3.1.25 \(\int \sec ^7(c+d x) (a+a \sin (c+d x))^2 \, dx\) [25]

Optimal. Leaf size=109 \[ \frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a^5}{12 d (a-a \sin (c+d x))^3}+\frac {a^4}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{16 d (a-a \sin (c+d x))}-\frac {a^3}{16 d (a+a \sin (c+d x))} \]

[Out]

1/4*a^2*arctanh(sin(d*x+c))/d+1/12*a^5/d/(a-a*sin(d*x+c))^3+1/8*a^4/d/(a-a*sin(d*x+c))^2+3/16*a^3/d/(a-a*sin(d
*x+c))-1/16*a^3/d/(a+a*sin(d*x+c))

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Rubi [A]
time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2746, 46, 212} \begin {gather*} \frac {a^5}{12 d (a-a \sin (c+d x))^3}+\frac {a^4}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{16 d (a-a \sin (c+d x))}-\frac {a^3}{16 d (a \sin (c+d x)+a)}+\frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*ArcTanh[Sin[c + d*x]])/(4*d) + a^5/(12*d*(a - a*Sin[c + d*x])^3) + a^4/(8*d*(a - a*Sin[c + d*x])^2) + (3*
a^3)/(16*d*(a - a*Sin[c + d*x])) - a^3/(16*d*(a + a*Sin[c + d*x]))

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 2746

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])

Rubi steps

\begin {align*} \int \sec ^7(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {a^7 \text {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \text {Subst}\left (\int \left (\frac {1}{4 a^2 (a-x)^4}+\frac {1}{4 a^3 (a-x)^3}+\frac {3}{16 a^4 (a-x)^2}+\frac {1}{16 a^4 (a+x)^2}+\frac {1}{4 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5}{12 d (a-a \sin (c+d x))^3}+\frac {a^4}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{16 d (a-a \sin (c+d x))}-\frac {a^3}{16 d (a+a \sin (c+d x))}+\frac {a^3 \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=\frac {a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a^5}{12 d (a-a \sin (c+d x))^3}+\frac {a^4}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{16 d (a-a \sin (c+d x))}-\frac {a^3}{16 d (a+a \sin (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 85, normalized size = 0.78 \begin {gather*} -\frac {a^2 \sec ^6(c+d x) (1+\sin (c+d x))^2 \left (-4-\sin (c+d x)+6 \sin ^2(c+d x)-3 \sin ^3(c+d x)+3 \tanh ^{-1}(\sin (c+d x)) (-1+\sin (c+d x))^3 (1+\sin (c+d x))\right )}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(a^2*Sec[c + d*x]^6*(1 + Sin[c + d*x])^2*(-4 - Sin[c + d*x] + 6*Sin[c + d*x]^2 - 3*Sin[c + d*x]^3 + 3*Ar
cTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])))/d

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Maple [A]
time = 0.27, size = 160, normalized size = 1.47

method	result	size
derivativedivides	\(\frac {a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {a^{2}}{3 \cos \left (d x +c \right )^{6}}+a^{2} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(160\)
default	\(\frac {a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {a^{2}}{3 \cos \left (d x +c \right )^{6}}+a^{2} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(160\)
risch	\(-\frac {i a^{2} \left (-12 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}-8 i {\mathrm e}^{4 i \left (d x +c \right )}-13 \,{\mathrm e}^{5 i \left (d x +c \right )}-12 i {\mathrm e}^{2 i \left (d x +c \right )}+13 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{6 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{4 d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{4 d}\)	\(162\)
norman	\(\frac {\frac {3 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {31 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {77 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {139 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {139 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {77 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {31 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {3 a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {4 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {80 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {52 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {52 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 d}+\frac {a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d}\)	\(357\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^2*(1/6*sin(d*x+c)^3/cos(d*x+c)^6+1/8*sin(d*x+c)^3/cos(d*x+c)^4+1/16*sin(d*x+c)^3/cos(d*x+c)^2+1/16*sin(
d*x+c)-1/16*ln(sec(d*x+c)+tan(d*x+c)))+1/3*a^2/cos(d*x+c)^6+a^2*(-(-1/6*sec(d*x+c)^5-5/24*sec(d*x+c)^3-5/16*se
c(d*x+c))*tan(d*x+c)+5/16*ln(sec(d*x+c)+tan(d*x+c))))

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Maxima [A]
time = 0.31, size = 108, normalized size = 0.99 \begin {gather*} \frac {3 \, a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, a^{2} \sin \left (d x + c\right )^{3} - 6 \, a^{2} \sin \left (d x + c\right )^{2} + a^{2} \sin \left (d x + c\right ) + 4 \, a^{2}\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{3} + 2 \, \sin \left (d x + c\right ) - 1}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*a^2*log(sin(d*x + c) + 1) - 3*a^2*log(sin(d*x + c) - 1) - 2*(3*a^2*sin(d*x + c)^3 - 6*a^2*sin(d*x + c)
^2 + a^2*sin(d*x + c) + 4*a^2)/(sin(d*x + c)^4 - 2*sin(d*x + c)^3 + 2*sin(d*x + c) - 1))/d

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Fricas [A]
time = 0.37, size = 203, normalized size = 1.86 \begin {gather*} -\frac {12 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2} - 3 \, {\left (a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2}\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(12*a^2*cos(d*x + c)^2 - 4*a^2 - 3*(a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c)^2*sin(d*x + c) - 2*a^2*cos(d
*x + c)^2)*log(sin(d*x + c) + 1) + 3*(a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c)^2*sin(d*x + c) - 2*a^2*cos(d*x +
c)^2)*log(-sin(d*x + c) + 1) - 2*(3*a^2*cos(d*x + c)^2 - 4*a^2)*sin(d*x + c))/(d*cos(d*x + c)^4 + 2*d*cos(d*x
+ c)^2*sin(d*x + c) - 2*d*cos(d*x + c)^2)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [A]
time = 5.43, size = 119, normalized size = 1.09 \begin {gather*} \frac {6 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 6 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {3 \, {\left (2 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}\right )}}{\sin \left (d x + c\right ) + 1} + \frac {11 \, a^{2} \sin \left (d x + c\right )^{3} - 42 \, a^{2} \sin \left (d x + c\right )^{2} + 57 \, a^{2} \sin \left (d x + c\right ) - 30 \, a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{3}}}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(6*a^2*log(abs(sin(d*x + c) + 1)) - 6*a^2*log(abs(sin(d*x + c) - 1)) - 3*(2*a^2*sin(d*x + c) + 3*a^2)/(si
n(d*x + c) + 1) + (11*a^2*sin(d*x + c)^3 - 42*a^2*sin(d*x + c)^2 + 57*a^2*sin(d*x + c) - 30*a^2)/(sin(d*x + c)
- 1)^3)/d

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Mupad [B]
time = 4.34, size = 94, normalized size = 0.86 \begin {gather*} \frac {a^2\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{4\,d}-\frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^3}{4}-\frac {a^2\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {a^2\,\sin \left (c+d\,x\right )}{12}+\frac {a^2}{3}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^3+2\,\sin \left (c+d\,x\right )-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*atanh(sin(c + d*x)))/(4*d) - ((a^2*sin(c + d*x))/12 + a^2/3 - (a^2*sin(c + d*x)^2)/2 + (a^2*sin(c + d*x)^
3)/4)/(d*(2*sin(c + d*x) - 2*sin(c + d*x)^3 + sin(c + d*x)^4 - 1))

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