3.6.27 \(\int (a+b \tan (c+d x))^{7/2} \, dx\) [527]

Optimal. Leaf size=167 \[ -\frac {i (a-i b)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b \left (3 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 167, 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 = {3563, 3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 b \left (3 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 {4 a b (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {i (a-i b)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^(7/2),x]

[Out]

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

Rule 65

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 214

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

Rule 3563

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 3609

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 3618

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]

Rule 3620

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 (a+b \tan (c+d x))^{7/2} \, dx &=\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\int (a+b \tan (c+d x))^{3/2} \left (a^2-b^2+2 a b \tan (c+d x)\right ) \, dx\\ &=\frac {4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\int \sqrt {a+b \tan (c+d x)} \left (a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {2 b \left (3 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\int \frac {a^4-6 a^2 b^2+b^4+4 a b \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 b \left (3 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {1}{2} (a-i b)^4 \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} (a+i b)^4 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 b \left (3 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {\left (i (a-i b)^4\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 (i (a+i b)^4\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 (3 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}-\frac {(a-i b)^4 \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 {(a+i b)^4 \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-i b)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b \left (3 a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 b (a+b \tan (c+d x))^{5/2}}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.99, size = 143, normalized size = 0.86 \begin {gather*} \frac {-15 i (a-i b)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+15 i (a+i b)^{7/2} \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)} \left (58 a^2-15 b^2+16 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)\right )}{15 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-15*I)*(a - I*b)^(7/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (15*I)*(a + I*b)^(7/2)*ArcTanh[Sqrt
[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]]*(58*a^2 - 15*b^2 + 16*a*b*Tan[c + d*x] + 3*
b^2*Tan[c + d*x]^2))/(15*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs. \(2(139)=278\).
time = 0.15, size = 958, normalized size = 5.74

