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

Optimal. Leaf size=151 \[ -\frac{2 b \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{(-b+i a) (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{(a+i b)^{5/2} (b+i a) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

[Out]

((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
an[c + d*x])^(5/2))/(5*d)

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Rubi [A]  time = 0.25285, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {3528, 12, 3482, 3539, 3537, 63, 208} \[ -\frac{2 b \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{(-b+i a) (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{(a+i b)^{5/2} (b+i a) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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
an[c + d*x])^(5/2))/(5*d)

Rule 3528

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]

Rule 12

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

Rule 3482

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
[a^2 + b^2, 0] && GtQ[n, 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 (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{5/2} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \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 )}{b d}+\frac{\left ((a-i b) (a+i b)^3\right ) \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 )}{b d}\\ &=\frac{(i a-b) (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{(a+i b)^{5/2} (i a+b) \tanh ^{-1}\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]  time = 1.25652, size = 193, normalized size = 1.28 \[ \frac{\cos (c+d x) (a-b \tan (c+d x)) \left (2 b \sqrt{a+b \tan (c+d x)} \left (-4 a^2+2 a b \tan (c+d x)+b^2 \tan ^2(c+d x)-5 b^2\right )+5 i (a+i b) (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )-5 i (a+i b)^{5/2} (a-i b) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )\right )}{5 d (a \cos (c+d x)-b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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]))

