3.554 \(\int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=194 \[ -\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^3 \sqrt {a+b \tan (c+d x)}}-\frac {4 a b}{3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{3/2}}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{7/2}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{7/2}} \]

[Out]

-I*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(7/2)/d+I*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2
))/(a+I*b)^(7/2)/d-2*b*(3*a^2-b^2)/(a^2+b^2)^3/d/(a+b*tan(d*x+c))^(1/2)-2/5*b/(a^2+b^2)/d/(a+b*tan(d*x+c))^(5/
2)-4/3*a*b/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(3/2)

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Rubi [A]  time = 0.39, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3483, 3529, 3539, 3537, 63, 208} \[ -\frac {2 b \left (3 a^2-b^2\right )}{d \left (a^2+b^2\right )^3 \sqrt {a+b \tan (c+d x)}}-\frac {4 a b}{3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{3/2}}-\frac {2 b}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{7/2}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^(-7/2),x]

[Out]

((-I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(7/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x
]]/Sqrt[a + I*b]])/((a + I*b)^(7/2)*d) - (2*b)/(5*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(5/2)) - (4*a*b)/(3*(a^2
+ b^2)^2*d*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*Sqrt[a + b*Tan[c + d*x]])

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]

Rule 3483

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)
*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3529

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

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 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]

Rubi steps

\begin {align*} \int \frac {1}{(a+b \tan (c+d x))^{7/2}} \, dx &=-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}+\frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx}{a^2+b^2}\\ &=-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}-\frac {4 a b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^{3/2}}+\frac {\int \frac {a^2-b^2-2 a b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx}{\left (a^2+b^2\right )^2}\\ &=-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}-\frac {4 a b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d \sqrt {a+b \tan (c+d x)}}+\frac {\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{\left (a^2+b^2\right )^3}\\ &=-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}-\frac {4 a b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d \sqrt {a+b \tan (c+d x)}}+\frac {\int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)^3}+\frac {\int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)^3}\\ &=-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}-\frac {4 a b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d \sqrt {a+b \tan (c+d x)}}-\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)^3 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)^3 d}\\ &=-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}-\frac {4 a b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d \sqrt {a+b \tan (c+d x)}}-\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)^3 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)^3 b d}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{7/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{7/2} d}-\frac {2 b}{5 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{5/2}}-\frac {4 a b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 0.25, size = 108, normalized size = 0.56 \[ \frac {i (a+i b) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {a+b \tan (c+d x)}{a-i b}\right )+(-b-i a) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {a+b \tan (c+d x)}{a+i b}\right )}{5 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x])^(-7/2),x]

[Out]

(I*(a + I*b)*Hypergeometric2F1[-5/2, 1, -3/2, (a + b*Tan[c + d*x])/(a - I*b)] + ((-I)*a - b)*Hypergeometric2F1
[-5/2, 1, -3/2, (a + b*Tan[c + d*x])/(a + I*b)])/(5*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(5/2))

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fricas [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(1/(a+b*tan(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

Timed out

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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(1/(a+b*tan(d*x+c))^(7/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.24, size = 3073, normalized size = 15.84 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(d*x+c))^(7/2),x)

[Out]

