3.547 \(\int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.494789, 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, 3635, 3630, 3539, 3537, 63, 208} \[ -\frac{2 a^2 \tan ^2(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{4 a^3 \left (2 a^2+5 b^2\right )}{3 b^3 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 \left (4 a^2+3 b^2\right ) \sqrt{a+b \tan (c+d x)}}{3 b^3 d \left (a^2+b^2\right )}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

((-I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x
]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (2*a^2*Tan[c + d*x]^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2))
+ (4*a^3*(2*a^2 + 5*b^2))/(3*b^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]]) + (2*(4*a^2 + 3*b^2)*Sqrt[a + b*Ta
n[c + d*x]])/(3*b^3*(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 3635

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

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

Mathematica [A]  time = 2.92477, size = 308, normalized size = 1.36 \[ \frac{-\frac{12 a b^4}{\left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}+\frac{2 a^2 \left (8 a^2+9 b^2\right )}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{3 \left (-b^2\right )^{3/2} \left (a^2+2 a \sqrt{-b^2}-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right )}{\left (a^2+b^2\right )^2 \sqrt{a-\sqrt{-b^2}}}+\frac{3 \left (-b^2\right )^{3/2} \left (-a^2+2 a \sqrt{-b^2}+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right )}{\left (a^2+b^2\right )^2 \sqrt{a+\sqrt{-b^2}}}+\frac{6 b^2 \tan ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}}+\frac{24 a b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}}}{3 b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

((3*(-b^2)^(3/2)*(a^2 - b^2 + 2*a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/((a^2 +
b^2)^2*Sqrt[a - Sqrt[-b^2]]) + (3*(-b^2)^(3/2)*(-a^2 + b^2 + 2*a*Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/
Sqrt[a + Sqrt[-b^2]]])/((a^2 + b^2)^2*Sqrt[a + Sqrt[-b^2]]) + (2*a^2*(8*a^2 + 9*b^2))/((a^2 + b^2)*(a + b*Tan[
c + d*x])^(3/2)) + (24*a*b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(3/2) + (6*b^2*Tan[c + d*x]^2)/(a + b*Tan[c + d*
x])^(3/2) - (12*a*b^4)/((a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]]))/(3*b^3*d)

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Maple [B]  time = 0.055, size = 2379, normalized size = 10.5 \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))^(5/2),x)

[Out]

1/4/d/b/(a^2+b^2)^(7/2)*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^5-1/d*b^3/(a^2+b^2)^(7/2)/(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^2-1/4/d/b/(a^2+b^2)^(7/2)*ln(b*ta
n(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^5+1/2/d*b/(a^2+b^2)^(7/2)*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/d/b/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*ta
n(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^4-2/d*b^3/(a^2+b^2)^3/(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-1/4/d/b/(a^2+b^2)^3*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/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^4-1/d/b/(a^2+b^2)^(7/2)/(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^6-4/d*b/(a^2+b^2)^(7/2)/(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^4+1/d/b/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arct
an((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^6-3/4/d*b^3/(a^2+
b^2)^(7/2)*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-2/d*b^5/(a^2+b^2)^(7/2)/(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)^(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))+2/d*b^5/(a^
2+b^2)^(7/2)/(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))-1/4/d*b^3/(a^2+b^2)^3*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)^3*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)+8/d/b*a^3/(a^2+b^2)^2/
(a+b*tan(d*x+c))^(1/2)-2/3/d/b^3*a^4/(a^2+b^2)/(a+b*tan(d*x+c))^(3/2)+4/d/b^3*a^5/(a^2+b^2)^2/(a+b*tan(d*x+c))
^(1/2)+1/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))+2/d*b^3/(a^2+b^2)^3/(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+1/4/d/b/(a^2+b^2)^3*ln(b*ta
n(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^4-2/d*b/(a^2+b^2)^3/(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+2/d*b/(a^2+b^2)^3/(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+3/4/d*b^3/(a^2+b^2)^(7/2)*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-1/2/d*b/(a^2+b^2)^(7/2)*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/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*ta
n(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^4+1/d*b^3/(a^2+b^2)^(7/2)/(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^2+2/d/b^3*(a+b*tan(d*x+c))^(1/2)

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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))^(5/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [B]  time = 7.02158, size = 22831, normalized size = 101.02 \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))^(5/2),x, algorithm="fricas")

[Out]

