3.2.24 \(\int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\) [124]

Optimal. Leaf size=107 \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{2 \sqrt {2} d}+\frac {a \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2754, 2728, 212} \begin {gather*} -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}+\frac {a \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

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

Rule 212

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]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2754

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]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.26, size = 130, normalized size = 1.21 \begin {gather*} \frac {\left (\frac {1}{12}+\frac {i}{12}\right ) a \sec ^3(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \sqrt {a (1+\sin (c+d x))} \left (6 (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3-(1-i) (-5+3 \sin (c+d x))\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]

[Out]

((1/12 + I/12)*a*Sec[c + d*x]^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*Sqrt[a*(1 + Sin[c + d*x])]*(6*(-1)^(3/
4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 - (1 - I)*(
-5 + 3*Sin[c + d*x])))/d

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Maple [A]
time = 0.46, size = 107, normalized size = 1.00

method	result	size
default	\(\frac {\left (1+\sin \left (d x +c \right )\right ) \left (6 a^{\frac {7}{2}} \sin \left (d x +c \right )-10 a^{\frac {7}{2}}+3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}\right )}{12 a^{\frac {3}{2}} \left (\sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/12/a^(3/2)*(1+sin(d*x+c))/(sin(d*x+c)-1)*(6*a^(7/2)*sin(d*x+c)-10*a^(7/2)+3*2^(1/2)*arctanh(1/2*(a-a*sin(d*x
+c))^(1/2)*2^(1/2)/a^(1/2))*a^2*(a-a*sin(d*x+c))^(3/2))/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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Maxima [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))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (88) = 176\).
time = 0.39, size = 215, normalized size = 2.01 \begin {gather*} \frac {3 \, {\left (\sqrt {2} a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \sqrt {2} a \cos \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sqrt {2} \cos \left (d x + c\right ) - \sqrt {2} \sin \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (3 \, a \sin \left (d x + c\right ) - 5 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{24 \, {\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))^(3/2),x, algorithm="fricas")

[Out]

1/24*(3*(sqrt(2)*a*cos(d*x + c)*sin(d*x + c) - sqrt(2)*a*cos(d*x + c))*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt
(a*sin(d*x + c) + a)*(sqrt(2)*cos(d*x + c) - sqrt(2)*sin(d*x + c) + sqrt(2))*sqrt(a) + 3*a*cos(d*x + c) - (a*c
os(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))
+ 4*(3*a*sin(d*x + c) - 5*a)*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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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))**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8009 deep

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Giac [A]
time = 6.10, size = 95, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {2} a^{\frac {3}{2}} {\left (\frac {2 \, {\left (3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - 3 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 3 \, \log \left (-\sin \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 )}{24 \, d} \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))^(3/2),x, algorithm="giac")

[Out]

-1/24*sqrt(2)*a^(3/2)*(2*(3*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 + 1)/sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 3*log(sin
(-1/4*pi + 1/2*d*x + 1/2*c) + 1) + 3*log(-sin(-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 )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(3/2)/cos(c + d*x)^4,x)

[Out]

int((a + a*sin(c + d*x))^(3/2)/cos(c + d*x)^4, x)

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