\(\int (a+b \sec (c+d x))^{3/2} (a^2-b^2 \sec ^2(c+d x)) \, dx\) [733]
Optimal result
Integrand size = 32, antiderivative size = 403 \[
\int (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (4 a^2-3 b^2\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{5 d}+\frac {2 \sqrt {a+b} \left (10 a^3-4 a^2 b-4 a b^2+3 b^3\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{5 d}-\frac {2 a^3 \sqrt {a+b} \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{d}-\frac {2 a b^2 \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{5 d}-\frac {2 b^2 (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}
\]
[Out]
-2/5*(a-b)*(4*a^2-3*b^2)*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1
/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/d+2/5*(10*a^3-4*a^2*b-4*a*b^2+3*b^3)*cot(d*
x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1
/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/d-2*a^3*cot(d*x+c)*EllipticPi((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,(
(a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/d-2/5*b^2*(a+b*
sec(d*x+c))^(3/2)*tan(d*x+c)/d-2/5*a*b^2*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d
Rubi [A] (verified)
Time = 0.68 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4127, 4003, 4141, 4143, 4006,
3869, 3917, 4089} \[
\int (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx=-\frac {2 a^3 \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {2 (a-b) \sqrt {a+b} \left (4 a^2-3 b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{5 d}+\frac {2 \sqrt {a+b} \left (10 a^3-4 a^2 b-4 a b^2+3 b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{5 d}-\frac {2 a b^2 \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d}-\frac {2 b^2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}
\]
[In]
Int[(a + b*Sec[c + d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2),x]
[Out]
(-2*(a - b)*Sqrt[a + b]*(4*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (
a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*d) + (2*Sqrt[
a + b]*(10*a^3 - 4*a^2*b - 4*a*b^2 + 3*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]
], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*d) - (2*a
^3*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b
)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*a*b^2*Sqrt[a + b*Sec[c
+ d*x]]*Tan[c + d*x])/(5*d) - (2*b^2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)
Rule 3869
Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b
*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b)*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b
*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Rule 3917
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 4003
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[(-b)
*d*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Dist[1/m, Int[(a + b*Csc[e + f*x])^(m - 2)*Simp[a^2
*c*m + (b^2*d*(m - 1) + 2*a*b*c*m + a^2*d*m)*Csc[e + f*x] + b*(b*c*m + a*d*(2*m - 1))*Csc[e + f*x]^2, x], x],
x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]
Rule 4006
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[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 4089
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Rule 4127
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Dist[
C/b^2, Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[-a + b*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x
] && EqQ[A*b^2 + a^2*C, 0]
Rule 4141
Int[((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/(f*(m + 1))), x] + Dist[1/(m + 1), I
nt[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)*Csc[e + f*x] + (b*B*(m + 1) +
a*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]
Rule 4143
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_
.) + (a_)], x_Symbol] :> Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Dist[C, Int[Csc[e + f*x
]*((1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0
]
Rubi steps \begin{align*}
\text {integral}& = -\int (-a+b \sec (c+d x)) (a+b \sec (c+d x))^{5/2} \, dx \\ & = -\frac {2 b^2 (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}-\frac {2}{5} \int \sqrt {a+b \sec (c+d x)} \left (-\frac {5 a^3}{2}-\frac {1}{2} b \left (5 a^2-3 b^2\right ) \sec (c+d x)+\frac {3}{2} a b^2 \sec ^2(c+d x)\right ) \, dx \\ & = -\frac {2 a b^2 \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{5 d}-\frac {2 b^2 (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}-\frac {4}{15} \int \frac {-\frac {15 a^4}{4}-\frac {3}{2} a b \left (5 a^2-2 b^2\right ) \sec (c+d x)-\frac {3}{4} b^2 \left (4 a^2-3 b^2\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = -\frac {2 a b^2 \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{5 d}-\frac {2 b^2 (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}-\frac {4}{15} \int \frac {-\frac {15 a^4}{4}+\left (\frac {3}{4} b^2 \left (4 a^2-3 b^2\right )-\frac {3}{2} a b \left (5 a^2-2 b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {1}{5} \left (b^2 \left (4 a^2-3 b^2\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = -\frac {2 (a-b) \sqrt {a+b} \left (4 a^2-3 b^2\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{5 d}-\frac {2 a b^2 \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{5 d}-\frac {2 b^2 (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{5 d}+a^4 \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {1}{5} \left (b \left (10 a^3-4 a^2 b-4 a b^2+3 b^3\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = -\frac {2 (a-b) \sqrt {a+b} \left (4 a^2-3 b^2\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{5 d}+\frac {2 \sqrt {a+b} \left (10 a^3-4 a^2 b-4 a b^2+3 b^3\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{5 d}-\frac {2 a^3 \sqrt {a+b} \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{d}-\frac {2 a b^2 \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{5 d}-\frac {2 b^2 (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{5 d} \\
\end{align*}
Mathematica [B] (warning: unable to verify)
Leaf count is larger than twice the leaf count of optimal. \(956\) vs. \(2(403)=806\).
