3.538 \(\int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=226 \[ -\frac{2 a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left (4 a^2+b^2\right ) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 d \left (a^2+b^2\right )}-\frac{2 a \left (8 a^2+5 b^2\right ) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left (a^2+b^2\right )}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{3/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{3/2}} \]

[Out]

((-I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x
]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*a^2*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) - (
2*a*(8*a^2 + 5*b^2)*Sqrt[a + b*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)*d) + (2*(4*a^2 + b^2)*Tan[c + d*x]*Sqrt[a + b
*Tan[c + d*x]])/(3*b^2*(a^2 + b^2)*d)

________________________________________________________________________________________

Rubi [A]  time = 0.450952, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3565, 3647, 3630, 3539, 3537, 63, 208} \[ -\frac{2 a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left (4 a^2+b^2\right ) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 d \left (a^2+b^2\right )}-\frac{2 a \left (8 a^2+5 b^2\right ) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left (a^2+b^2\right )}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{3/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^(3/2),x]

[Out]

((-I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x
]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*a^2*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) - (
2*a*(8*a^2 + 5*b^2)*Sqrt[a + b*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)*d) + (2*(4*a^2 + b^2)*Tan[c + d*x]*Sqrt[a + b
*Tan[c + d*x]])/(3*b^2*(a^2 + b^2)*d)

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, 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] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(C*(a + b*Tan[e + f*x])^m*(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 - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3630

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]

Rule 3539

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 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 63

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 208

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

Rubi steps

\begin{align*} \int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx &=-\frac{2 a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d \sqrt{a+b \tan (c+d x)}}+\frac{2 \int \frac{\tan (c+d x) \left (2 a^2-\frac{1}{2} a b \tan (c+d x)+\frac{1}{2} \left (4 a^2+b^2\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac{2 a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d \sqrt{a+b \tan (c+d x)}}+\frac{2 \left (4 a^2+b^2\right ) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 \left (a^2+b^2\right ) d}+\frac{4 \int \frac{-\frac{1}{2} a \left (4 a^2+b^2\right )-\frac{3}{4} b^3 \tan (c+d x)-\frac{1}{4} a \left (8 a^2+5 b^2\right ) \tan ^2(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )}\\ &=-\frac{2 a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d \sqrt{a+b \tan (c+d x)}}-\frac{2 a \left (8 a^2+5 b^2\right ) \sqrt{a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right ) d}+\frac{2 \left (4 a^2+b^2\right ) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 \left (a^2+b^2\right ) d}+\frac{4 \int \frac{\frac{3 a b^2}{4}-\frac{3}{4} b^3 \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )}\\ &=-\frac{2 a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d \sqrt{a+b \tan (c+d x)}}-\frac{2 a \left (8 a^2+5 b^2\right ) \sqrt{a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right ) d}+\frac{2 \left (4 a^2+b^2\right ) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 \left (a^2+b^2\right ) d}+\frac{\int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a-i b)}+\frac{\int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a+i b)}\\ &=-\frac{2 a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d \sqrt{a+b \tan (c+d x)}}-\frac{2 a \left (8 a^2+5 b^2\right ) \sqrt{a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right ) d}+\frac{2 \left (4 a^2+b^2\right ) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 \left (a^2+b^2\right ) d}+\frac{\operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (i a-b) d}-\frac{\operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (i a+b) d}\\ &=-\frac{2 a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d \sqrt{a+b \tan (c+d x)}}-\frac{2 a \left (8 a^2+5 b^2\right ) \sqrt{a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right ) d}+\frac{2 \left (4 a^2+b^2\right ) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 \left (a^2+b^2\right ) d}-\frac{\operatorname{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 )}{(a-i b) b d}-\frac{\operatorname{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 )}{(a+i b) b d}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{(a-i b)^{3/2} d}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{(a+i b)^{3/2} d}-\frac{2 a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d \sqrt{a+b \tan (c+d x)}}-\frac{2 a \left (8 a^2+5 b^2\right ) \sqrt{a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right ) d}+\frac{2 \left (4 a^2+b^2\right ) \tan (c+d x) \sqrt{a+b \tan (c+d x)}}{3 b^2 \left (a^2+b^2\right ) d}\\ \end{align*}

