3.341 \(\int (-a+b \tan (c+d x)) (a+b \tan (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=408 \[ -\frac{b \left (a^2+b^2\right ) \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \left (a^2+b^2\right ) \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 b (a+b \tan (c+d x))^{3/2}}{3 d} \]

[Out]

-((b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^
2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqr
t[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*(a^2 + b^2)*Log[
a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]
*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*(a^2 + b^2)*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqr
t[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*b*(a + b*Tan[c + d*x])^(
3/2))/(3*d)

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Rubi [A]  time = 0.473074, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3528, 12, 3485, 700, 1129, 634, 618, 206, 628} \[ -\frac{b \left (a^2+b^2\right ) \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \left (a^2+b^2\right ) \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}+\frac{2 b (a+b \tan (c+d x))^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^
2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqr
t[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*(a^2 + b^2)*Log[
a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]
*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*(a^2 + b^2)*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqr
t[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*b*(a + b*Tan[c + d*x])^(
3/2))/(3*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 3485

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

Rule 700

Int[Sqrt[(d_) + (e_.)*(x_)]/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2 + a*e^2 - 2*c*d
*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1129

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/
c, 2]}, Dist[1/(2*c*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[1/(2*c*r), Int[x^(m - 1)/(q + r*x + x^2),
x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [C]  time = 0.454474, size = 183, normalized size = 0.45 \[ \frac{(a-b \tan (c+d x)) \left (3 i \sqrt{a-i b} \left (a^2+b^2\right ) \cos (c+d x) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )-3 i \sqrt{a+i b} \left (a^2+b^2\right ) \cos (c+d x) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )+2 b \sqrt{a+b \tan (c+d x)} (a \cos (c+d x)+b \sin (c+d x))\right )}{3 d (a \cos (c+d x)-b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.089, size = 986, normalized size = 2.4 \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))^(3/2),x)

[Out]

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

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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))^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.17914, size = 9535, normalized size = 23.37 \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))^(3/2),x, algorithm="fricas")

[Out]

1/12*(12*sqrt(2)*d^5*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*((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((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4)*arctan(-(sqrt(2)*(
a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*si
n(d*x + c))/cos(d*x + c))*((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((a^
8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4) - sqrt(2)*d^5*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 +
a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))
*sqrt((sqrt(2)*(a^4*b^3 + 2*a^2*b^5 + b^7)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c)
+ b*sin(d*x + c))/cos(d*x + c))*((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)*co
s(d*x + c) + (a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*d^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)/d^4)*cos(d*x + c) + (a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b^1
0 + 6*a^3*b^12 + a*b^14)*cos(d*x + c) + (a^12*b^3 + 6*a^10*b^5 + 15*a^8*b^7 + 20*a^6*b^9 + 15*a^4*b^11 + 6*a^2
*b^13 + b^15)*sin(d*x + c))/cos(d*x + c))*((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((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4) + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a
^4*b^6 + 5*a^2*b^8 + b^10)*d^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)/d^4)*sqrt(
(a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4) + (a^15 + 7*a^13*b^2 + 21*a^11*b^4 + 35*a^9*b^6 + 35
*a^7*b^8 + 21*a^5*b^10 + 7*a^3*b^12 + a*b^14)*d^2*sqrt((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^
4))/(a^18*b^2 + 9*a^16*b^4 + 36*a^14*b^6 + 84*a^12*b^8 + 126*a^10*b^10 + 126*a^8*b^12 + 84*a^6*b^14 + 36*a^4*b
^16 + 9*a^2*b^18 + b^20))*cos(d*x + c) + 12*sqrt(2)*d^5*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*((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((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*
b^8 + b^10)/d^4)*arctan(-(sqrt(2)*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*d^5*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4
+ b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((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((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4) - sqrt(2)*d^5*sqrt((
a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a*d^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)
/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*sqrt(-(sqrt(2)*(a^4*b^3 + 2*a^2*b^5 + b^7)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a
^2*b^4 + b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a
^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((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)*cos(d*x + c) - (a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*d^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)/d^4)*cos(d*x + c) - (a^13*b^2 + 6*a^11*b^4 +
15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12 + a*b^14)*cos(d*x + c) - (a^12*b^3 + 6*a^10*b^5 + 15*a^8*b^
7 + 20*a^6*b^9 + 15*a^4*b^11 + 6*a^2*b^13 + b^15)*sin(d*x + c))/cos(d*x + c))*((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((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4) - (
a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^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)/d^4)*sqrt((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4) - (a^15 + 7*a^13
*b^2 + 21*a^11*b^4 + 35*a^9*b^6 + 35*a^7*b^8 + 21*a^5*b^10 + 7*a^3*b^12 + a*b^14)*d^2*sqrt((a^8*b^2 + 4*a^6*b^
4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)/d^4))/(a^18*b^2 + 9*a^16*b^4 + 36*a^14*b^6 + 84*a^12*b^8 + 126*a^10*b^10 + 1
26*a^8*b^12 + 84*a^6*b^14 + 36*a^4*b^16 + 9*a^2*b^18 + b^20))*cos(d*x + c) - 3*sqrt(2)*((a^5 + 2*a^3*b^2 + a*b
^4)*d^3*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)/d^4)*cos(d*x + c) - (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*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a
*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*(
(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((sqrt(2)*(a^4*b^3 + 2*a^2*b^5 +
b^7)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((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)^(3/4)*cos(d*x + c) + (a^8*b^2 + 4*a^6*b^4 + 6
*a^4*b^6 + 4*a^2*b^8 + b^10)*d^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)/d^4)*cos
(d*x + c) + (a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12 + a*b^14)*cos(d*x + c)
+ (a^12*b^3 + 6*a^10*b^5 + 15*a^8*b^7 + 20*a^6*b^9 + 15*a^4*b^11 + 6*a^2*b^13 + b^15)*sin(d*x + c))/cos(d*x +
c)) + 3*sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^3*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)/d^4)*cos(d*x + c) - (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*cos(d*x + c))*sqr
t((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*((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(-(sqrt(2)*(a^4*b^3 + 2*a^2*b^5 + b^7)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + a*d^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)/d^4))/(a^4*b^2 + 2*a^2*b^4 + b^6))*sqrt((a*cos(d*x +
c) + b*sin(d*x + c))/cos(d*x + c))*((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
)*cos(d*x + c) - (a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*d^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)/d^4)*cos(d*x + c) - (a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5
*b^10 + 6*a^3*b^12 + a*b^14)*cos(d*x + c) - (a^12*b^3 + 6*a^10*b^5 + 15*a^8*b^7 + 20*a^6*b^9 + 15*a^4*b^11 + 6
*a^2*b^13 + b^15)*sin(d*x + c))/cos(d*x + c)) + 8*((a^11*b + 5*a^9*b^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 +
a*b^11)*cos(d*x + c) + (a^10*b^2 + 5*a^8*b^4 + 10*a^6*b^6 + 10*a^4*b^8 + 5*a^2*b^10 + b^12)*sin(d*x + c))*sqr
t((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/((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*cos(d*x + c))

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

[Out]

-Integral(a**2*sqrt(a + b*tan(c + d*x)), x) - Integral(-b**2*sqrt(a + b*tan(c + d*x))*tan(c + d*x)**2, 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))^(3/2),x, algorithm="giac")

[Out]

Timed out