\(\int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) [505]

Optimal result

Integrand size = 23, antiderivative size = 439 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}-\frac {\left (128 a^4-144 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{280 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 a \left (8 a^2-3 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{35 d \sqrt {a+b \sin (c+d x)}}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d} \]

[Out]

1/7*sec(d*x+c)^7*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^(3/2)/d+3/70*sec(d*x+c)^5*(3*a*b+(4*a^2-b^2)*sin(d*x+c))*(a
+b*sin(d*x+c))^(1/2)/d-1/140*sec(d*x+c)^3*(4*a*b*(a^2-b^2)-(32*a^4-39*a^2*b^2+7*b^4)*sin(d*x+c))*(a+b*sin(d*x+
c))^(1/2)/(a^2-b^2)/d-1/280*sec(d*x+c)*(a*b*(32*a^4-59*a^2*b^2+27*b^4)-(128*a^6-272*a^4*b^2+165*a^2*b^4-21*b^6
)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/(a^2-b^2)^2/d+1/280*(128*a^4-144*a^2*b^2+21*b^4)*(sin(1/2*c+1/4*Pi+1/2*d*
x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*
x+c))^(1/2)/(a^2-b^2)/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-2/35*a*(8*a^2-3*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2
)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+
b))^(1/2)/d/(a+b*sin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2770, 2940, 2945, 2831, 2742, 2740, 2734, 2732} \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-b^2\right ) \sin (c+d x)+3 a b\right )}{70 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{35 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 d \left (a^2-b^2\right )}-\frac {\left (128 a^4-144 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{280 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 d \left (a^2-b^2\right )^2}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{7 d} \]

[In]

Int[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

(Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(7*d) - ((128*a^4 - 144*a^2*b^2 + 21*b^4)*Ell
ipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(280*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x
])/(a + b)]) + (2*a*(8*a^2 - 3*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a
+ b)])/(35*d*Sqrt[a + b*Sin[c + d*x]]) + (3*Sec[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]]*(3*a*b + (4*a^2 - b^2)*Sin
[c + d*x]))/(70*d) - (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(4*a*b*(a^2 - b^2) - (32*a^4 - 39*a^2*b^2 + 7*b^
4)*Sin[c + d*x]))/(140*(a^2 - b^2)*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b*(32*a^4 - 59*a^2*b^2 + 27*
b^4) - (128*a^6 - 272*a^4*b^2 + 165*a^2*b^4 - 21*b^6)*Sin[c + d*x]))/(280*(a^2 - b^2)^2*d)

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 2770

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-(g*C
os[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Dist[1/(g^2*(p +
1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*S
in[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (Integers
Q[2*m, 2*p] || IntegerQ[m])

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[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]

Rule 2940

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f
*g*(p + 1))), x] + Dist[1/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(p
+ 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2,
0] && GtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b
*x])