1/12*(12*sqrt(2)*((a^18*b^3 + a^16*b^5 - 20*a^14*b^7 - 84*a^12*b^9 - 154*a^10*b^11 - 154*a^8*b^13 - 84*a^6*b^1
5 - 20*a^4*b^17 + a^2*b^19 + b^21)*d^5*cos(d*x + c)^4 + 2*(3*a^16*b^5 + 20*a^14*b^7 + 56*a^12*b^9 + 84*a^10*b^
11 + 70*a^8*b^13 + 28*a^6*b^15 - 4*a^2*b^19 - b^21)*d^5*cos(d*x + c)^2 + (a^14*b^7 + 7*a^12*b^9 + 21*a^10*b^11
+ 35*a^8*b^13 + 35*a^6*b^15 + 21*a^4*b^17 + 7*a^2*b^19 + b^21)*d^5 + 4*((a^17*b^4 + 6*a^15*b^6 + 14*a^13*b^8
+ 14*a^11*b^10 - 14*a^7*b^14 - 14*a^5*b^16 - 6*a^3*b^18 - a*b^20)*d^5*cos(d*x + c)^3 + (a^15*b^6 + 7*a^13*b^8
+ 21*a^11*b^10 + 35*a^9*b^12 + 35*a^7*b^14 + 21*a^5*b^16 + 7*a^3*b^18 + a*b^20)*d^5*cos(d*x + c))*sin(d*x + c)
)*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*
a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((25*a^8*b^2
- 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*
b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((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^4))^(3/4)*arctan(((5*a^20 + 30*a^18*b^2 + 61*a^16*b^4 + 8*
a^14*b^6 - 182*a^12*b^8 - 364*a^10*b^10 - 350*a^8*b^12 - 184*a^6*b^14 - 47*a^4*b^16 - 2*a^2*b^18 + b^20)*d^4*s
qrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14
*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sq
rt(1/((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^4)) + (5*a^15 + 15*a^13*b^2 + a^11*b^4
- 45*a^9*b^6 - 65*a^7*b^8 - 35*a^5*b^10 - 5*a^3*b^12 + a*b^14)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b
^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210
*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) - sqrt(2)*((3*a^22 + 29*a^20*b^2 + 125*a^18
*b^4 + 315*a^16*b^6 + 510*a^14*b^8 + 546*a^12*b^10 + 378*a^10*b^12 + 150*a^8*b^14 + 15*a^6*b^16 - 15*a^4*b^18
- 7*a^2*b^20 - b^22)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^
2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10
*a^2*b^18 + b^20)*d^4))*sqrt(1/((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^4)) + 2*(a^1
7 + 8*a^15*b^2 + 28*a^13*b^4 + 56*a^11*b^6 + 70*a^9*b^8 + 56*a^7*b^10 + 28*a^5*b^12 + 8*a^3*b^14 + a*b^16)*d^5
*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^
14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))
*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a
^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(((25*a^14*b^2
- 25*a^12*b^4 - 115*a^10*b^6 + 35*a^8*b^8 + 171*a^6*b^10 + 53*a^4*b^12 - 17*a^2*b^14 + b^16)*d^2*sqrt(1/((a^1
0 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + sqrt(2)*(2*(25*a^15*b^3 - 25*
a^13*b^5 - 115*a^11*b^7 + 35*a^9*b^9 + 171*a^7*b^11 + 53*a^5*b^13 - 17*a^3*b^15 + a*b^17)*d^3*sqrt(1/((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^4))*cos(d*x + c) + (75*a^10*b^3 - 325*a^8*b^5 + 430*
a^6*b^7 - 170*a^4*b^9 + 23*a^2*b^11 - b^13)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 +
5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*
b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6
*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((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^4))^(1/4) + (25*a^9*b^2 - 100*a^7*b^4 + 110*a^5*b^6 - 20
*a^3*b^8 + a*b^10)*cos(d*x + c) + (25*a^8*b^3 - 100*a^6*b^5 + 110*a^4*b^7 - 20*a^2*b^9 + b^11)*sin(d*x + c))/c
os(d*x + c))*(1/((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^4))^(3/4) + sqrt(2)*((15*a^
26*b + 115*a^24*b^3 + 338*a^22*b^5 + 354*a^20*b^7 - 475*a^18*b^9 - 2055*a^16*b^11 - 3060*a^14*b^13 - 2484*a^12
*b^15 - 1047*a^10*b^17 - 75*a^8*b^19 + 130*a^6*b^21 + 50*a^4*b^23 + 3*a^2*b^25 - b^27)*d^7*sqrt((25*a^8*b^2 -
