3.5.98 \(\int (d \csc (e+f x))^m (b (c \tan (e+f x))^n)^p \, dx\) [498]

Optimal. Leaf size=104 \[ \frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} (d \csc (e+f x))^m \, _2F_1\left (\frac {1}{2} (1+n p),\frac {1}{2} (1-m+n p);\frac {1}{2} (3-m+n p);\sin ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)} \]

[Out]

(cos(f*x+e)^2)^(1/2*n*p+1/2)*(d*csc(f*x+e))^m*hypergeom([1/2*n*p+1/2, 1/2*n*p-1/2*m+1/2],[1/2*n*p-1/2*m+3/2],s
in(f*x+e)^2)*tan(f*x+e)*(b*(c*tan(f*x+e))^n)^p/f/(n*p-m+1)

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Rubi [A]
time = 0.14, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3740, 2698, 2682, 2657} \begin {gather*} \frac {\tan (e+f x) (d \csc (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (n p+1)} \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (\frac {1}{2} (n p+1),\frac {1}{2} (-m+n p+1);\frac {1}{2} (-m+n p+3);\sin ^2(e+f x)\right )}{f (-m+n p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[e + f*x]^2)^((1 + n*p)/2)*(d*Csc[e + f*x])^m*Hypergeometric2F1[(1 + n*p)/2, (1 - m + n*p)/2, (3 - m + n*
p)/2, Sin[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - m + 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 2682

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[a*Cos[e + f*
x]^(n + 1)*((b*Tan[e + f*x])^(n + 1)/(b*(a*Sin[e + f*x])^(n + 1))), 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]

Rule 2698

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Csc[e + f*
x])^FracPart[m]*(Sin[e + f*x]/a)^FracPart[m], Int[(b*Tan[e + f*x])^n/(Sin[e + f*x]/a)^m, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]

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 (d \csc (e+f x))^m \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 (d \csc (e+f x))^m (c \tan (e+f x))^{n p} \, dx\\ &=\left ((d \csc (e+f x))^m \left (\frac {\sin (e+f x)}{d}\right )^m (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-m} (c \tan (e+f x))^{n p} \, dx\\ &=\frac {\left (\cos ^{n p}(e+f x) (d \csc (e+f x))^{1+m} \sin (e+f x) \left (\frac {\sin (e+f x)}{d}\right )^{m-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \cos ^{-n p}(e+f x) \left (\frac {\sin (e+f x)}{d}\right )^{-m+n p} \, dx}{d}\\ &=\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} (d \csc (e+f x))^{1+m} \, _2F_1\left (\frac {1}{2} (1+n p),\frac {1}{2} (1-m+n p);\frac {1}{2} (3-m+n p);\sin ^2(e+f x)\right ) \sin (e+f x) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{d f (1-m+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 = 1.46, size = 319, normalized size = 3.07 \begin {gather*} -\frac {d (-3+m-n p) F_1\left (\frac {1}{2} (1-m+n p);n p,1-m;\frac {1}{2} (3-m+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (d \csc (e+f x))^{-1+m} \left (b (c \tan (e+f x))^n\right )^p}{f (-1+m-n p) \left ((-3+m-n p) F_1\left (\frac {1}{2} (1-m+n p);n p,1-m;\frac {1}{2} (3-m+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left ((-1+m) F_1\left (\frac {1}{2} (3-m+n p);n p,2-m;\frac {1}{2} (5-m+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n p F_1\left (\frac {1}{2} (3-m+n p);1+n p,1-m;\frac {1}{2} (5-m+n p);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((d*(-3 + m - n*p)*AppellF1[(1 - m + n*p)/2, n*p, 1 - m, (3 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/
2]^2]*(d*Csc[e + f*x])^(-1 + m)*(b*(c*Tan[e + f*x])^n)^p)/(f*(-1 + m - n*p)*((-3 + m - n*p)*AppellF1[(1 - m +
n*p)/2, n*p, 1 - m, (3 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*((-1 + m)*AppellF1[(3 - m +
n*p)/2, n*p, 2 - m, (5 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*p*AppellF1[(3 - m + n*p)/2,
1 + n*p, 1 - m, (5 - m + n*p)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))

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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \left (d \csc \left (f x +e \right )\right )^{m} \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*csc(f*x+e))^m*(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((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

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

[Out]

integral(((c*tan(f*x + e))^n*b)^p*(d*csc(f*x + e))^m, 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} \left (d \csc {\left (e + f x \right )}\right )^{m}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*(c*tan(e + f*x))**n)**p*(d*csc(e + f*x))**m, 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((d*csc(f*x+e))^m*(b*(c*tan(f*x+e))^n)^p,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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