-2/d*b^3/(a^2+b^2)^4/(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+1/4/d/b/(a^2+b^2)^4*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^5-3/4/d*b^3/(a^2+b^2)^4*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)^4*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^3-6/d*b/(a^2+b^2)^3/(a+b*tan(d*x+c))^(1/2)*a^2+1/d*b^5/(a^2+b^2)^4/(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^5/(a^2+b^2)^4/(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))+1/4/d*b^5/(a^2+b^2)^(9/2)*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^5/(a^2+b^2)^(9/2)*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)+2/d*b^3/(a^
2+b^2)^3/(a+b*tan(d*x+c))^(1/2)-3/d*b/(a^2+b^2)^4/(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^4+2/d*b/(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^3+
3/d*b/(a^2+b^2)^(9/2)/(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^5-5/d*b^3/(a^2+b^2)^(9/2)/(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^3-7/d*b^5/(a^2+b^2)^(9/2)/(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+1/4/d/b/(a^2+b^2)^(9/2)*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^6-5/4/d*b^3/(a^2+b^2)^(9/2)*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^2-1/d/b/(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^5+1/d/b/(a^2+b^2)^(9/2)/(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^7-5/d*b^3/(a^2+b^2)^(9/2)/(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^3-7/d*b^
5/(a^2+b^2)^(9/2)/(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/d*b/(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^3+3/d*b/(a^2+b^2)^(9/2)/(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^5-5/4/d*b/(a^2+b^2)^(9/2)*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+5/4/d*b/(a^2+b^2)^(9/2)*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+3/4/d*b^3/(a^2+b^2)^4*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
-2/d*b^3/(a^2+b^2)^4/(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^2-1/4/d/b/(a^2+b^2)^4*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^5+1/d/b/(a^2+b^2)^(9/2)/(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
^7-2/5*b/(a^2+b^2)/d/(a+b*tan(d*x+c))^(5/2)-1/d/b/(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^5+3/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-3/d*b/(a^2+b^2)^4/(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^4-1/4/d/b/(a^2+b^2)^(9/2)*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^6+1/2/d*b/(a^2+b^2)^4*l
n(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+3/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+5/4/d*b^3/(a^2+b^2)^(9/2)*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^2-4/3*a*b/(a^2+b^2)^2/
d/(a+b*tan(d*x+c))^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is b-a positive, negative or zero?

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mupad [B]  time = 20.89, size = 5307, normalized size = 27.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x))^(7/2),x)

[Out]