100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^
8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((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^4)) + 2*(5*a^21*b + 30*a^19*b^3 + 61*a^17*b^5 + 8*a^15*b
^7 - 182*a^13*b^9 - 364*a^11*b^11 - 350*a^9*b^13 - 184*a^7*b^15 - 47*a^5*b^17 - 2*a^3*b^19 + a*b^21)*d^5*sqrt(
(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6
+ 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt(
(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6
- 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b
*sin(d*x + c))/cos(d*x + c))*(1/((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^4))^(3/4))/
(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)) + 12*sqrt(2)*((a^18*b^3 + a^16*b^5 - 20*a^14*b^7
- 84*a^12*b^9 - 154*a^10*b^11 - 154*a^8*b^13 - 84*a^6*b^15 - 20*a^4*b^17 + a^2*b^19 + b^21)*d^5*cos(d*x + c)^
4 + 2*(3*a^16*b^5 + 20*a^14*b^7 + 56*a^12*b^9 + 84*a^10*b^11 + 70*a^8*b^13 + 28*a^6*b^15 - 4*a^2*b^19 - b^21)*
d^5*cos(d*x + c)^2 + (a^14*b^7 + 7*a^12*b^9 + 21*a^10*b^11 + 35*a^8*b^13 + 35*a^6*b^15 + 21*a^4*b^17 + 7*a^2*b
^19 + b^21)*d^5 + 4*((a^17*b^4 + 6*a^15*b^6 + 14*a^13*b^8 + 14*a^11*b^10 - 14*a^7*b^14 - 14*a^5*b^16 - 6*a^3*b
^18 - a*b^20)*d^5*cos(d*x + c)^3 + (a^15*b^6 + 7*a^13*b^8 + 21*a^11*b^10 + 35*a^9*b^12 + 35*a^7*b^14 + 21*a^5*
b^16 + 7*a^3*b^18 + a*b^20)*d^5*cos(d*x + c))*sin(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 +
5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*
b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6
*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a
^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 +
45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((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^
4))^(3/4)*arctan(-((5*a^20 + 30*a^18*b^2 + 61*a^16*b^4 + 8*a^14*b^6 - 182*a^12*b^8 - 364*a^10*b^10 - 350*a^8*b
^12 - 184*a^6*b^14 - 47*a^4*b^16 - 2*a^2*b^18 + b^20)*d^4*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^
2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12
+ 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((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^4)) + (5*a^15 + 15*a^13*b^2 + a^11*b^4 - 45*a^9*b^6 - 65*a^7*b^8 - 35*a^5*b^10 - 5*a^3*b^
12 + a*b^14)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a
^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^1
8 + b^20)*d^4)) + sqrt(2)*((3*a^22 + 29*a^20*b^2 + 125*a^18*b^4 + 315*a^16*b^6 + 510*a^14*b^8 + 546*a^12*b^10
+ 378*a^10*b^12 + 150*a^8*b^14 + 15*a^6*b^16 - 15*a^4*b^18 - 7*a^2*b^20 - b^22)*d^7*sqrt((25*a^8*b^2 - 100*a^6
*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252
*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((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^4)) + 2*(a^17 + 8*a^15*b^2 + 28*a^13*b^4 + 56*a^11*b^6 + 70*a^9
*b^8 + 56*a^7*b^10 + 28*a^5*b^12 + 8*a^3*b^14 + a*b^16)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*
a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^1
2 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 +
5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*
b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6
*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(((25*a^14*b^2 - 25*a^12*b^4 - 115*a^10*b^6 + 35*a^8*b^8 + 171*a^
6*b^10 + 53*a^4*b^12 - 17*a^2*b^14 + b^16)*d^2*sqrt(1/((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^4))*cos(d*x + c) - sqrt(2)*(2*(25*a^15*b^3 - 25*a^13*b^5 - 115*a^11*b^7 + 35*a^9*b^9 + 171*a^7*b^11