Rule 2945

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c -
b*d)*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p
+ 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}-\frac {1}{7} \int \sec ^6(c+d x) \sqrt {a+b \sin (c+d x)} \left (-6 a^2+\frac {3 b^2}{2}-\frac {9}{2} a b \sin (c+d x)\right ) \, dx \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}+\frac {1}{35} \int \frac {\sec ^4(c+d x) \left (\frac {3}{4} a \left (32 a^2-11 b^2\right )+\frac {21}{4} b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\int \frac {\sec ^2(c+d x) \left (-6 a \left (8 a^4-11 a^2 b^2+3 b^4\right )-\frac {9}{8} b \left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 \left (a^2-b^2\right )} \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d}+\frac {\int \frac {-\frac {3}{16} a b^2 \left (32 a^4-59 a^2 b^2+27 b^4\right )-\frac {3}{16} b \left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 \left (a^2-b^2\right )^2} \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d}+\frac {1}{35} \left (a \left (8 a^2-3 b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {\left (128 a^4-144 a^2 b^2+21 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{560 \left (a^2-b^2\right )} \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d}-\frac {\left (\left (128 a^4-144 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{560 \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (a \left (8 a^2-3 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{35 \sqrt {a+b \sin (c+d x)}} \\ & = \frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}-\frac {\left (128 a^4-144 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{280 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 a \left (8 a^2-3 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{35 d \sqrt {a+b \sin (c+d x)}}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.87 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.77 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\frac {\left (\left (128 a^4-144 a^2 b^2+21 b^4\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )-16 a \left (8 a^3-8 a^2 b-3 a b^2+3 b^3\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right )\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a-b}+\frac {\sec (c+d x) (a+b \sin (c+d x)) \left (-32 a^3 b+27 a b^3+128 a^4 \sin (c+d x)-144 a^2 b^2 \sin (c+d x)+21 b^4 \sin (c+d x)+2 \left (a^2-b^2\right ) \sec ^2(c+d x) \left (-4 a b+\left (32 a^2-7 b^2\right ) \sin (c+d x)\right )-4 \left (a^2-b^2\right ) \sec ^4(c+d x) \left (a b+3 \left (-4 a^2+b^2\right ) \sin (c+d x)\right )+40 \left (a^2-b^2\right ) \sec ^6(c+d x) \left (2 a b+\left (a^2+b^2\right ) \sin (c+d x)\right )\right )}{a^2-b^2}}{280 d \sqrt {a+b \sin (c+d x)}} \]

[In]

Integrate[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^(5/2),x]

[Out]

((((128*a^4 - 144*a^2*b^2 + 21*b^4)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 16*a*(8*a^3 - 8*a^2*b -
3*a*b^2 + 3*b^3)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a - b)
+ (Sec[c + d*x]*(a + b*Sin[c + d*x])*(-32*a^3*b + 27*a*b^3 + 128*a^4*Sin[c + d*x] - 144*a^2*b^2*Sin[c + d*x] +
21*b^4*Sin[c + d*x] + 2*(a^2 - b^2)*Sec[c + d*x]^2*(-4*a*b + (32*a^2 - 7*b^2)*Sin[c + d*x]) - 4*(a^2 - b^2)*S
ec[c + d*x]^4*(a*b + 3*(-4*a^2 + b^2)*Sin[c + d*x]) + 40*(a^2 - b^2)*Sec[c + d*x]^6*(2*a*b + (a^2 + b^2)*Sin[c
+ d*x])))/(a^2 - b^2))/(280*d*Sqrt[a + b*Sin[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1886\) vs. \(2(477)=954\).

Time = 6439.01 (sec) , antiderivative size = 1887, normalized size of antiderivative = 4.30

method	result	size
default	\(\text {Expression too large to display}\)	\(1887\)

[In]

