3.4.0 ∫ (−a + b tan(c + dx))(a + b tan(c + dx))5∕2 dx [340]

Integrand size = 27, antiderivative size = 151 \[ \int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx=\frac {(i a-b) (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (i a+b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d} \]

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

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

Rubi [A] (veriﬁed)

Time = 0.44 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3609, 12, 3563, 3620, 3618, 65, 214} \[ \int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx=-\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {(-b+i a) (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (b+i a) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d} \]

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

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

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

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

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,

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

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))

), x] + Int[(a^2 - b^2 + 2*a*b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*

((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x

], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

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]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\int \left (-a^2-b^2\right ) (a+b \tan (c+d x))^{3/2} \, dx \\ & = \frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\left (-a^2-b^2\right ) \int (a+b \tan (c+d x))^{3/2} \, dx \\ & = -\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\left (-a^2-b^2\right ) \int \frac {a^2-b^2+2 a b \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}-\frac {1}{2} \left ((a-i b)^2 \left (a^2+b^2\right )\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {1}{2} \left ((a+i b)^2 \left (a^2+b^2\right )\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {\left ((a+i b)^3 (i a+b)\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}+\frac {\left ((a+i b) (i a+b)^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d} \\ & = -\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {\left ((a-i b)^3 (a+i b)\right ) \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 {\left ((a-i b) (a+i b)^3\right ) \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 a-b) (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} (i a+b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {2 b \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 1.42 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.28 \[ \int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx=\frac {\cos (c+d x) (a-b \tan (c+d x)) \left (5 i (a-i b)^{5/2} (a+i b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-5 i (a-i b) (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 b \sqrt {a+b \tan (c+d x)} \left (-4 a^2-5 b^2+2 a b \tan (c+d x)+b^2 \tan ^2(c+d x)\right )\right )}{5 d (a \cos (c+d x)-b \sin (c+d x))} \]

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

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

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

+ d*x]]*(-4*a^2 - 5*b^2 + 2*a*b*Tan[c + d*x] + b^2*Tan[c + d*x]^2)))/(5*d*(a*Cos[c + d*x] - b*Sin[c + d*x]))

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1374\) vs. \(2(127)=254\).

Time = 0.08 (sec) , antiderivative size = 1375, normalized size of antiderivative = 9.11


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1375\)
default	\(\text {Expression too large to display}\)	\(1375\)
parts	\(\text {Expression too large to display}\)	\(2381\)

int((-a+b*tan(d*x+c))*(a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

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

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

n(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/d*b/(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+b^2)^(1/2)*a^2+1/d*b^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^2+b^2)^(1/2)-2/d*b/(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^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*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))*(a^2+b^2)^(

1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-1/4/d*b*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))*(a^2+b^2)^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/4/d/b*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^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)

-1/d*b/(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+b^2)^(1/2)*a^2-1/d*b^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^2+b^2)^(1/2)+2/d*b/(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+2/d*

b^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)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1472 vs. \(2 (121) = 242\).

Time = 0.29 (sec) , antiderivative size = 1472, normalized size of antiderivative = 9.75 \[ \int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]

integrate((-a+b*tan(d*x+c))*(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

-1/10*(5*d*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 + d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a

^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)*log(-(3*a^10*b + 11*a^8*b^3 + 14*a^6*b^5 + 6*a^4*b^7 - a^2

*b^9 - b^11)*sqrt(b*tan(d*x + c) + a) + (a*d^3*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^

4*b^10 - 2*a^2*b^12 + b^14)/d^4) + (3*a^6*b^2 + 5*a^4*b^4 + a^2*b^6 - b^8)*d)*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4

- 3*a*b^6 + d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^

4))/d^2)) - 5*d*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 + d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6

+ 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)*log(-(3*a^10*b + 11*a^8*b^3 + 14*a^6*b^5 + 6*a^4*b^7

- a^2*b^9 - b^11)*sqrt(b*tan(d*x + c) + a) - (a*d^3*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 -

9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4) + (3*a^6*b^2 + 5*a^4*b^4 + a^2*b^6 - b^8)*d)*sqrt(-(a^7 - a^5*b^2 - 5*a^

3*b^4 - 3*a*b^6 + d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^1

4)/d^4))/d^2)) - 5*d*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 - d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8

*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)*log(-(3*a^10*b + 11*a^8*b^3 + 14*a^6*b^5 + 6*a^4

*b^7 - a^2*b^9 - b^11)*sqrt(b*tan(d*x + c) + a) + (a*d^3*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*

b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4) - (3*a^6*b^2 + 5*a^4*b^4 + a^2*b^6 - b^8)*d)*sqrt(-(a^7 - a^5*b^2 -