+ 53*a^5*b^13 - 17*a^3*b^15 + a*b^17)*d^3*sqrt(1/((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^4))*cos(d*x + c) + (75*a^10*b^3 - 325*a^8*b^5 + 430*a^6*b^7 - 170*a^4*b^9 + 23*a^2*b^11 - b^13)*d*cos(d
*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^
4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*co
s(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((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^4))^(1/4) + (25*a^9*b^2 - 100*a^7*b^4 + 110*a^5*b^6 - 20*a^3*b^8 + a*b^10)*cos(d*x + c) + (25*a^8*b^3 - 100
*a^6*b^5 + 110*a^4*b^7 - 20*a^2*b^9 + b^11)*sin(d*x + c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 1
0*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4) - sqrt(2)*((15*a^26*b + 115*a^24*b^3 + 338*a^22*b^5 + 354*a^20*b^7 -
475*a^18*b^9 - 2055*a^16*b^11 - 3060*a^14*b^13 - 2484*a^12*b^15 - 1047*a^10*b^17 - 75*a^8*b^19 + 130*a^6*b^21
+ 50*a^4*b^23 + 3*a^2*b^25 - b^27)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^
20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 4
5*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((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^4)) + 2*(5*a^21*b + 30*a^19*b^3 + 61*a^17*b^5 + 8*a^15*b^7 - 182*a^13*b^9 - 364*a^11*b^11 - 350*a^9*b^13 -
184*a^7*b^15 - 47*a^5*b^17 - 2*a^3*b^19 + a*b^21)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^
8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 12
0*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2
*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*
d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6*b^4 +
110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((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^4))^(3/4))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^
8 + b^10)) + 3*sqrt(2)*((a^8*b^3 - 4*a^6*b^5 - 10*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cos(d*x + c)^4 + 2*(3*a^6*b^5
+ 5*a^4*b^7 + a^2*b^9 - b^11)*d*cos(d*x + c)^2 + (a^4*b^7 + 2*a^2*b^9 + b^11)*d + 4*((a^7*b^4 + a^5*b^6 - a^3*
b^8 - a*b^10)*d*cos(d*x + c)^3 + (a^5*b^6 + 2*a^3*b^8 + a*b^10)*d*cos(d*x + c))*sin(d*x + c) - ((a^13*b^3 - 14
*a^11*b^5 + 35*a^9*b^7 + 76*a^7*b^9 - 9*a^5*b^11 - 30*a^3*b^13 + 5*a*b^15)*d^3*cos(d*x + c)^4 + 2*(3*a^11*b^5
- 25*a^9*b^7 - 34*a^7*b^9 + 14*a^5*b^11 + 15*a^3*b^13 - 5*a*b^15)*d^3*cos(d*x + c)^2 + (a^9*b^7 - 8*a^7*b^9 -
14*a^5*b^11 + 5*a*b^15)*d^3 + 4*((a^12*b^4 - 9*a^10*b^6 - 6*a^8*b^8 + 14*a^6*b^10 + 5*a^4*b^12 - 5*a^2*b^14)*d
^3*cos(d*x + c)^3 + (a^10*b^6 - 8*a^8*b^8 - 14*a^6*b^10 + 5*a^2*b^14)*d^3*cos(d*x + c))*sin(d*x + c))*sqrt(1/(
(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^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 1
0*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*
b^12 + 5*a*b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^
2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*(1/((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^4))^(1/4)*log(((25*a^14*b^2 - 25*a^12*b^4 - 115*a^10*b^6 + 35*a^8*b^8 + 171*a^6*b^10 + 53*a^4*b^12
- 17*a^2*b^14 + b^16)*d^2*sqrt(1/((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^4))*cos(d
*x + c) + sqrt(2)*(2*(25*a^15*b^3 - 25*a^13*b^5 - 115*a^11*b^7 + 35*a^9*b^9 + 171*a^7*b^11 + 53*a^5*b^13 - 17*
a^3*b^15 + a*b^17)*d^3*sqrt(1/((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^4))*cos(d*x +
c) + (75*a^10*b^3 - 325*a^8*b^5 + 430*a^6*b^7 - 170*a^4*b^9 + 23*a^2*b^11 - b^13)*d*cos(d*x + c))*sqrt((a^10
+ 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*