int(sec(d*x+c)^8*(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/280*(-(-b*sin(d*x+c)-a)*cos(d*x+c)^2)^(1/2)/cos(d*x+c)^9/b/(a+b*sin(d*x+c))^(3/2)/(a^2-b^2)*(2*cos(d*x+c)^4*
(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*b^2*(4*a^4-5*a^2*b^2+b^4)+40*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos
(d*x+c)^2)^(1/2)*b^2*(3*a^4-2*a^2*b^2-b^4)+4*cos(d*x+c)^2*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*b^2
*(a^4-14*a^2*b^2+13*b^4)-cos(d*x+c)^8*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*b^2*(128*a^4-144*a^2*b^
2+21*b^4)+16*cos(d*x+c)^2*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*a*b*(3*a^4-4*a^2*b^2+b^4)*sin(d*x+c
)+40*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*a*b*(a^4+2*a^2*b^2-3*b^4)*sin(d*x+c)+16*cos(d*x+c)^6*(b*
cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*a*b*(8*a^4-11*a^2*b^2+3*b^4)*sin(d*x+c)+2*cos(d*x+c)^4*(b*cos(d*
x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*a*b*(32*a^4-43*a^2*b^2+11*b^4)*sin(d*x+c)+cos(d*x+c)^6*(b*cos(d*x+c)^2
*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*(128*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b
))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^6-272*
(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)
*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^4*b^2+165*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*
EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b
)*sin(d*x+c)+b/(a+b))^(1/2)*a^2*b^4-21*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b
))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*b^6-128*
(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)
*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^5*b+96*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*Ell
ipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*s
in(d*x+c)+b/(a+b))^(1/2)*a^4*b^2+176*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))
^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^3*b^3-11
7*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-
b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a^2*b^4-48*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)
*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+
b)*sin(d*x+c)+b/(a+b))^(1/2)*a*b^5+21*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b)
)^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*b^6+32*a^
4*b^2-39*a^2*b^4+7*b^6))/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.18 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.59 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (256 \, a^{5} - 384 \, a^{3} b^{2} + 123 \, a b^{4}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{7} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + \sqrt {2} {\left (256 \, a^{5} - 384 \, a^{3} b^{2} + 123 \, a b^{4}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{7} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 3 \, \sqrt {2} {\left (128 i \, a^{4} b - 144 i \, a^{2} b^{3} + 21 i \, b^{5}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{7} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {2} {\left (-128 i \, a^{4} b + 144 i \, a^{2} b^{3} - 21 i \, b^{5}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{7} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 6 \, {\left ({\left (32 \, a^{3} b^{2} - 27 \, a b^{4}\right )} \cos \left (d x + c\right )^{6} - 80 \, a^{3} b^{2} + 80 \, a b^{4} + 8 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (128 \, a^{4} b - 144 \, a^{2} b^{3} + 21 \, b^{5}\right )} \cos \left (d x + c\right )^{6} + 40 \, a^{4} b - 40 \, b^{5} + 2 \, {\left (32 \, a^{4} b - 39 \, a^{2} b^{3} + 7 \, b^{5}\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{1680 \, {\left (a^{2} b - b^{3}\right )} d \cos \left (d x + c\right )^{7}} \]

[In]

integrate(sec(d*x+c)^8*(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/1680*(sqrt(2)*(256*a^5 - 384*a^3*b^2 + 123*a*b^4)*sqrt(I*b)*cos(d*x + c)^7*weierstrassPInverse(-4/3*(4*a^2 -
3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b) + sqrt(2)
*(256*a^5 - 384*a^3*b^2 + 123*a*b^4)*sqrt(-I*b)*cos(d*x + c)^7*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -
8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) + 3*sqrt(2)*(128*I*a^4
*b - 144*I*a^2*b^3 + 21*I*b^5)*sqrt(I*b)*cos(d*x + c)^7*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a
^3 - 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*c
os(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b)) + 3*sqrt(2)*(-128*I*a^4*b + 144*I*a^2*b^3 - 21*I*b^5)*sqrt(-I*b)
*cos(d*x + c)^7*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, weierstrassPInvers
e(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I
*a)/b)) - 6*((32*a^3*b^2 - 27*a*b^4)*cos(d*x + c)^6 - 80*a^3*b^2 + 80*a*b^4 + 8*(a^3*b^2 - a*b^4)*cos(d*x + c)
^4 + 4*(a^3*b^2 - a*b^4)*cos(d*x + c)^2 - ((128*a^4*b - 144*a^2*b^3 + 21*b^5)*cos(d*x + c)^6 + 40*a^4*b - 40*b
^5 + 2*(32*a^4*b - 39*a^2*b^3 + 7*b^5)*cos(d*x + c)^4 + 12*(4*a^4*b - 5*a^2*b^3 + b^5)*cos(d*x + c)^2)*sin(d*x
+ c))*sqrt(b*sin(d*x + c) + a))/((a^2*b - b^3)*d*cos(d*x + c)^7)

Sympy [F(-1)]

Timed out. \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**8*(a+b*sin(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{8} \,d x } \]

[In]

integrate(sec(d*x+c)^8*(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^(5/2)*sec(d*x + c)^8, x)

Giac [F(-1)]

Timed out. \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)^8*(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Hanged} \]

[In]

int((a + b*sin(c + d*x))^(5/2)/cos(c + d*x)^8,x)

[Out]

\text{Hanged}