5*a^3*b^4 - 3*a*b^6 - d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12

+ b^14)/d^4))/d^2)) + 5*d*sqrt(-(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 - d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 3

1*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4))/d^2)*log(-(3*a^10*b + 11*a^8*b^3 + 14*a^6*b^5 +

6*a^4*b^7 - a^2*b^9 - b^11)*sqrt(b*tan(d*x + c) + a) - (a*d^3*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4

*a^6*b^8 - 9*a^4*b^10 - 2*a^2*b^12 + b^14)/d^4) - (3*a^6*b^2 + 5*a^4*b^4 + a^2*b^6 - b^8)*d)*sqrt(-(a^7 - a^5*

b^2 - 5*a^3*b^4 - 3*a*b^6 - d^2*sqrt(-(9*a^12*b^2 + 30*a^10*b^4 + 31*a^8*b^6 + 4*a^6*b^8 - 9*a^4*b^10 - 2*a^2*

b^12 + b^14)/d^4))/d^2)) - 4*(b^3*tan(d*x + c)^2 + 2*a*b^2*tan(d*x + c) - 4*a^2*b - 5*b^3)*sqrt(b*tan(d*x + c)

Sympy [F]

\[ \int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx=- \int a^{3} \sqrt {a + b \tan {\left (c + d x \right )}}\, dx - \int \left (- b^{3} \sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{3}{\left (c + d x \right )}\right )\, dx - \int \left (- a b^{2} \sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )}\right )\, dx - \int a^{2} b \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )}\, dx \]

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

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

tegral(-a*b**2*sqrt(a + b*tan(c + d*x))*tan(c + d*x)**2, x) - Integral(a**2*b*sqrt(a + b*tan(c + d*x))*tan(c +

Maxima [F(-2)]

Exception generated. \[ \int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx=\text {Exception raised: ValueError} \]

integrate((-a+b*tan(d*x+c))*(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")

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

Giac [F(-1)]

Timed out. \[ \int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]

integrate((-a+b*tan(d*x+c))*(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

Mupad [B] (veriﬁcation not implemented)

Time = 32.67 (sec) , antiderivative size = 3441, normalized size of antiderivative = 22.79 \[ \int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]

int(-(a + b*tan(c + d*x))^(5/2)*(a - b*tan(c + d*x)),x)

log((8*a^3*b^3*(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3 - ((((((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^7

*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2)*(64*a^2*b^5 + 64*a^4*b^3 + 32*a*b^2*d*(((-a^4*b^2*d^4*(5*a^4

+ b^4 - 10*a^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2

)))/(2*d) + (16*a^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(((-a^4*b^2*d^4

*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2))/2)*((20*a^6*b^8*d

^4 - a^4*b^10*d^4 - 110*a^8*b^6*d^4 + 100*a^10*b^4*d^4 - 25*a^12*b^2*d^4)^(1/2)/(4*d^4) - a^7/(4*d^2) - (5*a^3

*b^4)/(4*d^2) + (5*a^5*b^2)/(2*d^2))^(1/2) - log((8*a^3*b^3*(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3 - ((((-((-a^4*b^2

*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2)*(64*a^2*b^5 +

64*a^4*b^3 - 32*a*b^2*d*(-((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^

5*b^2*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/(2*d) - (16*a^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^6 - b^6 +

15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2

- 10*a^5*b^2*d^2)/d^4)^(1/2))/2)*(-(a^7*d^2 + (20*a^6*b^8*d^4 - a^4*b^10*d^4 - 110*a^8*b^6*d^4 + 100*a^10*b^4

*d^4 - 25*a^12*b^2*d^4)^(1/2) + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/(4*d^4))^(1/2) - log((8*a^3*b^3*(3*a^2 - b^2)*

(a^2 + b^2)^3)/d^3 - ((((((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5

*b^2*d^2)/d^4)^(1/2)*(64*a^2*b^5 + 64*a^4*b^3 - 32*a*b^2*d*(((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2)

- a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/(2*d) - (16*a^2*b^2*(a +

b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2

)^(1/2) - a^7*d^2 - 5*a^3*b^4*d^2 + 10*a^5*b^2*d^2)/d^4)^(1/2))/2)*(-(a^7*d^2 - (20*a^6*b^8*d^4 - a^4*b^10*d^4

- 110*a^8*b^6*d^4 + 100*a^10*b^4*d^4 - 25*a^12*b^2*d^4)^(1/2) + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/(4*d^4))^(1/2

) + log((8*a^3*b^3*(3*a^2 - b^2)*(a^2 + b^2)^3)/d^3 - ((((-((-a^4*b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2)