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Maple [B]  time = 0.104, size = 1375, normalized size = 9.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 22.9883, size = 17797, normalized size = 117.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(20*sqrt(2)*d^5*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^3 - 3*a*b^2)*d^
2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*
a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*
b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(3/4)*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)*arctan(((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^4*sqrt((a^14
+ 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*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^29 + 38*a^27*b^2 + 221*a
^25*b^4 + 780*a^23*b^6 + 1859*a^21*b^8 + 3146*a^19*b^10 + 3861*a^17*b^12 + 3432*a^15*b^14 + 2145*a^13*b^16 + 8
58*a^11*b^18 + 143*a^9*b^20 - 52*a^7*b^22 - 39*a^5*b^24 - 10*a^3*b^26 - a*b^28)*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) + sqrt(2)*(d^7*sqrt((a^14 + 7*a^12*b^2 +
21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*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) + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*s
qrt((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))*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^3 - 3*a*b^2)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b
^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*
a^2*b^8 + b^10))*sqrt(((9*a^14*b^2 + 39*a^12*b^4 + 61*a^10*b^6 + 35*a^8*b^8 - 5*a^6*b^10 - 11*a^4*b^12 - a^2*b
^14 + b^16)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b
^14)/d^4)*cos(d*x + c) + sqrt(2)*(2*(9*a^9*b^3 + 12*a^7*b^5 - 2*a^5*b^7 - 4*a^3*b^9 + a*b^11)*d^3*sqrt((a^14 +
7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + (9*
a^16*b^3 + 48*a^14*b^5 + 100*a^12*b^7 + 96*a^10*b^9 + 30*a^8*b^11 - 16*a^6*b^13 - 12*a^4*b^15 + b^19)*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^3 - 3*a*b^2)*d^2*sqrt((a^14 +
7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*b^2 + 12*a
^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^14 + 7*a^12*b
^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(1/4) + (9*a^21*b^2 + 66*a^
19*b^4 + 205*a^17*b^6 + 344*a^15*b^8 + 322*a^13*b^10 + 140*a^11*b^12 - 14*a^9*b^14 - 40*a^7*b^16 - 11*a^5*b^18
+ 2*a^3*b^20 + a*b^22)*cos(d*x + c) + (9*a^20*b^3 + 66*a^18*b^5 + 205*a^16*b^7 + 344*a^14*b^9 + 322*a^12*b^11
+ 140*a^10*b^13 - 14*a^8*b^15 - 40*a^6*b^17 - 11*a^4*b^19 + 2*a^2*b^21 + b^23)*sin(d*x + c))/cos(d*x + c))*((
a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(3/4) + sqrt
(2)*((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)*d^7*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b
^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*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) + 2*(3*a^17*b + 20*a^15*b^3 + 56*a^13*b^5 + 84*a^11*b^7
+ 70*a^9*b^9 + 28*a^7*b^11 - 4*a^3*b^15 - a*b^17)*d^5*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))*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^3 - 3*a*b^2)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12
+ b^14)/d^4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))
/cos(d*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/
d^4)^(3/4))/(9*a^34*b^2 + 129*a^32*b^4 + 856*a^30*b^6 + 3480*a^28*b^8 + 9660*a^26*b^10 + 19292*a^24*b^12 + 283
92*a^22*b^14 + 30888*a^20*b^16 + 24310*a^18*b^18 + 12870*a^16*b^20 + 3432*a^14*b^22 - 728*a^12*b^24 - 1092*a^1
0*b^26 - 420*a^8*b^28 - 40*a^6*b^30 + 24*a^4*b^32 + 9*a^2*b^34 + b^36))*cos(d*x + c)^2 + 20*sqrt(2)*d^5*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^3 - 3*a*b^2)*d^2*sqrt((a^14 + 7*a^12*b^2 +
21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4
*b^6 - 4*a^2*b^8 + b^10))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^
12 + b^14)/d^4)^(3/4)*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)*arctan(-((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^4*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4
+ 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*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^29 + 38*a^27*b^2 + 221*a^25*b^4 + 780*a^23*b^6 + 18
59*a^21*b^8 + 3146*a^19*b^10 + 3861*a^17*b^12 + 3432*a^15*b^14 + 2145*a^13*b^16 + 858*a^11*b^18 + 143*a^9*b^20
- 52*a^7*b^22 - 39*a^5*b^24 - 10*a^3*b^26 - a*b^28)*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) - sqrt(2)*(d^7*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 +
35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*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) + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*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))*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^3 - 3*a*b^2)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^
8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10))*sqrt(((9*a
^14*b^2 + 39*a^12*b^4 + 61*a^10*b^6 + 35*a^8*b^8 - 5*a^6*b^10 - 11*a^4*b^12 - a^2*b^14 + b^16)*d^2*sqrt((a^14
+ 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) - sq
rt(2)*(2*(9*a^9*b^3 + 12*a^7*b^5 - 2*a^5*b^7 - 4*a^3*b^9 + a*b^11)*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 +
35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + (9*a^16*b^3 + 48*a^14*b^5 + 10
0*a^12*b^7 + 96*a^10*b^9 + 30*a^8*b^11 - 16*a^6*b^13 - 12*a^4*b^15 + b^19)*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^3 - 3*a*b^2)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 +
