3.2.73 \(\int \csc ^3(e+f x) (b (c \tan (e+f x))^n)^p \, dx\) [173]

Optimal. Leaf size=92 \[ -\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} \csc ^2(e+f x) \, _2F_1\left (\frac {1}{2} (-2+n p),\frac {1}{2} (1+n p);\frac {n p}{2};\sin ^2(e+f x)\right ) \sec (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (2-n p)} \]

[Out]

-(cos(f*x+e)^2)^(1/2*n*p+1/2)*csc(f*x+e)^2*hypergeom([1/2*n*p-1, 1/2*n*p+1/2],[1/2*n*p],sin(f*x+e)^2)*sec(f*x+
e)*(b*(c*tan(f*x+e))^n)^p/f/(-n*p+2)

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Rubi [A]
time = 0.10, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3740, 2681, 2657} \begin {gather*} -\frac {\csc ^2(e+f x) \sec (e+f x) \cos ^2(e+f x)^{\frac {1}{2} (n p+1)} \, _2F_1\left (\frac {1}{2} (n p-2),\frac {1}{2} (n p+1);\frac {n p}{2};\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (2-n p)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]

[Out]

-(((Cos[e + f*x]^2)^((1 + n*p)/2)*Csc[e + f*x]^2*Hypergeometric2F1[(-2 + n*p)/2, (1 + n*p)/2, (n*p)/2, Sin[e +
f*x]^2]*Sec[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 - n*p)))

Rule 2657

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b^(2*IntPart[
(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^
2)^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rule 2681

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[Cos[e + f*x]
^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^n), Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(-1)]) || IntegersQ[m - 1/2, n -
1/2])

Rule 3740

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Tan[e + f*x
])^n)^FracPart[p]/(c*Tan[e + f*x])^(n*FracPart[p])), Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x]
)^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])

Rubi steps

\begin {align*} \int \csc ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \csc ^3(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\left (\cos ^{n p}(e+f x) \sin ^{-n p}(e+f x) \left (b (c \tan (e+f x))^n\right )^p\right ) \int \cos ^{-n p}(e+f x) \sin ^{-3+n p}(e+f x) \, dx\\ &=-\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} \csc ^2(e+f x) \, _2F_1\left (\frac {1}{2} (-2+n p),\frac {1}{2} (1+n p);\frac {n p}{2};\sin ^2(e+f x)\right ) \sec (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (2-n p)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 14.76, size = 1399, normalized size = 15.21 \begin {gather*} \frac {\cot ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (n p,-1+\frac {n p}{2};\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{n p} \left (b (c \tan (e+f x))^n\right )^p}{f (-8+4 n p)}+\frac {(4+n p) F_1\left (1+\frac {n p}{2};n p,1;2+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{8 f (2+n p) \left ((4+n p) F_1\left (1+\frac {n p}{2};n p,1;2+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+F_1\left (2+\frac {n p}{2};n p,2;3+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (-1+\cos (e+f x))+2 n p F_1\left (2+\frac {n p}{2};1+n p,1;3+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )}+\frac {\, _2F_1\left (n p,1+\frac {n p}{2};2+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{n p} \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (8+4 n p)}+\frac {\cot \left (\frac {1}{2} (e+f x)\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{n p} \left ((2+n p) \, _2F_1\left (\frac {n p}{2},n p;1+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n p F_1\left (1+\frac {n p}{2};n p,1;2+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^{n p}(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{8 f n p (2+n p) \left (\frac {\left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-1+n p} \left (-\sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)+\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )\right ) \left ((2+n p) \, _2F_1\left (\frac {n p}{2},n p;1+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n p F_1\left (1+\frac {n p}{2};n p,1;2+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^{n p}(e+f x)}{2 (2+n p)}+\frac {\left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{n p} \left (-n p F_1\left (1+\frac {n p}{2};n p,1;2+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )-n p \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left (-\frac {\left (1+\frac {n p}{2}\right ) F_1\left (2+\frac {n p}{2};n p,2;3+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{2+\frac {n p}{2}}+\frac {n p \left (1+\frac {n p}{2}\right ) F_1\left (2+\frac {n p}{2};1+n p,1;3+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{2+\frac {n p}{2}}\right )+\frac {1}{2} n p (2+n p) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \left (-\, _2F_1\left (\frac {n p}{2},n p;1+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+\left (1-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n p}\right )\right ) \tan ^{n p}(e+f x)}{2 n p (2+n p)}+\frac {\left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{n p} \sec ^2(e+f x) \left ((2+n p) \, _2F_1\left (\frac {n p}{2},n p;1+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n p F_1\left (1+\frac {n p}{2};n p,1;2+\frac {n p}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \tan ^{-1+n p}(e+f x)}{2 (2+n p)}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]

