\(\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) [646]
Optimal result
Integrand size = 25, antiderivative size = 363 \[
\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {4 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}
\]
[Out]
4/315*b*(57*a^4-62*a^2*b^2+5*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c)
,2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/a^2/d/(a+b*sec(d*x+c))^(1/2)+2/9*a^2
*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(7/2)+38/63*a*b*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)
^(5/2)+2/315*(49*a^2+75*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(3/2)+2/315*b*(163*a^2+5*b^2)*sin(
d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d/sec(d*x+c)^(1/2)+2/315*(147*a^4+279*a^2*b^2-10*b^4)*(cos(1/2*d*x+1/2*c)^2)^(
1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/a^2/d/((b
+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)
Rubi [A] (verified)
Time = 1.48 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.00, number of
steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3926, 4189, 4120, 3941, 2734,
2732, 3943, 2742, 2740} \[
\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (49 a^2+75 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{315 a d \sqrt {\sec (c+d x)}}+\frac {2 a^2 \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {38 a b \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{63 d \sec ^{\frac {5}{2}}(c+d x)}
\]
[In]
Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2),x]
[Out]
(4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sq
rt[Sec[c + d*x]])/(315*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*EllipticE[(c + d*
x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]
]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (38*a*b*Sqrt[a + b*Sec[c + d*x]]
*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(49*a^2 + 75*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d*
Sec[c + d*x]^(3/2)) + (2*b*(163*a^2 + 5*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sqrt[Sec[c + d*x]
])
Rule 2732
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2734
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] && !GtQ[a + b, 0]
Rule 2740
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2742
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] && !GtQ[a + b, 0]
Rule 3926
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^2*Co
t[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*n)), x] - Dist[1/(d*n), Int[(a + b*Csc[e + f*x]
)^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2*(n + 1))*Csc[e + f*x] - b*(b^2*
n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2]
&& ((IntegerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))
Rule 3941
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 3943
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[Sqrt[d*C
sc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 4120
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(
b_.) + (a_)]), x_Symbol] :> Dist[A/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[(A*b -
a*B)/(a*d), Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && Ne
Q[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Rule 4189
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1
)*((d*Csc[e + f*x])^n/(a*f*n)), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Rubi steps \begin{align*}
\text {integral}& = \frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {\frac {19 a^2 b}{2}+\frac {1}{2} a \left (7 a^2+27 b^2\right ) \sec (c+d x)+\frac {3}{2} b \left (2 a^2+3 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {4 \int \frac {-\frac {1}{4} a^2 \left (49 a^2+75 b^2\right )-\frac {1}{4} a b \left (137 a^2+63 b^2\right ) \sec (c+d x)-19 a^2 b^2 \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{63 a} \\ & = \frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {\frac {3}{8} a^2 b \left (163 a^2+5 b^2\right )+\frac {1}{8} a^3 \left (147 a^2+605 b^2\right ) \sec (c+d x)+\frac {1}{4} a^2 b \left (49 a^2+75 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{315 a^2} \\ & = \frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}-\frac {16 \int \frac {-\frac {3}{16} a^2 \left (147 a^4+279 a^2 b^2-10 b^4\right )-\frac {3}{16} a^3 b \left (261 a^2+155 b^2\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^3} \\ & = \frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}+\frac {\left (147 a^4+279 a^2 b^2-10 b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{315 a^2}+\frac {\left (2 b \left (57 a^4-62 a^2 b^2+5 b^4\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 a^2} \\ & = \frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}+\frac {\left (2 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{315 a^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4+279 a^2 b^2-10 b^4\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}}+\frac {\left (2 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{315 a^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4+279 a^2 b^2-10 b^4\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{315 a^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = \frac {4 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4+279 a^2 b^2-10 b^4\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a^2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {38 a b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2+75 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (163 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sqrt {\sec (c+d x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 2.21 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.79
\[
\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^{5/2} \left (16 \left (147 a^5+147 a^4 b+279 a^3 b^2+279 a^2 b^3-10 a b^4-10 b^5\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+32 b \left (57 a^4-62 a^2 b^2+5 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )+2 a \left (301 a^4+1984 a^2 b^2+40 b^4+4 a b \left (619 a^2+160 b^2\right ) \cos (c+d x)+8 \left (42 a^4+85 a^2 b^2\right ) \cos (2 (c+d x))+260 a^3 b \cos (3 (c+d x))+35 a^4 \cos (4 (c+d x))\right ) \sin (c+d x)\right )}{2520 a^2 d (b+a \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)}
\]
[In]
Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2),x]
[Out]
((a + b*Sec[c + d*x])^(5/2)*(16*(147*a^5 + 147*a^4*b + 279*a^3*b^2 + 279*a^2*b^3 - 10*a*b^4 - 10*b^5)*Sqrt[(b
+ a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 32*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*Sqrt[(b
+ a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(301*a^4 + 1984*a^2*b^2 + 40*b^4 + 4*a*
b*(619*a^2 + 160*b^2)*Cos[c + d*x] + 8*(42*a^4 + 85*a^2*b^2)*Cos[2*(c + d*x)] + 260*a^3*b*Cos[3*(c + d*x)] + 3
5*a^4*Cos[4*(c + d*x)])*Sin[c + d*x]))/(2520*a^2*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(3821\) vs. \(2(381)=762\).