35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*
b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b
^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(1/4) + (9*a^21*b^2 + 66*a^19*b^4 + 205*a^17*b^6 + 344
*a^15*b^8 + 322*a^13*b^10 + 140*a^11*b^12 - 14*a^9*b^14 - 40*a^7*b^16 - 11*a^5*b^18 + 2*a^3*b^20 + a*b^22)*cos
(d*x + c) + (9*a^20*b^3 + 66*a^18*b^5 + 205*a^16*b^7 + 344*a^14*b^9 + 322*a^12*b^11 + 140*a^10*b^13 - 14*a^8*b
^15 - 40*a^6*b^17 - 11*a^4*b^19 + 2*a^2*b^21 + b^23)*sin(d*x + c))/cos(d*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10
*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(3/4) - sqrt(2)*((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)*d^7*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^
8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*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) + 2*(3*a^17*b + 20*a^15*b^3 + 56*a^13*b^5 + 84*a^11*b^7 + 70*a^9*b^9 + 28*a^7*b^11
- 4*a^3*b^15 - a*b^17)*d^5*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))*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^3 - 3*a*b^2)*d^2*sqrt((a
^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*b^2 +
12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^14 + 7*a
^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(3/4))/(9*a^34*b^2 + 1
29*a^32*b^4 + 856*a^30*b^6 + 3480*a^28*b^8 + 9660*a^26*b^10 + 19292*a^24*b^12 + 28392*a^22*b^14 + 30888*a^20*b
^16 + 24310*a^18*b^18 + 12870*a^16*b^20 + 3432*a^14*b^22 - 728*a^12*b^24 - 1092*a^10*b^26 - 420*a^8*b^28 - 40*
a^6*b^30 + 24*a^4*b^32 + 9*a^2*b^34 + b^36))*cos(d*x + c)^2 + 5*sqrt(2)*((a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6)
*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*c
os(d*x + c)^2 - (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*
d*cos(d*x + c)^2)*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^3 - 3*a*b^2)*d^2*sq
rt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*
b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8
+ 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(1/4)*log(((9*a^14*b^2 + 39*a^12*b^4 + 61*a^10*b^6 + 35*a^8*b^8 - 5*a^
6*b^10 - 11*a^4*b^12 - a^2*b^14 + b^16)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 +
21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + sqrt(2)*(2*(9*a^9*b^3 + 12*a^7*b^5 - 2*a^5*b^7 - 4*a^3*b^
9 + a*b^11)*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b
^14)/d^4)*cos(d*x + c) + (9*a^16*b^3 + 48*a^14*b^5 + 100*a^12*b^7 + 96*a^10*b^9 + 30*a^8*b^11 - 16*a^6*b^13 -
12*a^4*b^15 + b^19)*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^3
- 3*a*b^2)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b
^14)/d^4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos
(d*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)
^(1/4) + (9*a^21*b^2 + 66*a^19*b^4 + 205*a^17*b^6 + 344*a^15*b^8 + 322*a^13*b^10 + 140*a^11*b^12 - 14*a^9*b^14
- 40*a^7*b^16 - 11*a^5*b^18 + 2*a^3*b^20 + a*b^22)*cos(d*x + c) + (9*a^20*b^3 + 66*a^18*b^5 + 205*a^16*b^7 +
344*a^14*b^9 + 322*a^12*b^11 + 140*a^10*b^13 - 14*a^8*b^15 - 40*a^6*b^17 - 11*a^4*b^19 + 2*a^2*b^21 + b^23)*si
n(d*x + c))/cos(d*x + c)) - 5*sqrt(2)*((a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6)*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*
a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c)^2 - (a^14 + 7*a^12*b^2
+ 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d*cos(d*x + c)^2)*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^3 - 3*a*b^2)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^
4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a
^2*b^8 + b^10))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)
/d^4)^(1/4)*log(((9*a^14*b^2 + 39*a^12*b^4 + 61*a^10*b^6 + 35*a^8*b^8 - 5*a^6*b^10 - 11*a^4*b^12 - a^2*b^14 +
b^16)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d
^4)*cos(d*x + c) - sqrt(2)*(2*(9*a^9*b^3 + 12*a^7*b^5 - 2*a^5*b^7 - 4*a^3*b^9 + a*b^11)*d^3*sqrt((a^14 + 7*a^1
2*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + (9*a^16*b
^3 + 48*a^14*b^5 + 100*a^12*b^7 + 96*a^10*b^9 + 30*a^8*b^11 - 16*a^6*b^13 - 12*a^4*b^15 + b^19)*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^3 - 3*a*b^2)*d^2*sqrt((a^14 + 7*a^1
2*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(9*a^8*b^2 + 12*a^6*b^4
- 2*a^4*b^6 - 4*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^14 + 7*a^12*b^2 + 2
1*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(1/4) + (9*a^21*b^2 + 66*a^19*b^4
+ 205*a^17*b^6 + 344*a^15*b^8 + 322*a^13*b^10 + 140*a^11*b^12 - 14*a^9*b^14 - 40*a^7*b^16 - 11*a^5*b^18 + 2*a
^3*b^20 + a*b^22)*cos(d*x + c) + (9*a^20*b^3 + 66*a^18*b^5 + 205*a^16*b^7 + 344*a^14*b^9 + 322*a^12*b^11 + 140
*a^10*b^13 - 14*a^8*b^15 - 40*a^6*b^17 - 11*a^4*b^19 + 2*a^2*b^21 + b^23)*sin(d*x + c))/cos(d*x + c)) - 8*(a^1
4*b^3 + 7*a^12*b^5 + 21*a^10*b^7 + 35*a^8*b^9 + 35*a^6*b^11 + 21*a^4*b^13 + 7*a^2*b^15 + b^17 - 2*(2*a^16*b +
17*a^14*b^3 + 63*a^12*b^5 + 133*a^10*b^7 + 175*a^8*b^9 + 147*a^6*b^11 + 77*a^4*b^13 + 23*a^2*b^15 + 3*b^17)*co
s(d*x + c)^2 + 2*(a^15*b^2 + 7*a^13*b^4 + 21*a^11*b^6 + 35*a^9*b^8 + 35*a^7*b^10 + 21*a^5*b^12 + 7*a^3*b^14 +
a*b^16)*cos(d*x + c)*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/((a^14 + 7*a^12*b^2 +
21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d*cos(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 +
d*x), x)

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

[Out]

Timed out