3.5.46 \(\int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\) [446]

Optimal. Leaf size=342 \[ \frac {2 \left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{105 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (82 a^2 A b-6 A b^3+63 a^3 B+21 a b^2 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+7 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2 A+3 A b^2+42 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}} \]

[Out]

2/105*(a^2-b^2)*(25*A*a^2-6*A*b^2+21*B*a*b)*(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)+2/7*a*A*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(5/2)+2/35*(8*A*b+7*B*a)*sin(d*x+c)*(a+b*sec(d*x+c))
^(1/2)/d/sec(d*x+c)^(3/2)+2/105*(25*A*a^2+3*A*b^2+42*B*a*b)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d/sec(d*x+c)^(
1/2)+2/105*(82*A*a^2*b-6*A*b^3+63*B*a^3+21*B*a*b^2)*(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)

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Rubi [A]
time = 0.74, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4110, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 \left (25 a^2 A+42 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{105 a d \sqrt {\sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \left (25 a^2 A+21 a b B-6 A b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (63 a^3 B+82 a^2 A b+21 a b^2 B-6 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (7 a B+8 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]

[Out]

(2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)
/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*
a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/
(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(8*
A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*
b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*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 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 4110

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

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*} \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx &=\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2}{7} \int \frac {-\frac {1}{2} a (8 A b+7 a B)-\frac {1}{2} \left (5 a^2 A+7 A b^2+14 a b B\right ) \sec (c+d x)-\frac {1}{2} b (4 a A+7 b B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+7 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 \int \frac {\frac {1}{4} a \left (25 a^2 A+3 A b^2+42 a b B\right )+\frac {1}{4} a \left (44 a A b+21 a^2 B+35 b^2 B\right ) \sec (c+d x)+\frac {1}{2} a b (8 A b+7 a B) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{35 a}\\ &=\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+7 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2 A+3 A b^2+42 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}-\frac {8 \int \frac {-\frac {1}{8} a \left (82 a^2 A b-6 A b^3+63 a^3 B+21 a b^2 B\right )-\frac {1}{8} a^2 \left (25 a^2 A+51 A b^2+84 a b B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{105 a^2}\\ &=\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+7 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2 A+3 A b^2+42 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{105 a^2}+\frac {\left (82 a^2 A b-6 A b^3+63 a^3 B+21 a b^2 B\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{105 a^2}\\ &=\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+7 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2 A+3 A b^2+42 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B\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}{105 a^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (82 a^2 A b-6 A b^3+63 a^3 B+21 a b^2 B\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{105 a^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+7 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2 A+3 A b^2+42 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B\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}{105 a^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (82 a^2 A b-6 A b^3+63 a^3 B+21 a b^2 B\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{105 a^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}\\ &=\frac {2 \left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{105 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (82 a^2 A b-6 A b^3+63 a^3 B+21 a b^2 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+7 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2 A+3 A b^2+42 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 1.96, size = 255, normalized size = 0.75 \begin {gather*} \frac {(a+b \sec (c+d x))^{3/2} \left (4 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \left (a^2 \left (25 a^2 A+51 A b^2+84 a b B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+\left (82 a^2 A b-6 A b^3+63 a^3 B+21 a b^2 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-b F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )\right )\right )+a (b+a \cos (c+d x)) \left (\left (115 a^2 A+12 A b^2+168 a b B\right ) \sin (c+d x)+3 a (2 (8 A b+7 a B) \sin (2 (c+d x))+5 a A \sin (3 (c+d x)))\right )\right )}{210 a^2 d (b+a \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]

[Out]

((a + b*Sec[c + d*x])^(3/2)*(4*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(25*a^2*A + 51*A*b^2 + 84*a*b*B)*Ellipt
icF[(c + d*x)/2, (2*a)/(a + b)] + (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*((a + b)*EllipticE[(c + d*x)/
2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[c + d*x])*((115*a^2*A + 12*A*b^2
+ 168*a*b*B)*Sin[c + d*x] + 3*a*(2*(8*A*b + 7*a*B)*Sin[2*(c + d*x)] + 5*a*A*Sin[3*(c + d*x)]))))/(210*a^2*d*(b
+ a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3751\) vs. \(2(366)=732\).
time = 14.16, size = 3752, normalized size = 10.97

method	result	size
default	\(\text {Expression too large to display}\)	\(3752\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/105/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(6*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^3-21*B*cos(d*x+c)*((a-b)/
(a+b))^(1/2)*a^2*b^2+21*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^3+39*A*cos(d*x+c)^4*((a-b)/(a+b))^(1/2)*a^3*b+27*
A*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^2*b^2+63*B*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^3*b+6*A*((b+a*cos(d*x+c))/(
1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(
-(a+b)/(a-b))^(1/2))*b^4*sin(d*x+c)-63*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2
)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*sin(d*x+c)+63*B*((b+a*cos
(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/si
n(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*sin(d*x+c)-82*A*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b
))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2
))*a^3*b+25*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))
*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*sin(d*x+c)+21*B*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+
c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x
+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2-21*B*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/
(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3-6*A
*cos(d*x+c)*((a-b)/(a+b))^(1/2)*b^4-63*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^4+15*A*cos(d*x+c)^5*((a-b)/(a+b))^(1
/2)*a^4+10*A*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^4-25*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^4+21*B*cos(d*x+c)^4*((
a-b)/(a+b))^(1/2)*a^4+42*B*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^4+51*A*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(
1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(
-(a+b)/(a-b))^(1/2))*a^2*b^2+6*A*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos
(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3+82*A*sin(
d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x
+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b-82*A*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(
1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(
-(a+b)/(a-b))^(1/2))*a^2*b^2-6*A*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos
(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3+84*B*sin(
d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x
+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b-21*B*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(
1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(
-(a+b)/(a-b))^(1/2))*a^2*b^2-63*B*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+co
s(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b-25*A*((a
-b)/(a+b))^(1/2)*a^3*b-82*A*((a-b)/(a+b))^(1/2)*a^2*b^2-3*A*((a-b)/(a+b))^(1/2)*a*b^3-63*B*((a-b)/(a+b))^(1/2)
*a^3*b-42*B*((a-b)/(a+b))^(1/2)*a^2*b^2-21*B*((a-b)/(a+b))^(1/2)*a*b^3+25*A*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*
x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d
*x+c),(-(a+b)/(a-b))^(1/2))*a^4+6*A*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+
cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*b^4-63*B*sin
(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*
x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4+63*B*sin(d*x+c)*cos(d*x+c)*((b+a*cos(d*x+c))/(1
+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-
(a+b)/(a-b))^(1/2))*a^4-82*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF(
(-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b*sin(d*x+c)+51*A*((b+a*cos(d*x+c))/(
1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(
-(a+b)/(a-b))^(1/2))*a^2*b^2*sin(d*x+c)+6*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(
1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3*sin(d*x+c)+82*A*((b+
a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/
2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b*sin(d*x+c)-82*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+
cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)^(7/2),x, algorithm="maxima")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)/sec(d*x + c)^(7/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.02, size = 589, normalized size = 1.72 \begin {gather*} \frac {\sqrt {2} {\left (-75 i \, A a^{4} - 126 i \, B a^{3} b + 11 i \, A a^{2} b^{2} + 42 i \, B a b^{3} - 12 i \, A b^{4}\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 (75 i \, A a^{4} + 126 i \, B a^{3} b - 11 i \, A a^{2} b^{2} - 42 i \, B a b^{3} + 12 i \, A b^{4}\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 (-63 i \, B a^{4} - 82 i \, A a^{3} b - 21 i \, B a^{2} b^{2} + 6 i \, A a b^{3}\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 (63 i \, B a^{4} + 82 i \, A a^{3} b + 21 i \, B a^{2} b^{2} - 6 i \, A a b^{3}\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 (15 \, A a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (7 \, B a^{4} + 8 \, A a^{3} b\right )} \cos \left (d x + c\right )^{2} + {\left (25 \, A a^{4} + 42 \, B a^{3} b + 3 \, A a^{2} b^{2}\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 )}}}{315 \, a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)^(7/2),x, algorithm="fricas")

[Out]

1/315*(sqrt(2)*(-75*I*A*a^4 - 126*I*B*a^3*b + 11*I*A*a^2*b^2 + 42*I*B*a*b^3 - 12*I*A*b^4)*sqrt(a)*weierstrassP
Inverse(-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)*(75*I*A*a^4 + 126*I*B*a^3*b - 11*I*A*a^2*b^2 - 42*I*B*a*b^3 + 12*I*A*b^4)*sqrt(a)*weierstrassPI
nverse(-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)*(-63*I*B*a^4 - 82*I*A*a^3*b - 21*I*B*a^2*b^2 + 6*I*A*a*b^3)*sqrt(a)*weierstrassZeta(-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)*(63*I*B*a^4 + 82*I*A*a^3*b + 21*I*B*
a^2*b^2 - 6*I*A*a*b^3)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstra
ssPInverse(-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*(15*A*a^4*cos(d*x + c)^3 + 3*(7*B*a^4 + 8*A*a^3*b)*cos(d*x + c)^2 + (25*A*a^4 + 42*B*a^3*b + 3*A*
a^2*b^2)*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4371 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)/sec(d*x + c)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(3/2))/(1/cos(c + d*x))^(7/2),x)

[Out]

int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(3/2))/(1/cos(c + d*x))^(7/2), x)

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