∫ sec3(c + dx)(a + a sin(c + dx))3∕2 dx [123]

Optimal. Leaf size=73 \[ \frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d} \]

1/2*sec(d*x+c)^2*(a+a*sin(d*x+c))^(3/2)/d+1/4*a^(3/2)*arctanh(1/2*(a+a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1

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Rubi [A]
time = 0.08, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2754, 2746, 65, 212} \begin {gather*} \frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \end {gather*}

(a^(3/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*d) + (Sec[c + d*x]^2*(a + a*Sin[c + d

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,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

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])

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,

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

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Mathematica [A]
time = 0.16, size = 72, normalized size = 0.99 \begin {gather*} \frac {a \left (\sqrt {2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a (1+\sin (c+d x))}}{\sqrt {2} \sqrt {a}}\right )-\frac {2 \sqrt {a (1+\sin (c+d x))}}{-1+\sin (c+d x)}\right )}{4 d} \end {gather*}

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

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method	result	size

default	\(\frac {2 a^{3} \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{4 a \left (a \sin \left (d x +c \right )-a \right )}+\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{d}\)	\(70\)

int(sec(d*x+c)^3*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

2*a^3*(-1/4*(a+a*sin(d*x+c))^(1/2)/a/(a*sin(d*x+c)-a)+1/8/a^(3/2)*2^(1/2)*arctanh(1/2*(a+a*sin(d*x+c))^(1/2)*2

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Maxima [A]
time = 0.52, size = 94, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {2} a^{\frac {5}{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 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{3}}{a \sin \left (d x + c\right ) - a}}{8 \, a d} \end {gather*}

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima")

-1/8*(sqrt(2)*a^(5/2)*log(-(sqrt(2)*sqrt(a) - sqrt(a*sin(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(a*sin(d*x + c)

+ a))) + 4*sqrt(a*sin(d*x + c) + a)*a^3/(a*sin(d*x + c) - a))/(a*d)

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Fricas [A]
time = 0.36, size = 99, normalized size = 1.36 \begin {gather*} \frac {{\left (\sqrt {2} a \sin \left (d x + c\right ) - \sqrt {2} a\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 \, \sqrt {a \sin \left (d x + c\right ) + a} a}{8 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \end {gather*}

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas")

1/8*((sqrt(2)*a*sin(d*x + c) - sqrt(2)*a)*sqrt(a)*log(-(a*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sq

rt(a) + 3*a)/(sin(d*x + c) - 1)) - 4*sqrt(a*sin(d*x + c) + a)*a)/(d*sin(d*x + c) - d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Exception raised: SystemError >> excessive stack use: stack is 5006 deep

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Giac [A]
time = 6.85, size = 91, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {2} a^{\frac {3}{2}} {\left (\frac {2 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{8 \, d} \end {gather*}

integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")

-1/8*sqrt(2)*a^(3/2)*(2*cos(-1/4*pi + 1/2*d*x + 1/2*c)/(cos(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1) - log(cos(-1/4*p

i + 1/2*d*x + 1/2*c) + 1) + log(-cos(-1/4*pi + 1/2*d*x + 1/2*c) + 1))*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \end {gather*}

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3.2.23 \(\int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\) [123]