Mathematica [A]  time = 3.09338, size = 252, normalized size = 1.12 \[ \frac{-\frac{2 a^2 \left (8 a^2+5 b^2\right )}{b^2 \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}+\frac{3 b^2 \left (\frac{a}{\sqrt{-b^2}}+1\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right )}{\left (a^2+b^2\right ) \sqrt{a-\sqrt{-b^2}}}+\frac{3 \left (a \sqrt{-b^2}+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right )}{\left (a^2+b^2\right ) \sqrt{a+\sqrt{-b^2}}}+\frac{2 \tan ^2(c+d x)}{\sqrt{a+b \tan (c+d x)}}-\frac{8 a \tan (c+d x)}{b \sqrt{a+b \tan (c+d x)}}}{3 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^(3/2),x]

[Out]

((3*b^2*(1 + a/Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/((a^2 + b^2)*Sqrt[a - Sqrt[
-b^2]]) + (3*(b^2 + a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/((a^2 + b^2)*Sqrt[a
+ Sqrt[-b^2]]) - (2*a^2*(8*a^2 + 5*b^2))/(b^2*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]]) - (8*a*Tan[c + d*x])/(b*Sq
rt[a + b*Tan[c + d*x]]) + (2*Tan[c + d*x]^2)/Sqrt[a + b*Tan[c + d*x]])/(3*b*d)

________________________________________________________________________________________

Maple [B]  time = 0.05, size = 1978, normalized size = 8.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^4/(a+b*tan(d*x+c))^(3/2),x)

[Out]

-1/4/d*b/(a^2+b^2)^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)+2*a)^(1/2)*a+4/d*b/(a^2+b^2)^(5/2)/(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))*a^3+1/4/d/b/(a^2+b^2)^(5/2)*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)+2*a)^(1/2)*a^4+1/
4/d/b/(a^2+b^2)^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^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/4/d*b/(a^2+b^2)^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^2+b^2)^(1/2)+2*a)^(1/2)*a-1/d*b/(a^2+b^2)^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*ar
ctan((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))*a^2-1/4/d/b/(a^2+
b^2)^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)+2*a)^(1/2)*a^3-4/d/b^3*a*(a+b*tan(d*x+c))^(1/2)+2/3/d/b^3*(a+b*tan(d*x+c))^(3/2)-1/d*b/(a^2+b^2)^(3/2)/(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))*a-1/d*b^3/(a^2+b^2)^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))+1/d*b^3/(a^2+b^2)^2/(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))-2/d/b^3*a^4/(a^2+b^2)/(a+b
*tan(d*x+c))^(1/2)+1/4/d*b^3/(a^2+b^2)^(5/2)*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^2+b^2)^(1/2)+2*a)^(1/2)-1/4/d*b^3/(a^2+b^2)^(5/2)*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)+2*a)^(1/2)+1/d*b/(a^2+b^2)^2/(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))*a^2-4/d*b/(a^2+b^2)^(5/2)/(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))*a^3-3/d*b^3/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*a
rctan(((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))*a-1/4/d/b/(a^2+b
^2)^(5/2)*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^2+b^2)
^(1/2)+2*a)^(1/2)*a^4-1/d/b/(a^2+b^2)^(3/2)/(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))*a^3+1/d/b/(a^2+b^2)^(5/2)/(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))*a^5+3/d*b^
3/(a^2+b^2)^(5/2)/(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))*a+1/d/b/(a^2+b^2)^(3/2)/(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))*a^3+1/d*b/(a^2+b^2)^(3/2)/(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
))*a-1/d/b/(a^2+b^2)^(5/2)/(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))*a^5

