3.545 \(\int (a+b \sin ^2(c+d x))^p \tan (c+d x) \, dx\)

Optimal. Leaf size=59 \[ \frac{\left (a+b \sin ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sin ^2(c+d x)+a}{a+b}\right )}{2 d (p+1) (a+b)} \]

[Out]

(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^2)/(a + b)]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)
*d*(1 + p))

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Rubi [A]  time = 0.0434174, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3194, 68} \[ \frac{\left (a+b \sin ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sin ^2(c+d x)+a}{a+b}\right )}{2 d (p+1) (a+b)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x],x]

[Out]

(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^2)/(a + b)]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)
*d*(1 + p))

Rule 3194

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff*x)^p)/(1 - ff*x)^((
m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \left (a+b \sin ^2(c+d x)\right )^p \tan (c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p}{1-x} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac{\, _2F_1\left (1,1+p;2+p;\frac{a+b \sin ^2(c+d x)}{a+b}\right ) \left (a+b \sin ^2(c+d x)\right )^{1+p}}{2 (a+b) d (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0615149, size = 61, normalized size = 1.03 \[ \frac{\left (a-b \cos ^2(c+d x)+b\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac{b \cos ^2(c+d x)}{a+b}\right )}{2 d (p+1) (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x],x]

[Out]

((a + b - b*Cos[c + d*x]^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 - (b*Cos[c + d*x]^2)/(a + b)])/(2*(a
+ b)*d*(1 + p))

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Maple [F]  time = 1.234, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ \left ( \sin \left ( dx+c \right ) \right ) ^{2}b \right ) ^{p}\tan \left ( dx+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+sin(d*x+c)^2*b)^p*tan(d*x+c),x)

[Out]

int((a+sin(d*x+c)^2*b)^p*tan(d*x+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c)^2 + a)^p*tan(d*x + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \tan \left (d x + c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x, algorithm="fricas")

[Out]

integral((-b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)**2)**p*tan(d*x+c),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \tan \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c)^2 + a)^p*tan(d*x + c), x)