Time = 18.69 (sec) , antiderivative size = 956, normalized size of antiderivative = 2.37
\[
\int (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx=-\frac {4 (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \left (-4 a^3 b \tan \left (\frac {1}{2} (c+d x)\right )-4 a^2 b^2 \tan \left (\frac {1}{2} (c+d x)\right )+3 a b^3 \tan \left (\frac {1}{2} (c+d x)\right )+3 b^4 \tan \left (\frac {1}{2} (c+d x)\right )+8 a^3 b \tan ^3\left (\frac {1}{2} (c+d x)\right )-6 a b^3 \tan ^3\left (\frac {1}{2} (c+d x)\right )-4 a^3 b \tan ^5\left (\frac {1}{2} (c+d x)\right )+4 a^2 b^2 \tan ^5\left (\frac {1}{2} (c+d x)\right )+3 a b^3 \tan ^5\left (\frac {1}{2} (c+d x)\right )-3 b^4 \tan ^5\left (\frac {1}{2} (c+d x)\right )+10 a^4 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+10 a^4 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+b \left (-4 a^3-4 a^2 b+3 a b^2+3 b^3\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-\left (5 a^4-10 a^3 b-4 a^2 b^2+4 a b^3+3 b^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}\right )}{5 d (b+a \cos (c+d x))^{3/2} \left (a^2-2 b^2+a^2 \cos (2 c+2 d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left (-1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}}}+\frac {\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \left (-\frac {4}{5} b \left (-4 a^2+3 b^2\right ) \sin (c+d x)-\frac {8}{5} a b^2 \tan (c+d x)-\frac {4}{5} b^3 \sec (c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x)) \left (a^2-2 b^2+a^2 \cos (2 c+2 d x)\right )}
\]
[In]
Integrate[(a + b*Sec[c + d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2),x]
[Out]
(-4*(a + b*Sec[c + d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2)*(-4*a^3*b*Tan[(c + d*x)/2] - 4*a^2*b^2*Tan[(c + d*x)
/2] + 3*a*b^3*Tan[(c + d*x)/2] + 3*b^4*Tan[(c + d*x)/2] + 8*a^3*b*Tan[(c + d*x)/2]^3 - 6*a*b^3*Tan[(c + d*x)/2
]^3 - 4*a^3*b*Tan[(c + d*x)/2]^5 + 4*a^2*b^2*Tan[(c + d*x)/2]^5 + 3*a*b^3*Tan[(c + d*x)/2]^5 - 3*b^4*Tan[(c +
d*x)/2]^5 + 10*a^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt
[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 10*a^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2
]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Ta
n[(c + d*x)/2]^2)/(a + b)] + b*(-4*a^3 - 4*a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b
)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c
+ d*x)/2]^2)/(a + b)] - (5*a^4 - 10*a^3*b - 4*a^2*b^2 + 4*a*b^3 + 3*b^4)*EllipticF[ArcSin[Tan[(c + d*x)/2]],
(a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*
Tan[(c + d*x)/2]^2)/(a + b)]))/(5*d*(b + a*Cos[c + d*x])^(3/2)*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*Sec[c + d*
x]^(7/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a
+ b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^3*(a + b*Sec[c +
d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2)*((-4*b*(-4*a^2 + 3*b^2)*Sin[c + d*x])/5 - (8*a*b^2*Tan[c + d*x])/5 - (
4*b^3*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x])*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x]))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(2776\) vs. \(2(364)=728\).