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [B]  time = 4.3246, size = 12614, normalized size = 55.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/12*(12*sqrt(2)*((a^10*b^3 + 3*a^8*b^5 + 2*a^6*b^7 - 2*a^4*b^9 - 3*a^2*b^11 - b^13)*d^5*cos(d*x + c)^3 + 2*(a
^9*b^4 + 4*a^7*b^6 + 6*a^5*b^8 + 4*a^3*b^10 + a*b^12)*d^5*cos(d*x + c)^2*sin(d*x + c) + (a^8*b^5 + 4*a^6*b^7 +
6*a^4*b^9 + 4*a^2*b^11 + b^13)*d^5*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 -
8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sq
rt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^1
2)*d^4))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*arctan(((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a
^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^4*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^
4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*
a^9 + 8*a^7*b^2 + 6*a^5*b^4 - a*b^8)*d^2*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + sqrt(2)*(2*(a^13 + 6*a^11*b^2 + 15*a^9*b^4 + 20*a^7*b^6
+ 15*a^5*b^8 + 6*a^3*b^10 + a*b^12)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (a^10 +
5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^1
0*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b
^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2
- 6*a^2*b^4 + b^6))*sqrt(((9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10)*d^2*sqrt(1/((a^6 + 3*a^4*b^
2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*a^8*b^3 + 12*a^6*b^5 - 2*a^4*b^7 - 4*a^2*b^9 + b^11)*d^3
*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + 2*(9*a^5*b^3 - 6*a^3*b^5 + a*b^7)*d*cos(d*x
+ c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a
^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x
+ c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*a^5*b^2 - 6*a^3*b^4 + a*b^6)*cos(d*x + c) + (9
*a^4*b^3 - 6*a^2*b^5 + b^7)*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4) +
sqrt(2)*(2*(3*a^15*b + 17*a^13*b^3 + 39*a^11*b^5 + 45*a^9*b^7 + 25*a^7*b^9 + 3*a^5*b^11 - 3*a^3*b^13 - a*b^15)
*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^1
0 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^12*b + 14*a^10*b^3 + 25*a^8*b^5 + 20*
a^6*b^7 + 5*a^4*b^9 - 2*a^2*b^11 - b^13)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b
^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^
5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b
^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))
/(9*a^4*b^2 - 6*a^2*b^4 + b^6)) + 12*sqrt(2)*((a^10*b^3 + 3*a^8*b^5 + 2*a^6*b^7 - 2*a^4*b^9 - 3*a^2*b^11 - b^1
3)*d^5*cos(d*x + c)^3 + 2*(a^9*b^4 + 4*a^7*b^6 + 6*a^5*b^8 + 4*a^3*b^10 + a*b^12)*d^5*cos(d*x + c)^2*sin(d*x +
c) + (a^8*b^5 + 4*a^6*b^7 + 6*a^4*b^9 + 4*a^2*b^11 + b^13)*d^5*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^
4*b^2 - 6*a^2*b^4 + b^6))*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 1
5*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*arctan(-((3*a^12 + 14
*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^4*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((
a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*d^4)) + (3*a^9 + 8*a^7*b^2 + 6*a^5*b^4 - a*b^8)*d^2*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^1
2 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*(2*(a^13 + 6*a^11*b
^2 + 15*a^9*b^4 + 20*a^7*b^6 + 15*a^5*b^8 + 6*a^3*b^10 + a*b^12)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12
+ 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a
^2*b^4 + b^6)*d^4)) + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^5*sqrt((9*a^4*b^2 - 6*
a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^
6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2
*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt(((9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10