method	result	size
derivativedivides	\(\frac {2 b \left (\frac {\left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 a^{2} \sqrt {a +b \tan \left (d x +c \right )}-b^{2} \sqrt {a +b \tan \left (d x +c \right )}+\frac {\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}-6 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-6 \sqrt {a^{2}+b^{2}}\, a^{2} b^{2}+2 b^{4} \sqrt {a^{2}+b^{2}}-\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}-6 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 b^{2}}+\frac {-\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}-6 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (6 \sqrt {a^{2}+b^{2}}\, a^{2} b^{2}-2 b^{4} \sqrt {a^{2}+b^{2}}+\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}-6 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 b^{2}}\right )}{d}\)	\(958\)
default	\(\frac {2 b \left (\frac {\left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 a^{2} \sqrt {a +b \tan \left (d x +c \right )}-b^{2} \sqrt {a +b \tan \left (d x +c \right )}+\frac {\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}-6 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-6 \sqrt {a^{2}+b^{2}}\, a^{2} b^{2}+2 b^{4} \sqrt {a^{2}+b^{2}}-\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}-6 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 b^{2}}+\frac {-\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}-6 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (6 \sqrt {a^{2}+b^{2}}\, a^{2} b^{2}-2 b^{4} \sqrt {a^{2}+b^{2}}+\frac {\left (-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}-6 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 b^{2}}\right )}{d}\)	\(958\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9021 vs. \(2 (133) = 266\).
time = 11.83, size = 9021, normalized size = 54.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(60*sqrt(2)*d^5*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 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*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))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8
+ 511*a^4*b^10 - 42*a^2*b^12 + b^14))*((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((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^
10 - 42*a^2*b^12 + b^14)/d^4)*arctan(-((7*a^26 + 35*a^24*b^2 - 14*a^22*b^4 - 526*a^20*b^6 - 1795*a^18*b^8 - 31
11*a^16*b^10 - 3060*a^14*b^12 - 1428*a^12*b^14 + 273*a^10*b^16 + 805*a^8*b^18 + 482*a^6*b^20 + 130*a^4*b^22 +
11*a^2*b^24 - b^26)*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((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12
+ b^14)/d^4) + (7*a^33 + 56*a^31*b^2 + 112*a^29*b^4 - 456*a^27*b^6 - 3380*a^25*b^8 - 10088*a^23*b^10 - 18304*a
^21*b^12 - 21736*a^19*b^14 - 16302*a^17*b^16 - 5720*a^15*b^18 + 2288*a^13*b^20 + 4264*a^11*b^22 + 2652*a^9*b^2
4 + 904*a^7*b^26 + 160*a^5*b^28 + 8*a^3*b^30 - a*b^32)*d^2*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1
484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4) + sqrt(2)*((21*a^14*b - 49*a^12*b^3 - 175*a^10*b^5 - 45*
a^8*b^7 + 111*a^6*b^9 + 29*a^4*b^11 - 21*a^2*b^13 + b^15)*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((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484
*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4) + 4*(7*a^21*b - 91*a^17*b^5 - 176*a^15*b^7 - 26*a^13*b^9 +
208*a^11*b^11 + 170*a^9*b^13 - 16*a^7*b^15 - 61*a^5*b^17 - 16*a^3*b^19 + a*b^21)*d^5*sqrt((49*a^12*b^2 - 490*a
^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/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 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^
6)*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)
)/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*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) + sqrt(2)*((3*a^2 - b^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((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 -
1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4) + 4*(a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((49*
a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/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 + (a^7 - 21*a^5*b^2 + 35*
a^3*b^4 - 7*a*b^6)*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))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^1
4))*sqrt(((49*a^20*b^2 - 294*a^18*b^4 - 147*a^16*b^6 + 1848*a^14*b^8 + 1778*a^12*b^10 - 1316*a^10*b^12 - 1518*
a^8*b^14 + 312*a^6*b^16 + 349*a^4*b^18 - 38*a^2*b^20 + b^22)*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)*(4*(49*a^15*b^3 - 539*a^13*b
^5 + 2009*a^11*b^7 - 3003*a^9*b^9 + 1995*a^7*b^11 - 553*a^5*b^13 + 43*a^3*b^15 - a*b^17)*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) + (147*a^2
2*b^3 - 931*a^20*b^5 - 147*a^18*b^7 + 5691*a^16*b^9 + 3486*a^14*b^11 - 5726*a^12*b^13 - 3238*a^10*b^15 + 2454*
a^8*b^17 + 735*a^6*b^19 - 463*a^4*b^21 + 41*a^2*b^23 - b^25)*d*cos(d*x + c))*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 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*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))/(4
9*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*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) + (49*a^27*b^2 - 147*a^25*b^4 - 882*a^23*b^6 + 574*a^21*b^8 + 6587*a^19*b^10
+ 9415*a^17*b^12 + 1716*a^15*b^14 - 6412*a^13*b^16 - 4585*a^11*b^18 + 427*a^9*b^20 + 1246*a^7*b^22 + 238*a^5*b
^24 - 35*a^3*b^26 + a*b^28)*cos(d*x + c) + (49*a^26*b^3 - 147*a^24*b^5 - 882*a^22*b^7 + 574*a^20*b^9 + 6587*a^
18*b^11 + 9415*a^16*b^13 + 1716*a^14*b^15 - 6412*a^12*b^17 - 4585*a^10*b^19 + 427*a^8*b^21 + 1246*a^6*b^23 + 2
38*a^4*b^25 - 35*a^2*b^27 + b^29)*sin(d*x + c))/((a^2 + b^2)*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...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [B]
time = 11.47, size = 2862, normalized size = 17.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(c + d*x))^(7/2),x)

[Out]