Time = 8.94 (sec) , antiderivative size = 3822, normalized size of antiderivative =
10.53
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\(3822\) |
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[In]
int((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(9/2),x,method=_RETURNVERBOSE)
[Out]
2/315/d/((a-b)/(a+b))^(1/2)/a^2*(-147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)
*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b*cos(d*x+c)^2-10*((a-b)/(a+
b))^(1/2)*b^5*sin(d*x+c)+155*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*Elliptic
F(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3*cos(d*x+c)^2+10*(1/(a+b)*(b+a*cos
(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c))
,(-(a+b)/(a-b))^(1/2))*a*b^4*cos(d*x+c)^2-294*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1
))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b*cos(d*x+c)+558*(1/
(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+
c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*b^2*cos(d*x+c)-558*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1
/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3*co
s(d*x+c)-20*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(
1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^4*cos(d*x+c)+522*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)
+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2
))*a^4*b*cos(d*x+c)-558*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(((a
-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*b^2*cos(d*x+c)+310*(1/(a+b)*(b+a*cos(d*x+c
))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+
b)/(a-b))^(1/2))*a^2*b^3*cos(d*x+c)+20*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2
)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^4*cos(d*x+c)-147*(1/(a+b)*(
b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1
/2))*(1/(cos(d*x+c)+1))^(1/2)*a^4*b+279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b)
)^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2-279*(1/(a+b)*(b+a*cos(
d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/
(cos(d*x+c)+1))^(1/2)*a^2*b^3-10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)
*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(d*x+c)+1))^(1/2)*a*b^4+261*(1/(a+b)*(b+a*cos(d*x+c))/(
cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(d*x+
c)+1))^(1/2)*a^4*b-279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x
+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2+155*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+
c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(d*x+c)+1))^(
1/2)*a^2*b^3+10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc
(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(d*x+c)+1))^(1/2)*a*b^4+147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1
/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^5*
cos(d*x+c)^2+10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b
))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^5*cos(d*x+c)^2-147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*
x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^
(1/2))*a^5*cos(d*x+c)^2+294*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE
(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^5*cos(d*x+c)+147*(1/(a+b)*(b+a*cos(d*x+c
))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(
d*x+c)+1))^(1/2)*a^5+10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*
x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(d*x+c)+1))^(1/2)*b^5-147*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+
1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(cos(d*x+c)+1))^(1/2
)*a^5+147*((a-b)/(a+b))^(1/2)*a^4*b*sin(d*x+c)+163*((a-b)/(a+b))^(1/2)*a^3*b^2*sin(d*x+c)+279*((a-b)/(a+b))^(1
/2)*a^2*b^3*sin(d*x+c)+5*((a-b)/(a+b))^(1/2)*a*b^4*sin(d*x+c)+35*((a-b)/(a+b))^(1/2)*a^5*cos(d*x+c)^5*sin(d*x+
c)+35*((a-b)/(a+b))^(1/2)*a^5*cos(d*x+c)^4*sin(d*x+c)+49*((a-b)/(a+b))^(1/2)*a^5*cos(d*x+c)^3*sin(d*x+c)+49*((
a-b)/(a+b))^(1/2)*a^5*cos(d*x+c)^2*sin(d*x+c)+147*((a-b)/(a+b))^(1/2)*a^5*cos(d*x+c)*sin(d*x+c)+20*(1/(a+b)*(b
