3.15 \(\int x \sec ^7(a+b x^2) \, dx\)
Optimal. Leaf size=90 \[ \frac{5 \tanh ^{-1}\left (\sin \left (a+b x^2\right )\right )}{32 b}+\frac{\tan \left (a+b x^2\right ) \sec ^5\left (a+b x^2\right )}{12 b}+\frac{5 \tan \left (a+b x^2\right ) \sec ^3\left (a+b x^2\right )}{48 b}+\frac{5 \tan \left (a+b x^2\right ) \sec \left (a+b x^2\right )}{32 b} \]
[Out]
(5*ArcTanh[Sin[a + b*x^2]])/(32*b) + (5*Sec[a + b*x^2]*Tan[a + b*x^2])/(32*b) + (5*Sec[a + b*x^2]^3*Tan[a + b*
x^2])/(48*b) + (Sec[a + b*x^2]^5*Tan[a + b*x^2])/(12*b)
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Rubi [A] time = 0.0733805, antiderivative size = 90, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used =
{4204, 3768, 3770} \[ \frac{5 \tanh ^{-1}\left (\sin \left (a+b x^2\right )\right )}{32 b}+\frac{\tan \left (a+b x^2\right ) \sec ^5\left (a+b x^2\right )}{12 b}+\frac{5 \tan \left (a+b x^2\right ) \sec ^3\left (a+b x^2\right )}{48 b}+\frac{5 \tan \left (a+b x^2\right ) \sec \left (a+b x^2\right )}{32 b} \]
Antiderivative was successfully verified.
[In]
Int[x*Sec[a + b*x^2]^7,x]
[Out]
(5*ArcTanh[Sin[a + b*x^2]])/(32*b) + (5*Sec[a + b*x^2]*Tan[a + b*x^2])/(32*b) + (5*Sec[a + b*x^2]^3*Tan[a + b*
x^2])/(48*b) + (Sec[a + b*x^2]^5*Tan[a + b*x^2])/(12*b)
Rule 4204
Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]
Rule 3768
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int x \sec ^7\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sec ^7(a+b x) \, dx,x,x^2\right )\\ &=\frac{\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b}+\frac{5}{12} \operatorname{Subst}\left (\int \sec ^5(a+b x) \, dx,x,x^2\right )\\ &=\frac{5 \sec ^3\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{48 b}+\frac{\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b}+\frac{5}{16} \operatorname{Subst}\left (\int \sec ^3(a+b x) \, dx,x,x^2\right )\\ &=\frac{5 \sec \left (a+b x^2\right ) \tan \left (a+b x^2\right )}{32 b}+\frac{5 \sec ^3\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{48 b}+\frac{\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b}+\frac{5}{32} \operatorname{Subst}\left (\int \sec (a+b x) \, dx,x,x^2\right )\\ &=\frac{5 \tanh ^{-1}\left (\sin \left (a+b x^2\right )\right )}{32 b}+\frac{5 \sec \left (a+b x^2\right ) \tan \left (a+b x^2\right )}{32 b}+\frac{5 \sec ^3\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{48 b}+\frac{\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b}\\ \end{align*}
Mathematica [A] time = 0.201372, size = 62, normalized size = 0.69 \[ \frac{15 \tanh ^{-1}\left (\sin \left (a+b x^2\right )\right )+\tan \left (a+b x^2\right ) \sec \left (a+b x^2\right ) \left (8 \sec ^4\left (a+b x^2\right )+10 \sec ^2\left (a+b x^2\right )+15\right )}{96 b} \]
Antiderivative was successfully verified.
[In]
Integrate[x*Sec[a + b*x^2]^7,x]
[Out]
(15*ArcTanh[Sin[a + b*x^2]] + Sec[a + b*x^2]*(15 + 10*Sec[a + b*x^2]^2 + 8*Sec[a + b*x^2]^4)*Tan[a + b*x^2])/(
96*b)
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Maple [A] time = 0.02, size = 92, normalized size = 1. \begin{align*}{\frac{ \left ( \sec \left ( b{x}^{2}+a \right ) \right ) ^{5}\tan \left ( b{x}^{2}+a \right ) }{12\,b}}+{\frac{5\, \left ( \sec \left ( b{x}^{2}+a \right ) \right ) ^{3}\tan \left ( b{x}^{2}+a \right ) }{48\,b}}+{\frac{5\,\sec \left ( b{x}^{2}+a \right ) \tan \left ( b{x}^{2}+a \right ) }{32\,b}}+{\frac{5\,\ln \left ( \sec \left ( b{x}^{2}+a \right ) +\tan \left ( b{x}^{2}+a \right ) \right ) }{32\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*sec(b*x^2+a)^7,x)
[Out]
1/12*sec(b*x^2+a)^5*tan(b*x^2+a)/b+5/48*sec(b*x^2+a)^3*tan(b*x^2+a)/b+5/32*sec(b*x^2+a)*tan(b*x^2+a)/b+5/32/b*
ln(sec(b*x^2+a)+tan(b*x^2+a))
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Maxima [B] time = 2.16545, size = 3831, normalized size = 42.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*sec(b*x^2+a)^7,x, algorithm="maxima")
[Out]
1/192*(4*(15*sin(11*b*x^2 + 11*a) + 85*sin(9*b*x^2 + 9*a) + 198*sin(7*b*x^2 + 7*a) - 198*sin(5*b*x^2 + 5*a) -
85*sin(3*b*x^2 + 3*a) - 15*sin(b*x^2 + a))*cos(12*b*x^2 + 12*a) - 60*(6*sin(10*b*x^2 + 10*a) + 15*sin(8*b*x^2
+ 8*a) + 20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*cos(11*b*x^2 + 11*a) + 24*(85*s
in(9*b*x^2 + 9*a) + 198*sin(7*b*x^2 + 7*a) - 198*sin(5*b*x^2 + 5*a) - 85*sin(3*b*x^2 + 3*a) - 15*sin(b*x^2 + a
))*cos(10*b*x^2 + 10*a) - 340*(15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2
*b*x^2 + 2*a))*cos(9*b*x^2 + 9*a) + 60*(198*sin(7*b*x^2 + 7*a) - 198*sin(5*b*x^2 + 5*a) - 85*sin(3*b*x^2 + 3*a
) - 15*sin(b*x^2 + a))*cos(8*b*x^2 + 8*a) - 792*(20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2
+ 2*a))*cos(7*b*x^2 + 7*a) - 80*(198*sin(5*b*x^2 + 5*a) + 85*sin(3*b*x^2 + 3*a) + 15*sin(b*x^2 + a))*cos(6*b*
x^2 + 6*a) + 2376*(5*sin(4*b*x^2 + 4*a) + 2*sin(2*b*x^2 + 2*a))*cos(5*b*x^2 + 5*a) - 300*(17*sin(3*b*x^2 + 3*a
) + 3*sin(b*x^2 + a))*cos(4*b*x^2 + 4*a) - 15*(2*(6*cos(10*b*x^2 + 10*a) + 15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*
x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*cos(12*b*x^2 + 12*a) + cos(12*b*x^2 + 12*a)^2 +
12*(15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*cos(10*
b*x^2 + 10*a) + 36*cos(10*b*x^2 + 10*a)^2 + 30*(20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2
+ 2*a) + 1)*cos(8*b*x^2 + 8*a) + 225*cos(8*b*x^2 + 8*a)^2 + 40*(15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) +
1)*cos(6*b*x^2 + 6*a) + 400*cos(6*b*x^2 + 6*a)^2 + 30*(6*cos(2*b*x^2 + 2*a) + 1)*cos(4*b*x^2 + 4*a) + 225*cos
