3.150 \(\int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=106 \[ -\frac {a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{2 d} \]

[Out]

-1/8*a^2*sec(d*x+c)^2*(a+a*sin(d*x+c))^(3/2)/d+1/2*a*sec(d*x+c)^4*(a+a*sin(d*x+c))^(5/2)/d-1/16*a^(7/2)*arctan
h(1/2*(a+a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d

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Rubi [A]  time = 0.17, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2676, 2675, 2667, 63, 206} \[ -\frac {a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

-(a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*d) - (a^2*Sec[c + d*x]^2*(a + a*Sin[
c + d*x])^(3/2))/(8*d) + (a*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2))/(2*d)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 2675

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(p + 1)), x] + Dist[(a*(m + p + 1))/(g^2*(p + 1)), Int[(
g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
&& GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ[m + 1/2, 2*p]

Rule 2676

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-2*b*
(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(p + 1)), x] + Dist[(b^2*(2*m + p - 1))/(g^2*(p +
1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2
- b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && IntegersQ[2*m, 2*p]

Rubi steps

\begin {align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}-\frac {1}{4} a^2 \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}-\frac {1}{16} a^3 \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}-\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{16 d}\\ &=-\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}-\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{8 d}\\ &=-\frac {a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 108, normalized size = 1.02 \[ \frac {2 a^3 (\sin (c+d x)+3) \sqrt {a (\sin (c+d x)+1)}-\sqrt {2} a^{7/2} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \tanh ^{-1}\left (\frac {\sqrt {a (\sin (c+d x)+1)}}{\sqrt {2} \sqrt {a}}\right )}{16 d (\sin (c+d x)-1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

(-(Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqrt[a])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])
^4) + 2*a^3*Sqrt[a*(1 + Sin[c + d*x])]*(3 + Sin[c + d*x]))/(16*d*(-1 + Sin[c + d*x])^2)

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fricas [A]  time = 0.78, size = 145, normalized size = 1.37 \[ \frac {{\left (\sqrt {2} a^{3} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} a^{3} \sin \left (d x + c\right ) - 2 \, \sqrt {2} a^{3}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (a^{3} \sin \left (d x + c\right ) + 3 \, a^{3}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{32 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

1/32*((sqrt(2)*a^3*cos(d*x + c)^2 + 2*sqrt(2)*a^3*sin(d*x + c) - 2*sqrt(2)*a^3)*sqrt(a)*log(-(a*sin(d*x + c) -
2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a) + 3*a)/(sin(d*x + c) - 1)) - 4*(a^3*sin(d*x + c) + 3*a^3)*sqrt(a*s
in(d*x + c) + a))/(d*cos(d*x + c)^2 + 2*d*sin(d*x + c) - 2*d)

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giac [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(d*x+c)^5*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.28, size = 75, normalized size = 0.71 \[ -\frac {2 a^{5} \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \left (3+\sin \left (d x +c \right )\right )}{16 \left (a \sin \left (d x +c \right )-a \right )^{2}}+\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{32 a^{\frac {3}{2}}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5*(a+a*sin(d*x+c))^(7/2),x)

[Out]

-2*a^5*(-1/16*(a+a*sin(d*x+c))^(1/2)*(3+sin(d*x+c))/(a*sin(d*x+c)-a)^2+1/32/a^(3/2)*2^(1/2)*arctanh(1/2*(a+a*s
in(d*x+c))^(1/2)*2^(1/2)/a^(1/2)))/d

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maxima [A]  time = 0.43, size = 132, normalized size = 1.25 \[ \frac {\sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5} + 2 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{6}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )} a + 4 \, a^{2}}}{32 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

1/32*(sqrt(2)*a^(9/2)*log(-(sqrt(2)*sqrt(a) - sqrt(a*sin(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(a*sin(d*x + c)
+ a))) + 4*((a*sin(d*x + c) + a)^(3/2)*a^5 + 2*sqrt(a*sin(d*x + c) + a)*a^6)/((a*sin(d*x + c) + a)^2 - 4*(a*s
in(d*x + c) + a)*a + 4*a^2))/(a*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(7/2)/cos(c + d*x)^5,x)

[Out]

int((a + a*sin(c + d*x))^(7/2)/cos(c + d*x)^5, x)

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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(d*x+c)**5*(a+a*sin(d*x+c))**(7/2),x)

[Out]

Timed out

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