\(\int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx\) [687]

Optimal result

Integrand size = 23, antiderivative size = 362 \[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\frac {3 \left (9 a^2+32 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{220 b^2 d}-\frac {9 a (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{44 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}+\frac {a \left (18 a^2+49 b^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{110 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {\left (9 a^4+23 a^2 b^2-32 b^4\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{55 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \]

[Out]

3/220*(9*a^2+32*b^2)*(a+b*sec(d*x+c))^(2/3)*tan(d*x+c)/b^2/d-9/44*a*(a+b*sec(d*x+c))^(5/3)*tan(d*x+c)/b^2/d+3/
11*sec(d*x+c)*(a+b*sec(d*x+c))^(5/3)*tan(d*x+c)/b/d+1/220*a*(18*a^2+49*b^2)*AppellF1(1/2,-2/3,1/2,3/2,b*(1-sec
(d*x+c))/(a+b),1/2-1/2*sec(d*x+c))*(a+b*sec(d*x+c))^(2/3)*tan(d*x+c)/b^3/d/((a+b*sec(d*x+c))/(a+b))^(2/3)*2^(1
/2)/(1+sec(d*x+c))^(1/2)-1/110*(9*a^4+23*a^2*b^2-32*b^4)*AppellF1(1/2,1/3,1/2,3/2,b*(1-sec(d*x+c))/(a+b),1/2-1
/2*sec(d*x+c))*((a+b*sec(d*x+c))/(a+b))^(1/3)*tan(d*x+c)/b^3/d/(a+b*sec(d*x+c))^(1/3)*2^(1/2)/(1+sec(d*x+c))^(
1/2)

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3950, 4167, 4087, 4092, 3919, 144, 143} \[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\frac {a \left (18 a^2+49 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{110 \sqrt {2} b^3 d \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}+\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{220 b^2 d}-\frac {\left (9 a^4+23 a^2 b^2-32 b^4\right ) \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{55 \sqrt {2} b^3 d \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{44 b^2 d}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d} \]

[In]

Int[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(2/3),x]

[Out]

(3*(9*a^2 + 32*b^2)*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(220*b^2*d) - (9*a*(a + b*Sec[c + d*x])^(5/3)*Tan
[c + d*x])/(44*b^2*d) + (3*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(11*b*d) + (a*(18*a^2 + 49*b^
2)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2
/3)*Tan[c + d*x])/(110*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - ((9*a^4 +
23*a^2*b^2 - 32*b^4)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a +
b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(55*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/
3))

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c
- a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 3919

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[Cot[e + f*x]/(f*Sqr
t[1 + Csc[e + f*x]]*Sqrt[1 - Csc[e + f*x]]), Subst[Int[(a + b*x)^m/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Csc[e + f
*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&  !IntegerQ[2*m]

Rule 3950

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-d^3)
*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 3)/(b*f*(m + n - 1))), x] + Dist[d^3/(b*(m +
n - 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 3)*Simp[a*(n - 3) + b*(m + n - 2)*Csc[e + f*x] - a*
(n - 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 3] && (Integ
erQ[n] || IntegersQ[2*m, 2*n]) &&  !IGtQ[m, 2]

Rule 4087

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))
, x_Symbol] :> Simp[(-B)*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), Int[Csc[e + f
*x]*(a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1))*Csc[e + f*x], x], x], x] /;
FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]

Rule 4092

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))
, x_Symbol] :> Dist[(A*b - a*B)/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] + Dist[B/b, Int[Csc[e + f*x
]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^
2, 0]

Rule 4167

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_
.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m
+ 2))), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) + (b*
B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}+\frac {3 \int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \left (a+\frac {8}{3} b \sec (c+d x)-2 a \sec ^2(c+d x)\right ) \, dx}{11 b} \\ & = -\frac {9 a (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{44 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}+\frac {9 \int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \left (-\frac {2 a b}{3}+\frac {2}{9} \left (9 a^2+32 b^2\right ) \sec (c+d x)\right ) \, dx}{88 b^2} \\ & = \frac {3 \left (9 a^2+32 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{220 b^2 d}-\frac {9 a (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{44 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}+\frac {27 \int \frac {\sec (c+d x) \left (\frac {2}{27} b \left (3 a^2+64 b^2\right )+\frac {2}{27} a \left (18 a^2+49 b^2\right ) \sec (c+d x)\right )}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{440 b^2} \\ & = \frac {3 \left (9 a^2+32 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{220 b^2 d}-\frac {9 a (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{44 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}+\frac {\left (a \left (18 a^2+49 b^2\right )\right ) \int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \, dx}{220 b^3}-\frac {\left (9 a^4+23 a^2 b^2-32 b^4\right ) \int \frac {\sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{110 b^3} \\ & = \frac {3 \left (9 a^2+32 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{220 b^2 d}-\frac {9 a (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{44 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}-\frac {\left (a \left (18 a^2+49 b^2\right ) \tan (c+d x)\right ) \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{220 b^3 d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}+\frac {\left (\left (9 a^4+23 a^2 b^2-32 b^4\right ) \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \sqrt [3]{a+b x}} \, dx,x,\sec (c+d x)\right )}{110 b^3 d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}} \\ & = \frac {3 \left (9 a^2+32 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{220 b^2 d}-\frac {9 a (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{44 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}-\frac {\left (a \left (18 a^2+49 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{2/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{220 b^3 d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{2/3}}+\frac {\left (\left (9 a^4+23 a^2 b^2-32 b^4\right ) \sqrt [3]{-\frac {a+b \sec (c+d x)}{-a-b}} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \sqrt [3]{-\frac {a}{-a-b}-\frac {b x}{-a-b}}} \, dx,x,\sec (c+d x)\right )}{110 b^3 d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \\ & = \frac {3 \left (9 a^2+32 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{220 b^2 d}-\frac {9 a (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{44 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}+\frac {a \left (18 a^2+49 b^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{110 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {\left (9 a^4+23 a^2 b^2-32 b^4\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{55 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(21877\) vs. \(2(362)=724\).

Time = 46.27 (sec) , antiderivative size = 21877, normalized size of antiderivative = 60.43 \[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\text {Result too large to show} \]

[In]

Integrate[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(2/3),x]

[Out]

Result too large to show

Maple [F]

\[\int \sec \left (d x +c \right )^{4} \left (a +b \sec \left (d x +c \right )\right )^{\frac {2}{3}}d x\]

[In]

int(sec(d*x+c)^4*(a+b*sec(d*x+c))^(2/3),x)

[Out]

int(sec(d*x+c)^4*(a+b*sec(d*x+c))^(2/3),x)

Fricas [F]

\[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(sec(d*x+c)^4*(a+b*sec(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

integral((b*sec(d*x + c) + a)^(2/3)*sec(d*x + c)^4, x)

Sympy [F]

\[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}} \sec ^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate(sec(d*x+c)**4*(a+b*sec(d*x+c))**(2/3),x)

[Out]

Integral((a + b*sec(c + d*x))**(2/3)*sec(c + d*x)**4, x)

Maxima [F]

\[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(sec(d*x+c)^4*(a+b*sec(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^(2/3)*sec(d*x + c)^4, x)

Giac [F]

\[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(sec(d*x+c)^4*(a+b*sec(d*x+c))^(2/3),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^(2/3)*sec(d*x + c)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}}{{\cos \left (c+d\,x\right )}^4} \,d x \]

[In]

int((a + b/cos(c + d*x))^(2/3)/cos(c + d*x)^4,x)

[Out]

int((a + b/cos(c + d*x))^(2/3)/cos(c + d*x)^4, x)