+ a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2)*(64*a^2*b^5 + 64*a^4*b^3 + 32*a*b^2*d*(-((-a^4*b^2*d^4*

(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x)

)^(1/2)))/(2*d) + (16*a^2*b^2*(a + b*tan(c + d*x))^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*(-((-a^4*

b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^7*d^2 + 5*a^3*b^4*d^2 - 10*a^5*b^2*d^2)/d^4)^(1/2))/2)*((5*a^5

*b^2)/(2*d^2) - a^7/(4*d^2) - (5*a^3*b^4)/(4*d^2) - (20*a^6*b^8*d^4 - a^4*b^10*d^4 - 110*a^8*b^6*d^4 + 100*a^1

0*b^4*d^4 - 25*a^12*b^2*d^4)^(1/2)/(4*d^4))^(1/2) + ((4*a^2*b)/d - (2*b*(a^2 + b^2))/d)*(a + b*tan(c + d*x))^(

1/2) - log(((((((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/d^

4)^(1/2)*(32*a^4*b^3 - 32*b^7 + 32*a*b^2*d*(((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + 5*a*b^6*d^2 - 10*

a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/(2*d) + (16*(a + b*tan(c + d*x))^(1/2)*(b^1

0 - 15*a^2*b^8 + 15*a^4*b^6 - a^6*b^4))/d^2)*(((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + 5*a*b^6*d^2 - 1

0*a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)^(1/2))/2 - (8*a*b^5*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3)*(((20*a^2*b^12*d^4 - b

^14*d^4 - 110*a^4*b^10*d^4 + 100*a^6*b^8*d^4 - 25*a^8*b^6*d^4)^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*

d^2)/(4*d^4))^(1/2) + log(- ((((((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + 5*a*b^6*d^2 - 10*a^3*b^4*d^2

+ a^5*b^2*d^2)/d^4)^(1/2)*(32*b^7 - 32*a^4*b^3 + 32*a*b^2*d*(((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) +

5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1/2)))/(2*d) + (16*(a + b*tan(c +

d*x))^(1/2)*(b^10 - 15*a^2*b^8 + 15*a^4*b^6 - a^6*b^4))/d^2)*(((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2)

+ 5*a*b^6*d^2 - 10*a^3*b^4*d^2 + a^5*b^2*d^2)/d^4)^(1/2))/2 - (8*a*b^5*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3)*((20*

a^2*b^12*d^4 - b^14*d^4 - 110*a^4*b^10*d^4 + 100*a^6*b^8*d^4 - 25*a^8*b^6*d^4)^(1/2)/(4*d^4) + (5*a*b^6)/(4*d^

2) - (5*a^3*b^4)/(2*d^2) + (a^5*b^2)/(4*d^2))^(1/2) - log(((((-((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2)

- 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2)*(32*a^4*b^3 - 32*b^7 + 32*a*b^2*d*(-((-b^6*d^4*(5*a^4

+ b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2)*(a + b*tan(c + d*x))^(1

/2)))/(2*d) + (16*(a + b*tan(c + d*x))^(1/2)*(b^10 - 15*a^2*b^8 + 15*a^4*b^6 - a^6*b^4))/d^2)*(-((-b^6*d^4*(5*

a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2))/2 - (8*a*b^5*(a^2 -

3*b^2)*(a^2 + b^2)^3)/d^3)*(-((20*a^2*b^12*d^4 - b^14*d^4 - 110*a^4*b^10*d^4 + 100*a^6*b^8*d^4 - 25*a^8*b^6*d

^4)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/(4*d^4))^(1/2) + log(- ((((-((-b^6*d^4*(5*a^4 + b^4 -

10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2)*(32*b^7 - 32*a^4*b^3 + 32*a*b^2*

d*(-((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2)*(a

+ b*tan(c + d*x))^(1/2)))/(2*d) + (16*(a + b*tan(c + d*x))^(1/2)*(b^10 - 15*a^2*b^8 + 15*a^4*b^6 - a^6*b^4))/

d^2)*(-((-b^6*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - 5*a*b^6*d^2 + 10*a^3*b^4*d^2 - a^5*b^2*d^2)/d^4)^(1/2)

)/2 - (8*a*b^5*(a^2 - 3*b^2)*(a^2 + b^2)^3)/d^3)*((5*a*b^6)/(4*d^2) - (20*a^2*b^12*d^4 - b^14*d^4 - 110*a^4*b^

10*d^4 + 100*a^6*b^8*d^4 - 25*a^8*b^6*d^4)^(1/2)/(4*d^4) - (5*a^3*b^4)/(2*d^2) + (a^5*b^2)/(4*d^2))^(1/2) + (2

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

\(\int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx\) [340]

Optimal result