\(\int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx\) [59]

Optimal result

Integrand size = 18, antiderivative size = 331 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {b^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {3 b^{5/2} e^{-3 a+\frac {3 b c}{d}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {b^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {3 b^{5/2} e^{3 a-\frac {3 b c}{d}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {16 b^2 \sinh (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cosh (a+b x) \sinh ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}-\frac {24 b^2 \sinh ^3(a+b x)}{5 d^3 \sqrt {c+d x}} \] Output:

-1/5*b^(5/2)*exp(-a+b*c/d)*Pi^(1/2)*erf(b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/d^( 
7/2)+3/5*b^(5/2)*exp(-3*a+3*b*c/d)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*b^(1/2)*(d 
*x+c)^(1/2)/d^(1/2))/d^(7/2)-1/5*b^(5/2)*exp(a-b*c/d)*Pi^(1/2)*erfi(b^(1/2 
)*(d*x+c)^(1/2)/d^(1/2))/d^(7/2)+3/5*b^(5/2)*exp(3*a-3*b*c/d)*3^(1/2)*Pi^( 
1/2)*erfi(3^(1/2)*b^(1/2)*(d*x+c)^(1/2)/d^(1/2))/d^(7/2)-16/5*b^2*sinh(b*x 
+a)/d^3/(d*x+c)^(1/2)-4/5*b*cosh(b*x+a)*sinh(b*x+a)^2/d^2/(d*x+c)^(3/2)-2/ 
5*sinh(b*x+a)^3/d/(d*x+c)^(5/2)-24/5*b^2*sinh(b*x+a)^3/d^3/(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.14 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\frac {e^{-3 a} \left (2 e^{6 a} \left (-d^2 e^{3 b x}-2 b e^{-\frac {3 b c}{d}} (c+d x) \left (e^{\frac {3 b (c+d x)}{d}} (d+6 b (c+d x))+6 \sqrt {3} d \left (-\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 b (c+d x)}{d}\right )\right )\right )+2 e^{4 a} \left (3 d^2 e^{b x}+2 b e^{-\frac {b c}{d}} (c+d x) \left (e^{b \left (\frac {c}{d}+x\right )} (d+2 b (c+d x))+2 d \left (-\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )\right )\right )+e^{2 a-b x} \left (-6 d^2+4 b d (c+d x)-8 b^2 (c+d x)^2+8 d^2 e^{b \left (\frac {c}{d}+x\right )} \left (\frac {b (c+d x)}{d}\right )^{5/2} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )\right )-2 e^{-3 b x} \left (-d^2+2 b (c+d x) \left (d-6 b (c+d x)+6 \sqrt {3} d e^{\frac {3 b (c+d x)}{d}} \left (\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {3 b (c+d x)}{d}\right )\right )\right )\right )}{40 d^3 (c+d x)^{5/2}} \] Input:

Integrate[Sinh[a + b*x]^3/(c + d*x)^(7/2),x]
 

Output:

(2*E^(6*a)*(-(d^2*E^(3*b*x)) - (2*b*(c + d*x)*(E^((3*b*(c + d*x))/d)*(d + 
6*b*(c + d*x)) + 6*Sqrt[3]*d*(-((b*(c + d*x))/d))^(3/2)*Gamma[1/2, (-3*b*( 
c + d*x))/d]))/E^((3*b*c)/d)) + 2*E^(4*a)*(3*d^2*E^(b*x) + (2*b*(c + d*x)* 
(E^(b*(c/d + x))*(d + 2*b*(c + d*x)) + 2*d*(-((b*(c + d*x))/d))^(3/2)*Gamm 
a[1/2, -((b*(c + d*x))/d)]))/E^((b*c)/d)) + E^(2*a - b*x)*(-6*d^2 + 4*b*d* 
(c + d*x) - 8*b^2*(c + d*x)^2 + 8*d^2*E^(b*(c/d + x))*((b*(c + d*x))/d)^(5 
/2)*Gamma[1/2, (b*(c + d*x))/d]) - (2*(-d^2 + 2*b*(c + d*x)*(d - 6*b*(c + 
d*x) + 6*Sqrt[3]*d*E^((3*b*(c + d*x))/d)*((b*(c + d*x))/d)^(3/2)*Gamma[1/2 
, (3*b*(c + d*x))/d])))/E^(3*b*x))/(40*d^3*E^(3*a)*(c + d*x)^(5/2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.53 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.45, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3042, 26, 3795, 26, 3042, 26, 3778, 3042, 3788, 26, 2611, 2633, 2634, 3794, 2009}

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.