Time = 18.56 (sec) , antiderivative size = 2777, normalized size of antiderivative =
6.89
| | |
| method | result | size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(2777\) |
| parts |
\(\text {Expression too large to display}\) |
\(3581\) |
| | |
|
|
|
|
[In]
int((a+b*sec(d*x+c))^(3/2)*(a^2-b^2*sec(d*x+c)^2),x,method=_RETURNVERBOSE)
[Out]
2/5/d*(a+b*sec(d*x+c))^(1/2)/(b+a*cos(d*x+c))/(cos(d*x+c)+1)*(-b^4*sec(d*x+c)*tan(d*x+c)+2*a^2*b^2*sin(d*x+c)-
3*a*b^3*sin(d*x+c)-3*a*b^3*tan(d*x+c)+4*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c
)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b+4*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/
(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^2-3*(c
os(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c
),((a-b)/(a+b))^(1/2))*a*b^3-4*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/
2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^2+4*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(
b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a*b^3-10*(cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)
/(a+b))^(1/2))*a^3*b-3*sin(d*x+c)*b^4+4*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(
d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*b*cos(d*x+c)^2+4*EllipticE(cot(d*x+c)-csc
(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)
*a^2*b^2*cos(d*x+c)^2-3*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)
*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^3*cos(d*x+c)^2-10*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a
+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b*cos(d*x+
c)^2-4*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-
csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^2*cos(d*x+c)^2+4*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x
+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a*b^3*cos(d*x+c)^2+8*EllipticE
(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d
*x+c)+1))^(1/2)*a^3*b*cos(d*x+c)+8*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c
)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^2*cos(d*x+c)-6*EllipticE(cot(d*x+c)-csc(d*x+
c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^
3*cos(d*x+c)-20*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(co
t(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b*cos(d*x+c)-8*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*co
s(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^2*cos(d*x+c)+8*(cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),
((a-b)/(a+b))^(1/2))*a*b^3*cos(d*x+c)+4*a^3*b*cos(d*x+c)*sin(d*x+c)-2*a^2*b^2*cos(d*x+c)*sin(d*x+c)-3*a*b^3*co
s(d*x+c)*sin(d*x+c)+3*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellipt
icF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^4+3*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c
))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^4*cos(d*x+c)^2+6*(cos(d*x+c)/(
cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a
+b))^(1/2))*b^4*cos(d*x+c)-10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c)
,-1,((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4-3*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/
2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^4+5*(1/(a+b)*(b+a*
cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)*a^4-b^4*tan(d*x+c)-10*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*
x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4*cos(d*x+c)^2-3*EllipticE(cot(d*x+c)-csc(d*x+
c),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^4*
cos(d*x+c)^2+5*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot
(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^4*cos(d*x+c)^2-20*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^
(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4*cos(d*x+c)-6*Elli
pticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(
cos(d*x+c)+1))^(1/2)*b^4*cos(d*x+c)+10*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^4*cos(d*x+c))
Fricas [F]
\[
\int (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx=\int { -{\left (b^{2} \sec \left (d x + c\right )^{2} - a^{2}\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x }
\]
[In]
integrate((a+b*sec(d*x+c))^(3/2)*(a^2-b^2*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
integral(-(b^3*sec(d*x + c)^3 + a*b^2*sec(d*x + c)^2 - a^2*b*sec(d*x + c) - a^3)*sqrt(b*sec(d*x + c) + a), x)
Sympy [F]
\[
\int (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx=\int \left (a - b \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx
\]
[In]
integrate((a+b*sec(d*x+c))**(3/2)*(a**2-b**2*sec(d*x+c)**2),x)
[Out]
Integral((a - b*sec(c + d*x))*(a + b*sec(c + d*x))**(5/2), x)
Maxima [F]
\[
\int (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx=\int { -{\left (b^{2} \sec \left (d x + c\right )^{2} - a^{2}\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x }
\]
[In]
integrate((a+b*sec(d*x+c))^(3/2)*(a^2-b^2*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
-integrate((b^2*sec(d*x + c)^2 - a^2)*(b*sec(d*x + c) + a)^(3/2), x)
Giac [F]
\[
\int (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx=\int { -{\left (b^{2} \sec \left (d x + c\right )^{2} - a^{2}\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x }
\]
[In]
integrate((a+b*sec(d*x+c))^(3/2)*(a^2-b^2*sec(d*x+c)^2),x, algorithm="giac")
[Out]
integrate(-(b^2*sec(d*x + c)^2 - a^2)*(b*sec(d*x + c) + a)^(3/2), x)
Mupad [F(-1)]
Timed out. \[
\int (a+b \sec (c+d x))^{3/2} \left (a^2-b^2 \sec ^2(c+d x)\right ) \, dx=-\int -\left (a^2-\frac {b^2}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x
\]
[In]
int((a^2 - b^2/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2),x)
[Out]
-int(-(a^2 - b^2/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2), x)