((8*a^2*b)/d - (2*b*(a^2 + b^2))/d)*(a + b*tan(c + d*x))^(1/2) - atan(((((32*(2*a^2*b^5*d^2 - b^7*d^2 + 3*a^4*
b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3
*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2))*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^
4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b^10 - 28*a^2*b^8 + 70*a^4*b^6
- 28*a^6*b^4 + a^8*b^2))/d^2)*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*35i + 2
1*a^5*b^2)/(4*d^2))^(1/2)*1i - (((32*(2*a^2*b^5*d^2 - b^7*d^2 + 3*a^4*b^3*d^2))/d^3 + 64*a*b^2*(a + b*tan(c +
d*x))^(1/2)*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2)
)^(1/2))*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(
1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^10 - 28*a^2*b^8 + 70*a^4*b^6 - 28*a^6*b^4 + a^8*b^2))/d^2)*((7*a*b^6
- a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2)*1i)/((((32*(2*
a^2*b^5*d^2 - b^7*d^2 + 3*a^4*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((7*a*b^6 - a^6*b*7i - a^7 +
b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2))*((7*a*b^6 - a^6*b*7i - a^7 + b^
7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b
^10 - 28*a^2*b^8 + 70*a^4*b^6 - 28*a^6*b^4 + a^8*b^2))/d^2)*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i
- 35*a^3*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) + (((32*(2*a^2*b^5*d^2 - b^7*d^2 + 3*a^4*b^3*d^2))/d^3
+ 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^
3*35i + 21*a^5*b^2)/(4*d^2))^(1/2))*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*3
5i + 21*a^5*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^10 - 28*a^2*b^8 + 70*a^4*b^6 - 28*a^6*b^4
+ a^8*b^2))/d^2)*((7*a*b^6 - a^6*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4
*d^2))^(1/2) + (64*(a*b^13 + 3*a^3*b^11 + 2*a^5*b^9 - 2*a^7*b^7 - 3*a^9*b^5 - a^11*b^3))/d^3))*((7*a*b^6 - a^6
*b*7i - a^7 + b^7*1i - a^2*b^5*21i - 35*a^3*b^4 + a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2)*2i - atan(((((32*(2
*a^2*b^5*d^2 - b^7*d^2 + 3*a^4*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((7*a*b^6 + a^6*b*7i - a^7
- b^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2))*((7*a*b^6 + a^6*b*7i - a^7 - b
^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(
b^10 - 28*a^2*b^8 + 70*a^4*b^6 - 28*a^6*b^4 + a^8*b^2))/d^2)*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b^5*21i
- 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2)*1i - (((32*(2*a^2*b^5*d^2 - b^7*d^2 + 3*a^4*b^3*d^2))
/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^
4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2))*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^4*b
^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^10 - 28*a^2*b^8 + 70*a^4*b^6 - 28*a^6*
b^4 + a^8*b^2))/d^2)*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2
)/(4*d^2))^(1/2)*1i)/((((32*(2*a^2*b^5*d^2 - b^7*d^2 + 3*a^4*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/
2)*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2))*
((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) - (1
6*(a + b*tan(c + d*x))^(1/2)*(b^10 - 28*a^2*b^8 + 70*a^4*b^6 - 28*a^6*b^4 + a^8*b^2))/d^2)*((7*a*b^6 + a^6*b*7
i - a^7 - b^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) + (((32*(2*a^2*b^5*d^2
- b^7*d^2 + 3*a^4*b^3*d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^
2*b^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2))*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b
^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^10 - 28*a^2
*b^8 + 70*a^4*b^6 - 28*a^6*b^4 + a^8*b^2))/d^2)*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b^5*21i - 35*a^3*b^4
- a^4*b^3*35i + 21*a^5*b^2)/(4*d^2))^(1/2) + (64*(a*b^13 + 3*a^3*b^11 + 2*a^5*b^9 - 2*a^7*b^7 - 3*a^9*b^5 - a
^11*b^3))/d^3))*((7*a*b^6 + a^6*b*7i - a^7 - b^7*1i + a^2*b^5*21i - 35*a^3*b^4 - a^4*b^3*35i + 21*a^5*b^2)/(4*
d^2))^(1/2)*2i + (2*b*(a + b*tan(c + d*x))^(5/2))/(5*d) + (4*a*b*(a + b*tan(c + d*x))^(3/2))/(3*d)

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