3.403 \(\int \sec (c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=80 \[ -\frac{3 a b^2 \sin (c+d x)}{d}+\frac{(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac{b^3 \sin ^2(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.105492, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2668, 702, 633, 31} \[ -\frac{3 a b^2 \sin (c+d x)}{d}+\frac{(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac{b^3 \sin ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2668

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

Rule 702

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \sec (c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^3}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (-3 a-x+\frac{a^3+3 a b^2+\left (3 a^2+b^2\right ) x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{3 a b^2 \sin (c+d x)}{d}-\frac{b^3 \sin ^2(c+d x)}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{a^3+3 a b^2+\left (3 a^2+b^2\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{3 a b^2 \sin (c+d x)}{d}-\frac{b^3 \sin ^2(c+d x)}{2 d}-\frac{(a-b)^3 \operatorname{Subst}\left (\int \frac{1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}+\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{b-x} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=-\frac{(a+b)^3 \log (1-\sin (c+d x))}{2 d}+\frac{(a-b)^3 \log (1+\sin (c+d x))}{2 d}-\frac{3 a b^2 \sin (c+d x)}{d}-\frac{b^3 \sin ^2(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.118111, size = 67, normalized size = 0.84 \[ -\frac{6 a b^2 \sin (c+d x)+(a-b)^3 (-\log (\sin (c+d x)+1))+(a+b)^3 \log (1-\sin (c+d x))+b^3 \sin ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b)^3*Log[1 - Sin[c + d*x]] - (a - b)^3*Log[1 + Sin[c + d*x]] + 6*a*b^2*Sin[c + d*x] + b^3*Sin[c + d*x]^
2)/(2*d)

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Maple [A]  time = 0.054, size = 108, normalized size = 1.4 \begin{align*}{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{{a}^{2}b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{a{b}^{2}\sin \left ( dx+c \right ) }{d}}+3\,{\frac{a{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)*(a+b*sin(d*x+c))^3,x)

[Out]

1/d*a^3*ln(sec(d*x+c)+tan(d*x+c))-3/d*a^2*b*ln(cos(d*x+c))-3*a*b^2*sin(d*x+c)/d+3/d*a*b^2*ln(sec(d*x+c)+tan(d*
x+c))-1/2*b^3*sin(d*x+c)^2/d-1/d*b^3*ln(cos(d*x+c))

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Maxima [A]  time = 0.945462, size = 123, normalized size = 1.54 \begin{align*} -\frac{b^{3} \sin \left (d x + c\right )^{2} + 6 \, a b^{2} \sin \left (d x + c\right ) -{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*sin(d*x + c)^2 + 6*a*b^2*sin(d*x + c) - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*log(sin(d*x + c) + 1) + (a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*log(sin(d*x + c) - 1))/d

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Fricas [A]  time = 2.44291, size = 221, normalized size = 2.76 \begin{align*} \frac{b^{3} \cos \left (d x + c\right )^{2} - 6 \, a b^{2} \sin \left (d x + c\right ) +{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*cos(d*x + c)^2 - 6*a*b^2*sin(d*x + c) + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*log(sin(d*x + c) + 1) - (a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*log(-sin(d*x + c) + 1))/d

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (c + d x \right )}\right )^{3} \sec{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a+b*sin(d*x+c))**3,x)

[Out]

Integral((a + b*sin(c + d*x))**3*sec(c + d*x), x)

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Giac [A]  time = 1.14519, size = 126, normalized size = 1.58 \begin{align*} -\frac{b^{3} \sin \left (d x + c\right )^{2} + 6 \, a b^{2} \sin \left (d x + c\right ) -{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) +{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*sin(d*x + c)^2 + 6*a*b^2*sin(d*x + c) - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*log(abs(sin(d*x + c) + 1)) +
(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(abs(sin(d*x + c) - 1)))/d