3.7.79 \(\int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx\) [679]

Optimal. Leaf size=163 \[ \frac {3 F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt [3]{\tan (c+d x)} \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 d \sqrt {a+b \tan (c+d x)}}+\frac {3 F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt [3]{\tan (c+d x)} \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 d \sqrt {a+b \tan (c+d x)}} \]

[Out]

3/2*AppellF1(1/3,1,1/2,4/3,-I*tan(d*x+c),-b*tan(d*x+c)/a)*(1+b*tan(d*x+c)/a)^(1/2)*tan(d*x+c)^(1/3)/d/(a+b*tan
(d*x+c))^(1/2)+3/2*AppellF1(1/3,1,1/2,4/3,I*tan(d*x+c),-b*tan(d*x+c)/a)*(1+b*tan(d*x+c)/a)^(1/2)*tan(d*x+c)^(1
/3)/d/(a+b*tan(d*x+c))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3656, 926, 129, 441, 440} \begin {gather*} \frac {3 \sqrt [3]{\tan (c+d x)} \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{2 d \sqrt {a+b \tan (c+d x)}}+\frac {3 \sqrt [3]{\tan (c+d x)} \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{2 d \sqrt {a+b \tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Tan[c + d*x]^(2/3)*Sqrt[a + b*Tan[c + d*x]]),x]

[Out]

(3*AppellF1[1/3, 1, 1/2, 4/3, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1/3)*Sqrt[1 + (b*Tan[c +
d*x])/a])/(2*d*Sqrt[a + b*Tan[c + d*x]]) + (3*AppellF1[1/3, 1, 1/2, 4/3, I*Tan[c + d*x], -((b*Tan[c + d*x])/a
)]*Tan[c + d*x]^(1/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*Sqrt[a + b*Tan[c + d*x]])

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 926

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 3656

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Wit
h[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + b*ff*x)^m*((c + d*ff*x)^n/(1 + ff^2*x^2)), x]
, x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^{2/3} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {i}{2 (i-x) x^{2/3} \sqrt {a+b x}}+\frac {i}{2 x^{2/3} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i \text {Subst}\left (\int \frac {1}{(i-x) x^{2/3} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \text {Subst}\left (\int \frac {1}{x^{2/3} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {(3 i) \text {Subst}\left (\int \frac {1}{\left (i-x^3\right ) \sqrt {a+b x^3}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d}+\frac {(3 i) \text {Subst}\left (\int \frac {1}{\left (i+x^3\right ) \sqrt {a+b x^3}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d}\\ &=\frac {\left (3 i \sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (i-x^3\right ) \sqrt {1+\frac {b x^3}{a}}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 i \sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (i+x^3\right ) \sqrt {1+\frac {b x^3}{a}}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d \sqrt {a+b \tan (c+d x)}}\\ &=\frac {3 F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt [3]{\tan (c+d x)} \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 d \sqrt {a+b \tan (c+d x)}}+\frac {3 F_1\left (\frac {1}{3};1,\frac {1}{2};\frac {4}{3};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt [3]{\tan (c+d x)} \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(6198\) vs. \(2(163)=326\).
time = 92.91, size = 6198, normalized size = 38.02 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Tan[c + d*x]^(2/3)*Sqrt[a + b*Tan[c + d*x]]),x]

[Out]

Result too large to show

________________________________________________________________________________________

Maple [F]
time = 0.41, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a +b \tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(2/3),x)

[Out]

int(1/(a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(2/3),x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(2/3),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*tan(d*x + c) + a)*tan(d*x + c)^(2/3)), x)

________________________________________________________________________________________

Fricas [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(1/(a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(2/3),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{\frac {2}{3}}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))**(1/2)/tan(d*x+c)**(2/3),x)

[Out]

Integral(1/(sqrt(a + b*tan(c + d*x))*tan(c + d*x)**(2/3)), x)

________________________________________________________________________________________

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(1/(a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(2/3),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{2/3}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(tan(c + d*x)^(2/3)*(a + b*tan(c + d*x))^(1/2)),x)

[Out]

int(1/(tan(c + d*x)^(2/3)*(a + b*tan(c + d*x))^(1/2)), x)

________________________________________________________________________________________