)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) - sqrt(2)*((9*a^8*b^3 + 12*a^6*b^5 - 2*a^
4*b^7 - 4*a^2*b^9 + b^11)*d^3*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + 2*(9*a^5*b^3 -
6*a^3*b^5 + a*b^7)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*
a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x +
c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*a^5*b^2 - 6*a^3*b
^4 + a*b^6)*cos(d*x + c) + (9*a^4*b^3 - 6*a^2*b^5 + b^7)*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6)*d^4))^(3/4) - sqrt(2)*(2*(3*a^15*b + 17*a^13*b^3 + 39*a^11*b^5 + 45*a^9*b^7 + 25*a^7*b^9 + 3*a^
5*b^11 - 3*a^3*b^13 - a*b^15)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6
*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^12*b + 1
4*a^10*b^3 + 25*a^8*b^5 + 20*a^6*b^7 + 5*a^4*b^9 - 2*a^2*b^11 - b^13)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/(
(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)
))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6)) + 3*sqrt(2)*((a^4*b^3 - b^7)*d*cos(d*x + c)^3 + 2
*(a^3*b^4 + a*b^6)*d*cos(d*x + c)^2*sin(d*x + c) + (a^2*b^5 + b^7)*d*cos(d*x + c) - ((a^7*b^3 - 3*a^5*b^5 - a^
3*b^7 + 3*a*b^9)*d^3*cos(d*x + c)^3 + 2*(a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8)*d^3*cos(d*x + c)^2*sin(d*x + c) + (a
^5*b^5 - 2*a^3*b^7 - 3*a*b^9)*d^3*cos(d*x + c))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((9*a^8
*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*
x + c) + sqrt(2)*((9*a^8*b^3 + 12*a^6*b^5 - 2*a^4*b^7 - 4*a^2*b^9 + b^11)*d^3*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2
*b^4 + b^6)*d^4))*cos(d*x + c) + 2*(9*a^5*b^3 - 6*a^3*b^5 + a*b^7)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a
^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/
(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a
^2*b^4 + b^6)*d^4))^(1/4) + (9*a^5*b^2 - 6*a^3*b^4 + a*b^6)*cos(d*x + c) + (9*a^4*b^3 - 6*a^2*b^5 + b^7)*sin(d
*x + c))/cos(d*x + c)) - 3*sqrt(2)*((a^4*b^3 - b^7)*d*cos(d*x + c)^3 + 2*(a^3*b^4 + a*b^6)*d*cos(d*x + c)^2*si
n(d*x + c) + (a^2*b^5 + b^7)*d*cos(d*x + c) - ((a^7*b^3 - 3*a^5*b^5 - a^3*b^7 + 3*a*b^9)*d^3*cos(d*x + c)^3 +
2*(a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8)*d^3*cos(d*x + c)^2*sin(d*x + c) + (a^5*b^5 - 2*a^3*b^7 - 3*a*b^9)*d^3*cos(
d*x + c))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 -
6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4
+ b^6))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*
b^8 + b^10)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) - sqrt(2)*((9*a^8*b^3 + 12*a^6*
b^5 - 2*a^4*b^7 - 4*a^2*b^9 + b^11)*d^3*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + 2*(9*
a^5*b^3 - 6*a^3*b^5 + a*b^7)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^9 - 6*a^5*b^4 - 8*a^
3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a
*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*a^5*b^2
- 6*a^3*b^4 + a*b^6)*cos(d*x + c) + (9*a^4*b^3 - 6*a^2*b^5 + b^7)*sin(d*x + c))/cos(d*x + c)) - 8*((8*a^5 + 2
*a^3*b^2 - 3*a*b^4)*cos(d*x + c)^3 + 3*(a^3*b^2 + a*b^4)*cos(d*x + c) - (a^2*b^3 + b^5 - (12*a^4*b + 10*a^2*b^
3 + b^5)*cos(d*x + c)^2)*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/((a^4*b^3 - b^7)*
d*cos(d*x + c)^3 + 2*(a^3*b^4 + a*b^6)*d*cos(d*x + c)^2*sin(d*x + c) + (a^2*b^5 + b^7)*d*cos(d*x + c))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{4}{\left (c + d x \right )}}{\left (a + b \tan{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**4/(a+b*tan(d*x+c))**(3/2),x)

[Out]

Integral(tan(c + d*x)**4/(a + b*tan(c + d*x))**(3/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{4}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate(tan(d*x + c)^4/(b*tan(d*x + c) + a)^(3/2), x)