Integrand size = 23, antiderivative size = 101 \[ \int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {3}{2},2+p,-p,\frac {5}{2},\tanh ^2(e+f x),\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \text {sech}^2(e+f x)^p \left (a+b \sinh ^2(e+f x)\right )^p \tanh ^3(e+f x) \left (1-\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{-p}}{3 f} \]
1/3*AppellF1(3/2,2+p,-p,5/2,tanh(f*x+e)^2,(a-b)*tanh(f*x+e)^2/a)*(sech(f*x +e)^2)^p*(a+b*sinh(f*x+e)^2)^p*tanh(f*x+e)^3/f/((1-(a-b)*tanh(f*x+e)^2/a)^ p)
\[ \int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx \]
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 25, 3653, 25, 395, 394}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\sin (i e+i f x)^2 \left (a-b \sin (i e+i f x)^2\right )^pdx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sin (i e+i f x)^2 \left (a-b \sin (i e+i f x)^2\right )^pdx\) |
\(\Big \downarrow \) 3653 |
\(\displaystyle -\frac {\text {sech}^2(e+f x)^p \left (a+b \sinh ^2(e+f x)\right )^p \left (a-(a-b) \tanh ^2(e+f x)\right )^{-p} \int -\tanh ^2(e+f x) \left (1-\tanh ^2(e+f x)\right )^{-p-2} \left (a-(a-b) \tanh ^2(e+f x)\right )^pd\tanh (e+f x)}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\text {sech}^2(e+f x)^p \left (a+b \sinh ^2(e+f x)\right )^p \left (a-(a-b) \tanh ^2(e+f x)\right )^{-p} \int \tanh ^2(e+f x) \left (1-\tanh ^2(e+f x)\right )^{-p-2} \left (a-(a-b) \tanh ^2(e+f x)\right )^pd\tanh (e+f x)}{f}\) |
\(\Big \downarrow \) 395 |
\(\displaystyle \frac {\text {sech}^2(e+f x)^p \left (a+b \sinh ^2(e+f x)\right )^p \left (1-\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{-p} \int \tanh ^2(e+f x) \left (1-\tanh ^2(e+f x)\right )^{-p-2} \left (1-\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^pd\tanh (e+f x)}{f}\) |
\(\Big \downarrow \) 394 |
\(\displaystyle \frac {\tanh ^3(e+f x) \text {sech}^2(e+f x)^p \left (a+b \sinh ^2(e+f x)\right )^p \left (1-\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},p+2,-p,\frac {5}{2},\tanh ^2(e+f x),\frac {(a-b) \tanh ^2(e+f x)}{a}\right )}{3 f}\) |
(AppellF1[3/2, 2 + p, -p, 5/2, Tanh[e + f*x]^2, ((a - b)*Tanh[e + f*x]^2)/ a]*(Sech[e + f*x]^2)^p*(a + b*Sinh[e + f*x]^2)^p*Tanh[e + f*x]^3)/(3*f*(1 - ((a - b)*Tanh[e + f*x]^2)/a)^p)
3.2.38.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 , -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ FracPart[p]) Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ [{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff*(a + b*Sin[e + f*x]^2)^p*((Sec[e + f*x]^2)^p/(f*(a + (a + b)*Tan[e + f*x]^2)^p)) Subst[Int[(a + (a + b)*ff^2*x^2)^p*((A + (A + B)*ff^2*x^2)/( 1 + ff^2*x^2)^(p + 2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, A, B}, x] && !IntegerQ[p]
\[\int \sinh \left (f x +e \right )^{2} \left (a +b \sinh \left (f x +e \right )^{2}\right )^{p}d x\]
\[ \int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \sinh \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
\[ \int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \sinh \left (f x + e\right )^{2} \,d x } \]
\[ \int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \sinh \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \sinh ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int {\mathrm {sinh}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]