\(\displaystyle \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \sin (i a+i b x)^3}{(c+d x)^{7/2}}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\sin (i a+i b x)^3}{(c+d x)^{7/2}}dx\)

\(\Big \downarrow \) 3795

\(\displaystyle i \left (\frac {12 b^2 \int -\frac {i \sinh ^3(a+b x)}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {8 b^2 \int \frac {i \sinh (a+b x)}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (-\frac {12 i b^2 \int \frac {\sinh ^3(a+b x)}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {8 i b^2 \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (-\frac {8 i b^2 \int -\frac {i \sin (i a+i b x)}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {12 i b^2 \int \frac {i \sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (-\frac {8 b^2 \int \frac {\sin (i a+i b x)}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {12 b^2 \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 3778

\(\displaystyle i \left (\frac {12 b^2 \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {8 b^2 \left (\frac {2 i b \int \frac {\cosh (a+b x)}{\sqrt {c+d x}}dx}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle i \left (\frac {12 b^2 \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {8 b^2 \left (\frac {2 i b \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 3788

\(\displaystyle i \left (\frac {12 b^2 \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {8 b^2 \left (\frac {2 i b \left (\frac {1}{2} i \int -\frac {i e^{a+b x}}{\sqrt {c+d x}}dx-\frac {1}{2} i \int \frac {i e^{-a-b x}}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle i \left (\frac {12 b^2 \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {8 b^2 \left (\frac {2 i b \left (\frac {1}{2} \int \frac {e^{-a-b x}}{\sqrt {c+d x}}dx+\frac {1}{2} \int \frac {e^{a+b x}}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle i \left (\frac {12 b^2 \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {8 b^2 \left (\frac {2 i b \left (\frac {\int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}+\frac {\int e^{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{d}\right )}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle i \left (-\frac {8 b^2 \left (\frac {2 i b \left (\frac {\int e^{-a-\frac {b (c+d x)}{d}+\frac {b c}{d}}d\sqrt {c+d x}}{d}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {12 b^2 \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle i \left (\frac {12 b^2 \int \frac {\sin (i a+i b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {8 b^2 \left (\frac {2 i b \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 3794

\(\displaystyle i \left (\frac {12 b^2 \left (\frac {6 i b \int \left (\frac {\cosh (a+b x)}{4 \sqrt {c+d x}}-\frac {\cosh (3 a+3 b x)}{4 \sqrt {c+d x}}\right )dx}{d}+\frac {2 i \sinh ^3(a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {8 b^2 \left (\frac {2 i b \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle i \left (\frac {12 b^2 \left (\frac {6 i b \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{3}} e^{\frac {3 b c}{d}-3 a} \text {erf}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{3}} e^{3 a-\frac {3 b c}{d}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 \sqrt {b} \sqrt {d}}\right )}{d}+\frac {2 i \sinh ^3(a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {8 b^2 \left (\frac {2 i b \left (\frac {\sqrt {\pi } e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 i \sinh (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {4 i b \sinh ^2(a+b x) \cosh (a+b x)}{5 d^2 (c+d x)^{3/2}}+\frac {2 i \sinh ^3(a+b x)}{5 d (c+d x)^{5/2}}\right )\)

Input:

Int[Sinh[a + b*x]^3/(c + d*x)^(7/2),x]
 

Output:

I*((((4*I)/5)*b*Cosh[a + b*x]*Sinh[a + b*x]^2)/(d^2*(c + d*x)^(3/2)) + ((( 
2*I)/5)*Sinh[a + b*x]^3)/(d*(c + d*x)^(5/2)) - (8*b^2*(((2*I)*b*((E^(-a + 
(b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d] 
) + (E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sq 
rt[b]*Sqrt[d])))/d - ((2*I)*Sinh[a + b*x])/(d*Sqrt[c + d*x])))/(5*d^2) + ( 
12*b^2*(((6*I)*b*((E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/S 
qrt[d]])/(8*Sqrt[b]*Sqrt[d]) - (E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[ 
3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(a - (b*c)/d) 
*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) - (E^ 
(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] 
)/(8*Sqrt[b]*Sqrt[d])))/d + ((2*I)*Sinh[a + b*x]^3)/(d*Sqrt[c + d*x])))/(5 
*d^2))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

rule 3794

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si 
mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 
)))   Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n 
- 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & 
& LtQ[m, -1]
 

rule 3795

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ 
b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) 
*(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2)))   Int[(c + 
d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* 
(m + 2)))   Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, 
c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
 

Maple [F]

\[\int \frac {\sinh \left (b x +a \right )^{3}}{\left (d x +c \right )^{\frac {7}{2}}}d x\]

Input:

int(sinh(b*x+a)^3/(d*x+c)^(7/2),x)
 

Output:

int(sinh(b*x+a)^3/(d*x+c)^(7/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3286 vs. \(2 (253) = 506\).

Time = 0.18 (sec) , antiderivative size = 3286, normalized size of antiderivative = 9.93 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate(sinh(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(sinh(b*x+a)**3/(d*x+c)**(7/2),x)
 

Output:

Integral(sinh(a + b*x)**3/(c + d*x)**(7/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.60 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\frac {3 \, {\left (\frac {3 \, \sqrt {3} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {5}{2}, \frac {3 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} - \frac {3 \, \sqrt {3} \left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} \Gamma \left (-\frac {5}{2}, -\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} - \frac {\left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {5}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}} + \frac {\left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {5}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {5}{2}}}\right )}}{8 \, d} \] Input:

integrate(sinh(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="maxima")
 

Output:

3/8*(3*sqrt(3)*((d*x + c)*b/d)^(5/2)*e^(3*(b*c - a*d)/d)*gamma(-5/2, 3*(d* 
x + c)*b/d)/(d*x + c)^(5/2) - 3*sqrt(3)*(-(d*x + c)*b/d)^(5/2)*e^(-3*(b*c 
- a*d)/d)*gamma(-5/2, -3*(d*x + c)*b/d)/(d*x + c)^(5/2) - ((d*x + c)*b/d)^ 
(5/2)*e^(-a + b*c/d)*gamma(-5/2, (d*x + c)*b/d)/(d*x + c)^(5/2) + (-(d*x + 
 c)*b/d)^(5/2)*e^(a - b*c/d)*gamma(-5/2, -(d*x + c)*b/d)/(d*x + c)^(5/2))/ 
d
 

Giac [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int { \frac {\sinh \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(sinh(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="giac")
 

Output:

integrate(sinh(b*x + a)^3/(d*x + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:

int(sinh(a + b*x)^3/(c + d*x)^(7/2),x)
 

Output:

int(sinh(a + b*x)^3/(c + d*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\sinh \left (b x +a \right )^{3}}{\sqrt {d x +c}\, c^{3}+3 \sqrt {d x +c}\, c^{2} d x +3 \sqrt {d x +c}\, c \,d^{2} x^{2}+\sqrt {d x +c}\, d^{3} x^{3}}d x \] Input:

int(sinh(b*x+a)^3/(d*x+c)^(7/2),x)
 

Output:

int(sinh(a + b*x)**3/(sqrt(c + d*x)*c**3 + 3*sqrt(c + d*x)*c**2*d*x + 3*sq 
rt(c + d*x)*c*d**2*x**2 + sqrt(c + d*x)*d**3*x**3),x)