(log((((a + b*tan(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*d^3 - 964
8*a^8*b^18*d^3 - 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1344*a^20*
b^6*d^3 + 96*a^22*b^4*d^3 - 16*a^24*b^2*d^3) - ((-1/(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b
^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2))^(1/2)*(((-1/(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d
^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2))^(1/2)*(a + b*tan(c +
d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7*b^26*d^5 + 87360*a^9*b^24*d^5 +
192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^17*b^16*d^5 + 320320*a^19*b^14*
d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a^27*b^6*d^5 + 960*a^29*b^4*d^5 +
64*a^31*b^2*d^5))/2 - 128*a*b^29*d^4 - 1408*a^3*b^27*d^4 - 6912*a^5*b^25*d^4 - 19712*a^7*b^23*d^4 - 35200*a^9
*b^21*d^4 - 38016*a^11*b^19*d^4 - 16896*a^13*b^17*d^4 + 16896*a^15*b^15*d^4 + 38016*a^17*b^13*d^4 + 35200*a^19
*b^11*d^4 + 19712*a^21*b^9*d^4 + 6912*a^23*b^7*d^4 + 1408*a^25*b^5*d^4 + 128*a^27*b^3*d^4))/2)*(-1/(a^7*d^2 +
b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2)
)^(1/2))/2 - 8*b^23*d^2 - 48*a^2*b^21*d^2 - 72*a^4*b^19*d^2 + 192*a^6*b^17*d^2 + 1008*a^8*b^15*d^2 + 2016*a^10
*b^13*d^2 + 2352*a^12*b^11*d^2 + 1728*a^14*b^9*d^2 + 792*a^16*b^7*d^2 + 208*a^18*b^5*d^2 + 24*a^20*b^3*d^2)*(-
1/(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21
*a^5*b^2*d^2))^(1/2))/2 - log(192*a^6*b^17*d^2 - 8*b^23*d^2 - 48*a^2*b^21*d^2 - 72*a^4*b^19*d^2 - (-1/(4*(a^7*
d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^
2*d^2)))^(1/2)*((a + b*tan(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*
d^3 - 9648*a^8*b^18*d^3 - 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1
344*a^20*b^6*d^3 + 96*a^22*b^4*d^3 - 16*a^24*b^2*d^3) - (-1/(4*(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2
*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2)))^(1/2)*(128*a*b^29*d^4 + (-1/(4*(a
^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^4*d^2 + a^4*b^3*d^2*35i - 21*a^5
*b^2*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7
*b^26*d^5 + 87360*a^9*b^24*d^5 + 192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a
^17*b^16*d^5 + 320320*a^19*b^14*d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a
^27*b^6*d^5 + 960*a^29*b^4*d^5 + 64*a^31*b^2*d^5) + 1408*a^3*b^27*d^4 + 6912*a^5*b^25*d^4 + 19712*a^7*b^23*d^4
+ 35200*a^9*b^21*d^4 + 38016*a^11*b^19*d^4 + 16896*a^13*b^17*d^4 - 16896*a^15*b^15*d^4 - 38016*a^17*b^13*d^4
- 35200*a^19*b^11*d^4 - 19712*a^21*b^9*d^4 - 6912*a^23*b^7*d^4 - 1408*a^25*b^5*d^4 - 128*a^27*b^3*d^4)) + 1008
*a^8*b^15*d^2 + 2016*a^10*b^13*d^2 + 2352*a^12*b^11*d^2 + 1728*a^14*b^9*d^2 + 792*a^16*b^7*d^2 + 208*a^18*b^5*
d^2 + 24*a^20*b^3*d^2)*(-1/(4*(a^7*d^2 + b^7*d^2*1i - 7*a*b^6*d^2 - a^6*b*d^2*7i - a^2*b^5*d^2*21i + 35*a^3*b^
4*d^2 + a^4*b^3*d^2*35i - 21*a^5*b^2*d^2)))^(1/2) + atan(-((-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^
6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*((a + b*tan(c + d*x))^(
1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*d^3 - 9648*a^8*b^18*d^3 - 8640*a^10*b^
16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1344*a^20*b^6*d^3 + 96*a^22*b^4*d^3 - 1
6*a^24*b^2*d^3) - (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35
i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*((-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21
*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^
32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7*b^26*d^5 + 87360*a^9*b^24*d^5 + 192192*a^11*b^22*d^5
+ 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^17*b^16*d^5 + 320320*a^19*b^14*d^5 + 192192*a^21*b^1
2*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a^27*b^6*d^5 + 960*a^29*b^4*d^5 + 64*a^31*b^2*d^5) - 1
28*a*b^29*d^4 - 1408*a^3*b^27*d^4 - 6912*a^5*b^25*d^4 - 19712*a^7*b^23*d^4 - 35200*a^9*b^21*d^4 - 38016*a^11*b
^19*d^4 - 16896*a^13*b^17*d^4 + 16896*a^15*b^15*d^4 + 38016*a^17*b^13*d^4 + 35200*a^19*b^11*d^4 + 19712*a^21*b
^9*d^4 + 6912*a^23*b^7*d^4 + 1408*a^25*b^5*d^4 + 128*a^27*b^3*d^4))*1i + (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6
