3.223 \(\int (d \sec (a+b x))^{7/2} \sin (a+b x) \, dx\)

Optimal. Leaf size=20 \[ \frac {2 d (d \sec (a+b x))^{5/2}}{5 b} \]

[Out]

2/5*d*(d*sec(b*x+a))^(5/2)/b

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2622, 30} \[ \frac {2 d (d \sec (a+b x))^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Sec[a + b*x])^(7/2)*Sin[a + b*x],x]

[Out]

(2*d*(d*Sec[a + b*x])^(5/2))/(5*b)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2622

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

Rubi steps

\begin {align*} \int (d \sec (a+b x))^{7/2} \sin (a+b x) \, dx &=\frac {d \operatorname {Subst}\left (\int x^{3/2} \, dx,x,d \sec (a+b x)\right )}{b}\\ &=\frac {2 d (d \sec (a+b x))^{5/2}}{5 b}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 20, normalized size = 1.00 \[ \frac {2 d (d \sec (a+b x))^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Sec[a + b*x])^(7/2)*Sin[a + b*x],x]

[Out]

(2*d*(d*Sec[a + b*x])^(5/2))/(5*b)

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fricas [A]  time = 0.88, size = 28, normalized size = 1.40 \[ \frac {2 \, d^{3} \sqrt {\frac {d}{\cos \left (b x + a\right )}}}{5 \, b \cos \left (b x + a\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(7/2)*sin(b*x+a),x, algorithm="fricas")

[Out]

2/5*d^3*sqrt(d/cos(b*x + a))/(b*cos(b*x + a)^2)

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giac [B]  time = 1.65, size = 154, normalized size = 7.70 \[ -\frac {4 \, {\left (5 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{4} d - 10 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{2} d^{2} + d^{3}\right )} d^{3} \mathrm {sgn}\left (\cos \left (b x + a\right )\right )}{5 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} - \sqrt {-d}\right )}^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(7/2)*sin(b*x+a),x, algorithm="giac")

[Out]

-4/5*(5*(sqrt(-d)*tan(1/2*b*x + 1/2*a)^2 - sqrt(-d*tan(1/2*b*x + 1/2*a)^4 + d))^4*d - 10*(sqrt(-d)*tan(1/2*b*x
+ 1/2*a)^2 - sqrt(-d*tan(1/2*b*x + 1/2*a)^4 + d))^2*d^2 + d^3)*d^3*sgn(cos(b*x + a))/((sqrt(-d)*tan(1/2*b*x +
1/2*a)^2 - sqrt(-d*tan(1/2*b*x + 1/2*a)^4 + d) - sqrt(-d))^5*b)

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maple [A]  time = 0.14, size = 17, normalized size = 0.85 \[ \frac {2 d \left (d \sec \left (b x +a \right )\right )^{\frac {5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*sec(b*x+a))^(7/2)*sin(b*x+a),x)

[Out]

2/5*d*(d*sec(b*x+a))^(5/2)/b

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maxima [A]  time = 0.72, size = 23, normalized size = 1.15 \[ \frac {2 \, \left (\frac {d}{\cos \left (b x + a\right )}\right )^{\frac {7}{2}} \cos \left (b x + a\right )}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sec(b*x+a))^(7/2)*sin(b*x+a),x, algorithm="maxima")

[Out]

2/5*(d/cos(b*x + a))^(7/2)*cos(b*x + a)/b

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mupad [B]  time = 1.59, size = 77, normalized size = 3.85 \[ \frac {8\,d^3\,\sqrt {\frac {d}{\cos \left (a+b\,x\right )}}\,\left (4\,\cos \left (2\,a+2\,b\,x\right )+\cos \left (4\,a+4\,b\,x\right )+3\right )}{5\,b\,\left (15\,\cos \left (2\,a+2\,b\,x\right )+6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )+10\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)*(d/cos(a + b*x))^(7/2),x)

[Out]

(8*d^3*(d/cos(a + b*x))^(1/2)*(4*cos(2*a + 2*b*x) + cos(4*a + 4*b*x) + 3))/(5*b*(15*cos(2*a + 2*b*x) + 6*cos(4
*a + 4*b*x) + cos(6*a + 6*b*x) + 10))

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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((d*sec(b*x+a))**(7/2)*sin(b*x+a),x)

[Out]

Timed out

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