3.2.49 \(\int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx\) [149]

Optimal. Leaf size=61 \[ -\frac {8 a^2 \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}+\frac {2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{d} \]

[Out]

-8/3*a^2*sec(d*x+c)^3*(a+a*sin(d*x+c))^(3/2)/d+2*a*sec(d*x+c)^3*(a+a*sin(d*x+c))^(5/2)/d

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Rubi [A]
time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2753, 2752} \begin {gather*} \frac {2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-\frac {8 a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

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

Rule 2752

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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]

Rule 2753

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]

Rubi steps

\begin {align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac {2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-(4 a) \int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\\ &=-\frac {8 a^2 \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}+\frac {2 a \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2}}{d}\\ \end {align*}

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Mathematica [A]
time = 5.19, size = 82, normalized size = 1.34 \begin {gather*} \frac {2 a^3 \sqrt {a (1+\sin (c+d x))} (-1+3 \sin (c+d x))}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2),x]

[Out]

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

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Maple [A]
time = 0.41, size = 57, normalized size = 0.93

method	result	size
default	\(-\frac {2 a^{4} \left (1+\sin \left (d x +c \right )\right ) \left (3 \sin \left (d x +c \right )-1\right )}{3 \left (\sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(a+a*sin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (55) = 110\).
time = 0.59, size = 320, normalized size = 5.25 \begin {gather*} \frac {2 \, {\left (a^{\frac {7}{2}} - \frac {6 \, a^{\frac {7}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {24 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {10 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {10 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {24 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {5 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {6 \, a^{\frac {7}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {a^{\frac {7}{2}} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )}}{3 \, d {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

2/3*(a^(7/2) - 6*a^(7/2)*sin(d*x + c)/(cos(d*x + c) + 1) + 5*a^(7/2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 24*
a^(7/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 10*a^(7/2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 36*a^(7/2)*sin(
d*x + c)^5/(cos(d*x + c) + 1)^5 + 10*a^(7/2)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 24*a^(7/2)*sin(d*x + c)^7/(
cos(d*x + c) + 1)^7 + 5*a^(7/2)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 6*a^(7/2)*sin(d*x + c)^9/(cos(d*x + c) +
1)^9 + a^(7/2)*sin(d*x + c)^10/(cos(d*x + c) + 1)^10)/(d*(3*sin(d*x + c)/(cos(d*x + c) + 1) - 3*sin(d*x + c)^
2/(cos(d*x + c) + 1)^2 + sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^(7
/2))

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Fricas [A]
time = 0.34, size = 57, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (3 \, a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(a+a*sin(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [A]
time = 5.54, size = 76, normalized size = 1.25 \begin {gather*} \frac {\sqrt {2} {\left (3 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{3 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")

[Out]

1/3*sqrt(2)*(3*a^3*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - a^3*sgn(cos(-1/4*pi
+ 1/2*d*x + 1/2*c)))*sqrt(a)/(d*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3)

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Mupad [B]
time = 8.53, size = 118, normalized size = 1.93 \begin {gather*} -\frac {a^3\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,\left (3-3\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,2{}\mathrm {i}\right )\,4{}\mathrm {i}}{3\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )\,{\left (1+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(7/2)/cos(c + d*x)^4,x)

[Out]

-(a^3*exp(c*1i + d*x*1i)*(a + a*((exp(- c*1i - d*x*1i)*1i)/2 - (exp(c*1i + d*x*1i)*1i)/2))^(1/2)*(exp(c*1i + d
*x*1i)*2i - 3*exp(c*2i + d*x*2i) + 3)*4i)/(3*d*(exp(c*1i + d*x*1i) + 1i)*(exp(c*1i + d*x*1i)*1i + 1)^3)

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