(4*b*x^2 + 4*a)^2 + 36*cos(2*b*x^2 + 2*a)^2 + 2*(6*sin(10*b*x^2 + 10*a) + 15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x
^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin(12*b*x^2 + 12*a) + sin(12*b*x^2 + 12*a)^2 + 12*(
15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin(10*b*x^2 + 1
0*a) + 36*sin(10*b*x^2 + 10*a)^2 + 30*(20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*s
in(8*b*x^2 + 8*a) + 225*sin(8*b*x^2 + 8*a)^2 + 120*(5*sin(4*b*x^2 + 4*a) + 2*sin(2*b*x^2 + 2*a))*sin(6*b*x^2 +
6*a) + 400*sin(6*b*x^2 + 6*a)^2 + 225*sin(4*b*x^2 + 4*a)^2 + 180*sin(4*b*x^2 + 4*a)*sin(2*b*x^2 + 2*a) + 36*s
in(2*b*x^2 + 2*a)^2 + 12*cos(2*b*x^2 + 2*a) + 1)*log((cos(b*x^2 + 2*a)^2 + cos(a)^2 - 2*cos(a)*sin(b*x^2 + 2*a
) + sin(b*x^2 + 2*a)^2 + 2*cos(b*x^2 + 2*a)*sin(a) + sin(a)^2)/(cos(b*x^2 + 2*a)^2 + cos(a)^2 + 2*cos(a)*sin(b
*x^2 + 2*a) + sin(b*x^2 + 2*a)^2 - 2*cos(b*x^2 + 2*a)*sin(a) + sin(a)^2)) - 4*(15*cos(11*b*x^2 + 11*a) + 85*co
s(9*b*x^2 + 9*a) + 198*cos(7*b*x^2 + 7*a) - 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) - 15*cos(b*x^2 + a)
)*sin(12*b*x^2 + 12*a) + 60*(6*cos(10*b*x^2 + 10*a) + 15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*x^2 + 6*a) + 15*cos(4
*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*sin(11*b*x^2 + 11*a) - 24*(85*cos(9*b*x^2 + 9*a) + 198*cos(7*b*x^2 +
7*a) - 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) - 15*cos(b*x^2 + a))*sin(10*b*x^2 + 10*a) + 340*(15*cos
(8*b*x^2 + 8*a) + 20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*sin(9*b*x^2 + 9*a)
- 60*(198*cos(7*b*x^2 + 7*a) - 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) - 15*cos(b*x^2 + a))*sin(8*b*x^
2 + 8*a) + 792*(20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*sin(7*b*x^2 + 7*a) +
80*(198*cos(5*b*x^2 + 5*a) + 85*cos(3*b*x^2 + 3*a) + 15*cos(b*x^2 + a))*sin(6*b*x^2 + 6*a) - 792*(15*cos(4*b*
x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*sin(5*b*x^2 + 5*a) + 300*(17*cos(3*b*x^2 + 3*a) + 3*cos(b*x^2 + a))*sin
(4*b*x^2 + 4*a) - 340*(6*cos(2*b*x^2 + 2*a) + 1)*sin(3*b*x^2 + 3*a) + 2040*cos(3*b*x^2 + 3*a)*sin(2*b*x^2 + 2*
a) + 360*cos(b*x^2 + a)*sin(2*b*x^2 + 2*a) - 360*cos(2*b*x^2 + 2*a)*sin(b*x^2 + a) - 60*sin(b*x^2 + a))/(b*cos
(12*b*x^2 + 12*a)^2 + 36*b*cos(10*b*x^2 + 10*a)^2 + 225*b*cos(8*b*x^2 + 8*a)^2 + 400*b*cos(6*b*x^2 + 6*a)^2 +
225*b*cos(4*b*x^2 + 4*a)^2 + 36*b*cos(2*b*x^2 + 2*a)^2 + b*sin(12*b*x^2 + 12*a)^2 + 36*b*sin(10*b*x^2 + 10*a)^
2 + 225*b*sin(8*b*x^2 + 8*a)^2 + 400*b*sin(6*b*x^2 + 6*a)^2 + 225*b*sin(4*b*x^2 + 4*a)^2 + 180*b*sin(4*b*x^2 +
4*a)*sin(2*b*x^2 + 2*a) + 36*b*sin(2*b*x^2 + 2*a)^2 + 2*(6*b*cos(10*b*x^2 + 10*a) + 15*b*cos(8*b*x^2 + 8*a) +