+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d
*x+c)),(-(a+b)/(a-b))^(1/2))*b^5*cos(d*x+c)-294*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)
+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^5*cos(d*x+c)+130*((a
-b)/(a+b))^(1/2)*a^4*b*cos(d*x+c)^4*sin(d*x+c)+130*((a-b)/(a+b))^(1/2)*a^4*b*cos(d*x+c)^3*sin(d*x+c)+170*((a-b
)/(a+b))^(1/2)*a^3*b^2*cos(d*x+c)^3*sin(d*x+c)+212*((a-b)/(a+b))^(1/2)*a^4*b*cos(d*x+c)^2*sin(d*x+c)+170*((a-b
)/(a+b))^(1/2)*a^3*b^2*cos(d*x+c)^2*sin(d*x+c)+80*((a-b)/(a+b))^(1/2)*a^2*b^3*cos(d*x+c)^2*sin(d*x+c)+212*((a-
b)/(a+b))^(1/2)*a^4*b*cos(d*x+c)*sin(d*x+c)+442*((a-b)/(a+b))^(1/2)*a^3*b^2*cos(d*x+c)*sin(d*x+c)+80*((a-b)/(a
+b))^(1/2)*a^2*b^3*cos(d*x+c)*sin(d*x+c)-5*((a-b)/(a+b))^(1/2)*a*b^4*cos(d*x+c)*sin(d*x+c)+279*(1/(a+b)*(b+a*c
os(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c
)),(-(a+b)/(a-b))^(1/2))*a^3*b^2*cos(d*x+c)^2-279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+
c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3*cos(d*x+c)^2
-10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(-c
ot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^4*cos(d*x+c)^2+261*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(
1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4
*b*cos(d*x+c)^2-279*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/
(a+b))^(1/2)*(-cot(d*x+c)+csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*b^2*cos(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/(b+a*
cos(d*x+c))/sec(d*x+c)^(1/2)/(cos(d*x+c)+1)
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.49
\[
\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left (-489 i \, a^{4} b + 93 i \, a^{2} b^{3} - 20 i \, b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (489 i \, a^{4} b - 93 i \, a^{2} b^{3} + 20 i \, b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-147 i \, a^{5} - 279 i \, a^{3} b^{2} + 10 i \, a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (147 i \, a^{5} + 279 i \, a^{3} b^{2} - 10 i \, a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (35 \, a^{5} \cos \left (d x + c\right )^{4} + 95 \, a^{4} b \cos \left (d x + c\right )^{3} + {\left (49 \, a^{5} + 75 \, a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (163 \, a^{4} b + 5 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{945 \, a^{3} d}
\]
[In]
integrate((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(9/2),x, algorithm="fricas")
[Out]
1/945*(sqrt(2)*(-489*I*a^4*b + 93*I*a^2*b^3 - 20*I*b^5)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2,
8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) + sqrt(2)*(489*I*a^4*b - 93*I
*a^2*b^3 + 20*I*b^5)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*
a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(-147*I*a^5 - 279*I*a^3*b^2 + 10*I*a*b^4)*sqrt(a)*we
ierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a
^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(147*I*a^5 +
279*I*a^3*b^2 - 10*I*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, wei
erstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x +
c) + 2*b)/a)) + 6*(35*a^5*cos(d*x + c)^4 + 95*a^4*b*cos(d*x + c)^3 + (49*a^5 + 75*a^3*b^2)*cos(d*x + c)^2 + (
163*a^4*b + 5*a^2*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/
(a^3*d)
Sympy [F(-1)]
Timed out. \[
\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out}
\]
[In]
integrate((a+b*sec(d*x+c))**(5/2)/sec(d*x+c)**(9/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x }
\]
[In]
integrate((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(9/2),x, algorithm="maxima")
[Out]
integrate((b*sec(d*x + c) + a)^(5/2)/sec(d*x + c)^(9/2), x)
Giac [F]
\[
\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x }
\]
[In]
integrate((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(9/2),x, algorithm="giac")
[Out]
integrate((b*sec(d*x + c) + a)^(5/2)/sec(d*x + c)^(9/2), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x
\]
[In]
int((a + b/cos(c + d*x))^(5/2)/(1/cos(c + d*x))^(9/2),x)
[Out]
int((a + b/cos(c + d*x))^(5/2)/(1/cos(c + d*x))^(9/2), x)