*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*((a + b*t
an(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*d^3 - 9648*a^8*b^18*d^3
- 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1344*a^20*b^6*d^3 + 96*a^
22*b^4*d^3 - 16*a^24*b^2*d^3) - (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 +
a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(128*a*b^29*d^4 + (-1i/(4*(a^7*d^2*1i + b^7*d^2 -
a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(a +
b*tan(c + d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7*b^26*d^5 + 87360*a^9*
b^24*d^5 + 192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^17*b^16*d^5 + 320320*
a^19*b^14*d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a^27*b^6*d^5 + 960*a^29
*b^4*d^5 + 64*a^31*b^2*d^5) + 1408*a^3*b^27*d^4 + 6912*a^5*b^25*d^4 + 19712*a^7*b^23*d^4 + 35200*a^9*b^21*d^4
+ 38016*a^11*b^19*d^4 + 16896*a^13*b^17*d^4 - 16896*a^15*b^15*d^4 - 38016*a^17*b^13*d^4 - 35200*a^19*b^11*d^4
- 19712*a^21*b^9*d^4 - 6912*a^23*b^7*d^4 - 1408*a^25*b^5*d^4 - 128*a^27*b^3*d^4))*1i)/((-1i/(4*(a^7*d^2*1i + b
^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(
1/2)*((a + b*tan(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20*d^3 - 9648
*a^8*b^18*d^3 - 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 + 1344*a^20*b
^6*d^3 + 96*a^22*b^4*d^3 - 16*a^24*b^2*d^3) - (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*
a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*((-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b
^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(a + b*
tan(c + d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*a^7*b^26*d^5 + 87360*a^9*b^2
4*d^5 + 192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 411840*a^17*b^16*d^5 + 320320*a^1
9*b^14*d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 6720*a^27*b^6*d^5 + 960*a^29*b^
4*d^5 + 64*a^31*b^2*d^5) - 128*a*b^29*d^4 - 1408*a^3*b^27*d^4 - 6912*a^5*b^25*d^4 - 19712*a^7*b^23*d^4 - 35200
*a^9*b^21*d^4 - 38016*a^11*b^19*d^4 - 16896*a^13*b^17*d^4 + 16896*a^15*b^15*d^4 + 38016*a^17*b^13*d^4 + 35200*
a^19*b^11*d^4 + 19712*a^21*b^9*d^4 + 6912*a^23*b^7*d^4 + 1408*a^25*b^5*d^4 + 128*a^27*b^3*d^4)) - (-1i/(4*(a^7
*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d
^2*21i)))^(1/2)*((a + b*tan(c + d*x))^(1/2)*(16*b^26*d^3 - 96*a^2*b^24*d^3 - 1344*a^4*b^22*d^3 - 5152*a^6*b^20
*d^3 - 9648*a^8*b^18*d^3 - 8640*a^10*b^16*d^3 + 8640*a^14*b^12*d^3 + 9648*a^16*b^10*d^3 + 5152*a^18*b^8*d^3 +
1344*a^20*b^6*d^3 + 96*a^22*b^4*d^3 - 16*a^24*b^2*d^3) - (-1i/(4*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*
b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d^2*21i)))^(1/2)*(128*a*b^29*d^4 + (-1i/(4
*(a^7*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*
b^2*d^2*21i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^32*d^5 + 960*a^3*b^30*d^5 + 6720*a^5*b^28*d^5 + 29120*
a^7*b^26*d^5 + 87360*a^9*b^24*d^5 + 192192*a^11*b^22*d^5 + 320320*a^13*b^20*d^5 + 411840*a^15*b^18*d^5 + 41184
0*a^17*b^16*d^5 + 320320*a^19*b^14*d^5 + 192192*a^21*b^12*d^5 + 87360*a^23*b^10*d^5 + 29120*a^25*b^8*d^5 + 672
0*a^27*b^6*d^5 + 960*a^29*b^4*d^5 + 64*a^31*b^2*d^5) + 1408*a^3*b^27*d^4 + 6912*a^5*b^25*d^4 + 19712*a^7*b^23*
d^4 + 35200*a^9*b^21*d^4 + 38016*a^11*b^19*d^4 + 16896*a^13*b^17*d^4 - 16896*a^15*b^15*d^4 - 38016*a^17*b^13*d
^4 - 35200*a^19*b^11*d^4 - 19712*a^21*b^9*d^4 - 6912*a^23*b^7*d^4 - 1408*a^25*b^5*d^4 - 128*a^27*b^3*d^4)) - 1
6*b^23*d^2 - 96*a^2*b^21*d^2 - 144*a^4*b^19*d^2 + 384*a^6*b^17*d^2 + 2016*a^8*b^15*d^2 + 4032*a^10*b^13*d^2 +
4704*a^12*b^11*d^2 + 3456*a^14*b^9*d^2 + 1584*a^16*b^7*d^2 + 416*a^18*b^5*d^2 + 48*a^20*b^3*d^2))*(-1i/(4*(a^7
*d^2*1i + b^7*d^2 - a*b^6*d^2*7i - 7*a^6*b*d^2 - 21*a^2*b^5*d^2 + a^3*b^4*d^2*35i + 35*a^4*b^3*d^2 - a^5*b^2*d
^2*21i)))^(1/2)*2i - ((2*b)/(5*(a^2 + b^2)) + (4*a*b*(a + b*tan(c + d*x)))/(3*(a^4 + b^4 + 2*a^2*b^2)) + (2*b*
(3*a^2 - b^2)*(a + b*tan(c + d*x))^2)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/(d*(a + b*tan(c + d*x))^(5/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))**(7/2),x)

[Out]

Integral((a + b*tan(c + d*x))**(-7/2), x)

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