a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d
*x + c))/cos(d*x + c))*(1/((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^4))^(1/4) + (25*a
^9*b^2 - 100*a^7*b^4 + 110*a^5*b^6 - 20*a^3*b^8 + a*b^10)*cos(d*x + c) + (25*a^8*b^3 - 100*a^6*b^5 + 110*a^4*b
^7 - 20*a^2*b^9 + b^11)*sin(d*x + c))/cos(d*x + c)) - 3*sqrt(2)*((a^8*b^3 - 4*a^6*b^5 - 10*a^4*b^7 - 4*a^2*b^9
+ b^11)*d*cos(d*x + c)^4 + 2*(3*a^6*b^5 + 5*a^4*b^7 + a^2*b^9 - b^11)*d*cos(d*x + c)^2 + (a^4*b^7 + 2*a^2*b^9
+ b^11)*d + 4*((a^7*b^4 + a^5*b^6 - a^3*b^8 - a*b^10)*d*cos(d*x + c)^3 + (a^5*b^6 + 2*a^3*b^8 + a*b^10)*d*cos
(d*x + c))*sin(d*x + c) - ((a^13*b^3 - 14*a^11*b^5 + 35*a^9*b^7 + 76*a^7*b^9 - 9*a^5*b^11 - 30*a^3*b^13 + 5*a*
b^15)*d^3*cos(d*x + c)^4 + 2*(3*a^11*b^5 - 25*a^9*b^7 - 34*a^7*b^9 + 14*a^5*b^11 + 15*a^3*b^13 - 5*a*b^15)*d^3
*cos(d*x + c)^2 + (a^9*b^7 - 8*a^7*b^9 - 14*a^5*b^11 + 5*a*b^15)*d^3 + 4*((a^12*b^4 - 9*a^10*b^6 - 6*a^8*b^8 +
14*a^6*b^10 + 5*a^4*b^12 - 5*a^2*b^14)*d^3*cos(d*x + c)^3 + (a^10*b^6 - 8*a^8*b^8 - 14*a^6*b^10 + 5*a^2*b^14)
*d^3*cos(d*x + c))*sin(d*x + c))*sqrt(1/((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^4))
)*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*
a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*(1/((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^4))^(1/4)*log(((25*a^14*b^2 - 25*a^12*b^4 - 115*a^10*b^6
+ 35*a^8*b^8 + 171*a^6*b^10 + 53*a^4*b^12 - 17*a^2*b^14 + b^16)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 1
0*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) - sqrt(2)*(2*(25*a^15*b^3 - 25*a^13*b^5 - 115*a^11*b^7 + 35*a
^9*b^9 + 171*a^7*b^11 + 53*a^5*b^13 - 17*a^3*b^15 + a*b^17)*d^3*sqrt(1/((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^4))*cos(d*x + c) + (75*a^10*b^3 - 325*a^8*b^5 + 430*a^6*b^7 - 170*a^4*b^9 + 23*a^2
*b^11 - b^13)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^15 - 5*
a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((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^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^
8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((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^4))^(1/4) + (25*a^9*b^2 - 100*a^7*b^4 + 110*a^5*b^6 - 20*a^3*b^8 + a*b^10)*cos(d*x + c
) + (25*a^8*b^3 - 100*a^6*b^5 + 110*a^4*b^7 - 20*a^2*b^9 + b^11)*sin(d*x + c))/cos(d*x + c)) + 8*(3*a^4*b^4 +
6*a^2*b^6 + 3*b^8 + (8*a^8 - 18*a^6*b^2 - 65*a^4*b^4 - 12*a^2*b^6 + 3*b^8)*cos(d*x + c)^4 + (35*a^6*b^2 + 65*a
^4*b^4 + 6*a^2*b^6 - 6*b^8)*cos(d*x + c)^2 + 2*(2*(7*a^7*b + 10*a^5*b^3 - 6*a^3*b^5 - 3*a*b^7)*cos(d*x + c)^3
+ 3*(3*a^5*b^3 + 6*a^3*b^5 + 2*a*b^7)*cos(d*x + c))*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d
*x + c)))/((a^8*b^3 - 4*a^6*b^5 - 10*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cos(d*x + c)^4 + 2*(3*a^6*b^5 + 5*a^4*b^7 +
a^2*b^9 - b^11)*d*cos(d*x + c)^2 + (a^4*b^7 + 2*a^2*b^9 + b^11)*d + 4*((a^7*b^4 + a^5*b^6 - a^3*b^8 - a*b^10)
*d*cos(d*x + c)^3 + (a^5*b^6 + 2*a^3*b^8 + a*b^10)*d*cos(d*x + c))*sin(d*x + c))

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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{5}{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))**(5/2),x)

[Out]

Integral(tan(c + d*x)**4/(a + b*tan(c + d*x))**(5/2), x)

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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{5}{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))^(5/2),x, algorithm="giac")

[Out]

integrate(tan(d*x + c)^4/(b*tan(d*x + c) + a)^(5/2), x)