3.116 \(\int \sec ^7(a+b x) \tan ^5(a+b x) \, dx\)

Optimal. Leaf size=46 \[ \frac {\sec ^{11}(a+b x)}{11 b}-\frac {2 \sec ^9(a+b x)}{9 b}+\frac {\sec ^7(a+b x)}{7 b} \]

[Out]

1/7*sec(b*x+a)^7/b-2/9*sec(b*x+a)^9/b+1/11*sec(b*x+a)^11/b

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Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2606, 270} \[ \frac {\sec ^{11}(a+b x)}{11 b}-\frac {2 \sec ^9(a+b x)}{9 b}+\frac {\sec ^7(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^7*Tan[a + b*x]^5,x]

[Out]

Sec[a + b*x]^7/(7*b) - (2*Sec[a + b*x]^9)/(9*b) + Sec[a + b*x]^11/(11*b)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

\begin {align*} \int \sec ^7(a+b x) \tan ^5(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\sec ^7(a+b x)}{7 b}-\frac {2 \sec ^9(a+b x)}{9 b}+\frac {\sec ^{11}(a+b x)}{11 b}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 46, normalized size = 1.00 \[ \frac {\sec ^{11}(a+b x)}{11 b}-\frac {2 \sec ^9(a+b x)}{9 b}+\frac {\sec ^7(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^7*Tan[a + b*x]^5,x]

[Out]

Sec[a + b*x]^7/(7*b) - (2*Sec[a + b*x]^9)/(9*b) + Sec[a + b*x]^11/(11*b)

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fricas [A]  time = 0.45, size = 35, normalized size = 0.76 \[ \frac {99 \, \cos \left (b x + a\right )^{4} - 154 \, \cos \left (b x + a\right )^{2} + 63}{693 \, b \cos \left (b x + a\right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^12*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

1/693*(99*cos(b*x + a)^4 - 154*cos(b*x + a)^2 + 63)/(b*cos(b*x + a)^11)

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giac [B]  time = 0.30, size = 204, normalized size = 4.43 \[ \frac {16 \, {\left (\frac {11 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {55 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac {297 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {1485 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac {2079 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {2541 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - \frac {1155 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} + \frac {462 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + 1\right )}}{693 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^12*sin(b*x+a)^5,x, algorithm="giac")

[Out]

16/693*(11*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 55*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 - 297*(cos(b*x
+ a) - 1)^3/(cos(b*x + a) + 1)^3 + 1485*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 - 2079*(cos(b*x + a) - 1)^5
/(cos(b*x + a) + 1)^5 + 2541*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^6 - 1155*(cos(b*x + a) - 1)^7/(cos(b*x +
a) + 1)^7 + 462*(cos(b*x + a) - 1)^8/(cos(b*x + a) + 1)^8 + 1)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)^
11)

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maple [B]  time = 0.04, size = 142, normalized size = 3.09 \[ \frac {\frac {\sin ^{6}\left (b x +a \right )}{11 \cos \left (b x +a \right )^{11}}+\frac {5 \left (\sin ^{6}\left (b x +a \right )\right )}{99 \cos \left (b x +a \right )^{9}}+\frac {5 \left (\sin ^{6}\left (b x +a \right )\right )}{231 \cos \left (b x +a \right )^{7}}+\frac {\sin ^{6}\left (b x +a \right )}{231 \cos \left (b x +a \right )^{5}}-\frac {\sin ^{6}\left (b x +a \right )}{693 \cos \left (b x +a \right )^{3}}+\frac {\sin ^{6}\left (b x +a \right )}{231 \cos \left (b x +a \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (b x +a \right )+\frac {4 \left (\sin ^{2}\left (b x +a \right )\right )}{3}\right ) \cos \left (b x +a \right )}{231}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^12*sin(b*x+a)^5,x)

[Out]

1/b*(1/11*sin(b*x+a)^6/cos(b*x+a)^11+5/99*sin(b*x+a)^6/cos(b*x+a)^9+5/231*sin(b*x+a)^6/cos(b*x+a)^7+1/231*sin(
b*x+a)^6/cos(b*x+a)^5-1/693*sin(b*x+a)^6/cos(b*x+a)^3+1/231*sin(b*x+a)^6/cos(b*x+a)+1/231*(8/3+sin(b*x+a)^4+4/
3*sin(b*x+a)^2)*cos(b*x+a))

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maxima [A]  time = 0.45, size = 35, normalized size = 0.76 \[ \frac {99 \, \cos \left (b x + a\right )^{4} - 154 \, \cos \left (b x + a\right )^{2} + 63}{693 \, b \cos \left (b x + a\right )^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)^12*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

1/693*(99*cos(b*x + a)^4 - 154*cos(b*x + a)^2 + 63)/(b*cos(b*x + a)^11)

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mupad [B]  time = 1.02, size = 35, normalized size = 0.76 \[ \frac {99\,{\cos \left (a+b\,x\right )}^4-154\,{\cos \left (a+b\,x\right )}^2+63}{693\,b\,{\cos \left (a+b\,x\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^5/cos(a + b*x)^12,x)

[Out]

(99*cos(a + b*x)^4 - 154*cos(a + b*x)^2 + 63)/(693*b*cos(a + b*x)^11)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x+a)**12*sin(b*x+a)**5,x)

[Out]

Timed out

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