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

Optimal. Leaf size=89 \[ \frac{64 a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d} \]

[Out]

(64*a^3*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (16*a^2*Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(3*d)
- (2*a*Sec[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(3*d)

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Rubi [A]  time = 0.173659, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ \frac{64 a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(64*a^3*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (16*a^2*Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(3*d)
- (2*a*Sec[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(3*d)

Rule 2674

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 - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

Rule 2673

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 - 1))/(f*g*(m - 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=-\frac{2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}+\frac{1}{3} (8 a) \int \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\\ &=-\frac{16 a^2 \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}+\frac{1}{3} \left (32 a^2\right ) \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac{64 a^3 \sec (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{16 a^2 \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a+a \sin (c+d x))^{5/2}}{3 d}\\ \end{align*}

Mathematica [A]  time = 5.47737, size = 48, normalized size = 0.54 \[ \frac{a^3 \sec (c+d x) \sqrt{a (\sin (c+d x)+1)} (-20 \sin (c+d x)+\cos (2 (c+d x))+45)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sec[c + d*x]*(45 + Cos[2*(c + d*x)] - 20*Sin[c + d*x])*Sqrt[a*(1 + Sin[c + d*x])])/(3*d)

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Maple [A]  time = 0.101, size = 55, normalized size = 0.6 \begin{align*} -{\frac{2\,{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{2}+10\,\sin \left ( dx+c \right ) -23 \right ) }{3\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*a^4*(1+sin(d*x+c))*(sin(d*x+c)^2+10*sin(d*x+c)-23)/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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Maxima [B]  time = 1.67178, size = 320, normalized size = 3.6 \begin{align*} -\frac{2 \,{\left (23 \, a^{\frac{7}{2}} - \frac{20 \, a^{\frac{7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{88 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{60 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{130 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{60 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{88 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{20 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{23 \, a^{\frac{7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}}{3 \, d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(23*a^(7/2) - 20*a^(7/2)*sin(d*x + c)/(cos(d*x + c) + 1) + 88*a^(7/2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2
- 60*a^(7/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 130*a^(7/2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 60*a^(7/
2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 88*a^(7/2)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 20*a^(7/2)*sin(d*x +
c)^7/(cos(d*x + c) + 1)^7 + 23*a^(7/2)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8)/(d*(sin(d*x + c)/(cos(d*x + c) +
1) - 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^(7/2))

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Fricas [A]  time = 1.66787, size = 134, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - 10 \, a^{3} \sin \left (d x + c\right ) + 22 \, a^{3}\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{3 \, d \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a^3*cos(d*x + c)^2 - 10*a^3*sin(d*x + c) + 22*a^3)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError