\(\int \frac {\tan ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [530]

Optimal result

Integrand size = 23, antiderivative size = 140 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d}-\frac {4 a \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {2 \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3647, 3711, 12, 3620, 3618, 65, 214} \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {4 a \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {2 \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 214

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

Rule 3618

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

Rule 3620

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

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.14 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {3 \sqrt {a-i b} (a+i b) b^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-(a-i b) \left (-3 \sqrt {a+i b} b^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 (a+i b) (2 a-b \tan (c+d x)) \sqrt {a+b \tan (c+d x)}\right )}{3 b^2 \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

(3*Sqrt[a - I*b]*(a + I*b)*b^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (a - I*b)*(-3*Sqrt[a + I*b]*b
^2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*(a + I*b)*(2*a - b*Tan[c + d*x])*Sqrt[a + b*Tan[c + d*x
]]))/(3*b^2*(a^2 + b^2)*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(376\) vs. \(2(116)=232\).

Time = 0.09 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.69

method	result	size
derivativedivides	\(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 a \sqrt {a +b \tan \left (d x +c \right )}+2 b^{2} \left (\frac {-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (\sqrt {a^{2}+b^{2}}-a \right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \sqrt {a^{2}+b^{2}}}+\frac {\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (a -\sqrt {a^{2}+b^{2}}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{2}}\)	\(377\)
default	\(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-2 a \sqrt {a +b \tan \left (d x +c \right )}+2 b^{2} \left (\frac {-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (\sqrt {a^{2}+b^{2}}-a \right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \sqrt {a^{2}+b^{2}}}+\frac {\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (a -\sqrt {a^{2}+b^{2}}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{2}}\)	\(377\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (112) = 224\).

Time = 0.25 (sec) , antiderivative size = 743, normalized size of antiderivative = 5.31 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {3 \, b^{2} d \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left ({\left ({\left (a^{2} + b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a d\right )} \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) - 3 \, b^{2} d \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (-{\left ({\left (a^{2} + b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a d\right )} \sqrt {\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a}{{\left (a^{2} + b^{2}\right )} d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) - 3 \, b^{2} d \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left ({\left ({\left (a^{2} + b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a d\right )} \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) + 3 \, b^{2} d \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (-{\left ({\left (a^{2} + b^{2}\right )} d^{3} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} + a d\right )} \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} d^{2} \sqrt {-\frac {b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} - a}{{\left (a^{2} + b^{2}\right )} d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) - 4 \, \sqrt {b \tan \left (d x + c\right ) + a} {\left (b \tan \left (d x + c\right ) - 2 \, a\right )}}{6 \, b^{2} d} \]

[In]

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

[Out]

-1/6*(3*b^2*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))*log(((a^2
+ b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2
+ b^4)*d^4)) + a)/((a^2 + b^2)*d^2)) + sqrt(b*tan(d*x + c) + a)) - 3*b^2*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a
^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))*log(-((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d
^4)) - a*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2)) + sqrt(b*ta
n(d*x + c) + a)) - 3*b^2*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d
^2))*log(((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a
^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2)) + sqrt(b*tan(d*x + c) + a)) + 3*b^2*d*sqrt(-((a^2 + b^2)*d
^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2))*log(-((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*
a^2*b^2 + b^4)*d^4)) + a*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*
d^2)) + sqrt(b*tan(d*x + c) + a)) - 4*sqrt(b*tan(d*x + c) + a)*(b*tan(d*x + c) - 2*a))/(b^2*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 5.68 (sec) , antiderivative size = 827, normalized size of antiderivative = 5.91 \[ \int \frac {\tan ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{3\,b^2\,d}-\frac {2\,a\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{b^2\,d}+\mathrm {atan}\left (-\frac {b^2\,\sqrt {\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{\frac {16\,b^2}{d}-\frac {64\,a^2\,b^2\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a\,b^3\,d^2\,64{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}+\frac {a^2\,b^2\,\sqrt {\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,128{}\mathrm {i}}{\frac {64\,b^4}{d}+\frac {64\,a^2\,b^2}{d}-\frac {256\,a^2\,b^4\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a^3\,b^3\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}-\frac {256\,a^4\,b^2\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a\,b^5\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}+\frac {128\,a\,b^3\,\sqrt {\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {64\,b^4}{d}+\frac {64\,a^2\,b^2}{d}-\frac {256\,a^2\,b^4\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a^3\,b^3\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}-\frac {256\,a^4\,b^2\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a\,b^5\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}\right )\,\sqrt {\frac {a-b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {b^2\,\sqrt {\frac {1}{a\,d^2-b\,d^2\,1{}\mathrm {i}}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,16{}\mathrm {i}}{\frac {16\,b^2}{d}-\frac {16\,a\,b^2\,d^2}{a\,d^3-b\,d^3\,1{}\mathrm {i}}}+\frac {a\,b^2\,\sqrt {\frac {1}{a\,d^2-b\,d^2\,1{}\mathrm {i}}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,16{}\mathrm {i}}{\frac {b^3\,16{}\mathrm {i}}{d}-\frac {16\,a\,b^2}{d}-\frac {a\,b^3\,d^2\,16{}\mathrm {i}}{a\,d^3-b\,d^3\,1{}\mathrm {i}}+\frac {16\,a^2\,b^2\,d^2}{a\,d^3-b\,d^3\,1{}\mathrm {i}}}\right )\,\sqrt {\frac {1}{a\,d^2-b\,d^2\,1{}\mathrm {i}}}\,1{}\mathrm {i} \]

[In]

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

[Out]

atan((a^2*b^2*(a/(4*a^2*d^2 + 4*b^2*d^2) - (b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*12
8i)/((64*b^4)/d + (64*a^2*b^2)/d - (256*a^2*b^4*d^2)/(4*a^2*d^3 + 4*b^2*d^3) + (a^3*b^3*d^2*256i)/(4*a^2*d^3 +
4*b^2*d^3) - (256*a^4*b^2*d^2)/(4*a^2*d^3 + 4*b^2*d^3) + (a*b^5*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3)) - (b^2*(a/
(4*a^2*d^2 + 4*b^2*d^2) - (b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((16*b^2)/d -
(64*a^2*b^2*d^2)/(4*a^2*d^3 + 4*b^2*d^3) + (a*b^3*d^2*64i)/(4*a^2*d^3 + 4*b^2*d^3)) + (128*a*b^3*(a/(4*a^2*d^2
+ 4*b^2*d^2) - (b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((64*b^4)/d + (64*a^2*b^2)/d
- (256*a^2*b^4*d^2)/(4*a^2*d^3 + 4*b^2*d^3) + (a^3*b^3*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a^4*b^2*d^2)/
(4*a^2*d^3 + 4*b^2*d^3) + (a*b^5*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3)))*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2
)*2i - atan((b^2*(1/(a*d^2 - b*d^2*1i))^(1/2)*(a + b*tan(c + d*x))^(1/2)*16i)/((16*b^2)/d - (16*a*b^2*d^2)/(a*
d^3 - b*d^3*1i)) + (a*b^2*(1/(a*d^2 - b*d^2*1i))^(1/2)*(a + b*tan(c + d*x))^(1/2)*16i)/((b^3*16i)/d - (16*a*b^
2)/d - (a*b^3*d^2*16i)/(a*d^3 - b*d^3*1i) + (16*a^2*b^2*d^2)/(a*d^3 - b*d^3*1i)))*(1/(a*d^2 - b*d^2*1i))^(1/2)
*1i + (2*(a + b*tan(c + d*x))^(3/2))/(3*b^2*d) - (2*a*(a + b*tan(c + d*x))^(1/2))/(b^2*d)