20*b*cos(6*b*x^2 + 6*a) + 15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) + b)*cos(12*b*x^2 + 12*a) + 12*(15
*b*cos(8*b*x^2 + 8*a) + 20*b*cos(6*b*x^2 + 6*a) + 15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) + b)*cos(10
*b*x^2 + 10*a) + 30*(20*b*cos(6*b*x^2 + 6*a) + 15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) + b)*cos(8*b*x
^2 + 8*a) + 40*(15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) + b)*cos(6*b*x^2 + 6*a) + 30*(6*b*cos(2*b*x^2
+ 2*a) + b)*cos(4*b*x^2 + 4*a) + 12*b*cos(2*b*x^2 + 2*a) + 2*(6*b*sin(10*b*x^2 + 10*a) + 15*b*sin(8*b*x^2 + 8
*a) + 20*b*sin(6*b*x^2 + 6*a) + 15*b*sin(4*b*x^2 + 4*a) + 6*b*sin(2*b*x^2 + 2*a))*sin(12*b*x^2 + 12*a) + 12*(1
5*b*sin(8*b*x^2 + 8*a) + 20*b*sin(6*b*x^2 + 6*a) + 15*b*sin(4*b*x^2 + 4*a) + 6*b*sin(2*b*x^2 + 2*a))*sin(10*b*
x^2 + 10*a) + 30*(20*b*sin(6*b*x^2 + 6*a) + 15*b*sin(4*b*x^2 + 4*a) + 6*b*sin(2*b*x^2 + 2*a))*sin(8*b*x^2 + 8*
a) + 120*(5*b*sin(4*b*x^2 + 4*a) + 2*b*sin(2*b*x^2 + 2*a))*sin(6*b*x^2 + 6*a) + b)
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Fricas [A] time = 1.70787, size = 254, normalized size = 2.82 \begin{align*} \frac{15 \, \cos \left (b x^{2} + a\right )^{6} \log \left (\sin \left (b x^{2} + a\right ) + 1\right ) - 15 \, \cos \left (b x^{2} + a\right )^{6} \log \left (-\sin \left (b x^{2} + a\right ) + 1\right ) + 2 \,{\left (15 \, \cos \left (b x^{2} + a\right )^{4} + 10 \, \cos \left (b x^{2} + a\right )^{2} + 8\right )} \sin \left (b x^{2} + a\right )}{192 \, b \cos \left (b x^{2} + a\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*sec(b*x^2+a)^7,x, algorithm="fricas")
[Out]
1/192*(15*cos(b*x^2 + a)^6*log(sin(b*x^2 + a) + 1) - 15*cos(b*x^2 + a)^6*log(-sin(b*x^2 + a) + 1) + 2*(15*cos(
b*x^2 + a)^4 + 10*cos(b*x^2 + a)^2 + 8)*sin(b*x^2 + a))/(b*cos(b*x^2 + a)^6)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sec ^{7}{\left (a + b x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*sec(b*x**2+a)**7,x)
[Out]
Integral(x*sec(a + b*x**2)**7, x)
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Giac [A] time = 1.28508, size = 115, normalized size = 1.28 \begin{align*} -\frac{\frac{2 \,{\left (15 \, \sin \left (b x^{2} + a\right )^{5} - 40 \, \sin \left (b x^{2} + a\right )^{3} + 33 \, \sin \left (b x^{2} + a\right )\right )}}{{\left (\sin \left (b x^{2} + a\right )^{2} - 1\right )}^{3}} - 15 \, \log \left (\sin \left (b x^{2} + a\right ) + 1\right ) + 15 \, \log \left (-\sin \left (b x^{2} + a\right ) + 1\right )}{192 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*sec(b*x^2+a)^7,x, algorithm="giac")
[Out]
-1/192*(2*(15*sin(b*x^2 + a)^5 - 40*sin(b*x^2 + a)^3 + 33*sin(b*x^2 + a))/(sin(b*x^2 + a)^2 - 1)^3 - 15*log(si
n(b*x^2 + a) + 1) + 15*log(-sin(b*x^2 + a) + 1))/b