Optimal. Leaf size=199 \[ \frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n}+\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[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

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Rubi [A]  time = 0.13, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n}+\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {2 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{b n} \]

Antiderivative was successfully verified.

[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 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 211

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 329

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 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 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]

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 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 3473

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 3476

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*} \int \frac {\tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 \operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\tan ^{-1}\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*}

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Mathematica [A]  time = 0.24, size = 175, normalized size = 0.88 \[ \frac {2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )-2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )+\sqrt {2} \log \left (\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )-\sqrt {2} \log \left (\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )+8 \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}{4 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   catdef: division by zero

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.03, size = 161, normalized size = 0.81 \[ \frac {2 \left (\sqrt {\tan }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b n}-\frac {\arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right ) \sqrt {2}}{2 b n}-\frac {\arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right ) \sqrt {2}}{2 b n}-\frac {\sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{4 b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 3.31, size = 78, normalized size = 0.39 \[ \frac {2\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{b\,n}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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