[Out]

(Cot[(e + f*x)/2]^2*Hypergeometric2F1[n*p, -1 + (n*p)/2, (n*p)/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f
*x)/2]^2)^(n*p)*(b*(c*Tan[e + f*x])^n)^p)/(f*(-8 + 4*n*p)) + ((4 + n*p)*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p
)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p)/(8*f*(2 + n*p)*((4 + n*
p)*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + Ap
pellF1[2 + (n*p)/2, n*p, 2, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(-1 + Cos[e + f*x]) + 2*n*p*
AppellF1[2 + (n*p)/2, 1 + n*p, 1, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^2)) +
(Hypergeometric2F1[n*p, 1 + (n*p)/2, 2 + (n*p)/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)
*Tan[(e + f*x)/2]^2*(b*(c*Tan[e + f*x])^n)^p)/(f*(8 + 4*n*p)) + (Cot[(e + f*x)/2]*(Cos[e + f*x]*Sec[(e + f*x)/
2]^2)^(n*p)*((2 + n*p)*Hypergeometric2F1[(n*p)/2, n*p, 1 + (n*p)/2, Tan[(e + f*x)/2]^2] - n*p*AppellF1[1 + (n*
p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^(n*p)*(b*
(c*Tan[e + f*x])^n)^p)/(8*f*n*p*(2 + n*p)*(((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n*p)*(-(Sec[(e + f*x)/2]^2
*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])*((2 + n*p)*Hypergeometric2F1[(n*p)/2, n*p,
1 + (n*p)/2, Tan[(e + f*x)/2]^2] - n*p*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e
+ f*x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^(n*p))/(2*(2 + n*p)) + ((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*
(-(n*p*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*
Tan[(e + f*x)/2]) - n*p*Tan[(e + f*x)/2]^2*(-(((1 + (n*p)/2)*AppellF1[2 + (n*p)/2, n*p, 2, 3 + (n*p)/2, Tan[(e
+ f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(2 + (n*p)/2)) + (n*p*(1 + (n*p)/2)*Ap
pellF1[2 + (n*p)/2, 1 + n*p, 1, 3 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(
e + f*x)/2])/(2 + (n*p)/2)) + (n*p*(2 + n*p)*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]*(-Hypergeometric2F1[(n*p)/2, n*
p, 1 + (n*p)/2, Tan[(e + f*x)/2]^2] + (1 - Tan[(e + f*x)/2]^2)^(-(n*p))))/2)*Tan[e + f*x]^(n*p))/(2*n*p*(2 + n
*p)) + ((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*Sec[e + f*x]^2*((2 + n*p)*Hypergeometric2F1[(n*p)/2, n*p, 1 +
(n*p)/2, Tan[(e + f*x)/2]^2] - n*p*AppellF1[1 + (n*p)/2, n*p, 1, 2 + (n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*
x)/2]^2]*Tan[(e + f*x)/2]^2)*Tan[e + f*x]^(-1 + n*p))/(2*(2 + n*p))))

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (\csc ^{3}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x)

[Out]

int(csc(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

integrate(((c*tan(f*x + e))^n*b)^p*csc(f*x + e)^3, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm="fricas")

[Out]

integral(((c*tan(f*x + e))^n*b)^p*csc(f*x + e)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \csc ^{3}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**3*(b*(c*tan(f*x+e))**n)**p,x)

[Out]

Integral((b*(c*tan(e + f*x))**n)**p*csc(e + f*x)**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^3*(b*(c*tan(f*x+e))^n)^p,x, algorithm="giac")

[Out]

integrate(((c*tan(f*x + e))^n*b)^p*csc(f*x + e)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p}{{\sin \left (e+f\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*(c*tan(e + f*x))^n)^p/sin(e + f*x)^3,x)

[Out]

int((b*(c*tan(e + f*x))^n)^p/sin(e + f*x)^3, x)

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